Normalized solutions for Schrödinger equations with critical Sobolev exponent and mixed nonlinearities
aa r X i v : . [ m a t h . A P ] F e b NORMALIZED SOLUTIONS FOR SCHR ¨ODINGER EQUATIONSWITH CRITICAL SOBOLEV EXPONENT AND MIXEDNONLINEARITIES
JUNCHENG WEI AND YUANZE WU
Abstract.
In this paper, we consider the following nonlinear Schr¨odingerequations with mixed nonlinearities: − ∆ u = λu + µ | u | q − u + | u | ∗ − u in R N ,u ∈ H ( R N ) , Z R N u = a , where N ≥ µ > λ ∈ R and 2 < q < ∗ . We prove in this paper(1) Existence of solutions of mountain-pass type for N = 3 and 2 < q < N .(2) Existence and nonexistence of ground states for 2 + N ≤ q < ∗ with µ > µ → µ goes to its upper bound.Our studies answer some questions proposed by Soave in [49, Remarks 1.1, 1.2and 8.1]. Keywords:
Normalized solution; Ground state; Variational method; Criticalnonlinearity; Mixed nonlinearities.
AMS
Subject Classification 2010: 35B09; 35B33; 35B40; 35J20. Introduction
In this paper, we consider the following nonlinear scalar field equation: ( − ∆ u = λu + µ | u | q − u + | u | ∗ − u in R N ,u ∈ H ( R N ) , (1.1)where N ≥ µ > λ ∈ R and 2 < q < ∗ = NN − .(1.1) is a special case of the following model, ( − ∆ u = λu + f ( u ) in R N ,u ∈ H ( R N ) , (1.2)which is related to finding the stationary waves of nonlinear Schr¨odinger equations: iψ t + ∆ ψ + g ( | ψ | ) ψ = 0 in R N . (1.3)Indeed, a stationary wave of (1.3) is of the form ψ ( t, x ) = e iλt u ( x ) where λ ∈ R is a constant and u ( x ) is a time-independent function, then it is well-known that ψ is a solution of (1.3) if and only if u is a solution of (1.2) with f ( u ) = g ( | u | ) u .As pointed out in [49, 50], in general, the function u is complex valued and thus, (1.2) can be regarded as a complex valued system coupled by the nonlinearities f ( u ) = g ( | u | ) u .As pointed out in [29, 36], the studies on (1.2) can be traced back to the semi-classical papers [12, 13, 42, 43, 52]. In these studies, there are two different methodsto study (1.2). The first one is to fix the number λ < u to be real valued. In this case, (1.2) is a single equation and under somemild assumptions on f ( u ), the solutions of (1.2) are critical points of the functional J ( u ) = 12 Z R N ( |∇ u | − λ | u | ) dx − Z R N F ( u ) dx in the usual Sobolev space H ( R N ), where F ( u ) = R u f ( t ) dt . In this case, particularattention is devoted to least action solutions, namely solutions minimizing J ( u )among all non-trivial solutions. We also refer the readers to [2–5] and the referencestherein for the recent results on the special case (1.1) in this direction. Anotherone is to fix the L norm of the unknown u , that is, to find solutions of (1.2) withprescribed mass. In this case, (1.2) is always rewritten as follows − ∆ u = λu + f ( u ) in R N ,u ∈ H ( R N ) , Z R N u = a , (1.4)where R R N u = a is the prescribed mass, and in this case, λ ∈ R is a part of theunknown which appears in (1.4) as a Lagrange multiplier. In particular, (1.1) isrewritten as − ∆ u = λu + µ | u | q − u + | u | ∗ − u in R N ,u ∈ H ( R N ) , Z R N u = a . (1.5)Similar to the first case, under some mild assumptions on f ( u ), the solutions of(1.4) are critical points of the functional F µ ( u ) = Z R N (cid:0) |∇ u | − F ( u ) (cid:1) dx on the smooth manifold S a = { u ∈ H ( R N ) | k u k = a } . In particular, the solutions of (1.5) are critical points of the C -functional E µ ( u ) = 12 k∇ u k − µq k u k qq − ∗ k u k ∗ ∗ on S a , where we denote the usual norm in L p ( R N ) by k · k p . In this case, solutionsof (1.4) are always called normalized solutions which are particularly relevant forthe nonlinear Schr¨odinger equation (1.3) since the mass is preserved along the timeevolution in (1.3). Thus, normalized solutions of (1.4) seems to be particularlymeaningful from the physical viewpoint, moreover, these solutions often offer agood insight of the dynamical properties of the stationary solutions for the nonlinearSchr¨odinger equation (1.3), such as stability or instability (cf. [11,18]). In this case,particular attention is also devoted to least action solutions which are also calledground states for normalized solutions, namely solutions minimizing F µ ( u ) amongall non-trivial solutions. The studies on normalized solutions of (1.4) is a hot ORMALIZED SOLUTIONS 3 topic in the community of nonlinear PDEs nowadays, thus, it is impossible for usto provide a complete references. We just refer the readers to [1, 8–10, 15, 17, 23,33, 35–37, 40, 46–51] and the references therein. In these references, we would liketo highlight [36, 49, 50] to the readers for their detail introductions and referenceson normalized solutions of (1.4) and new directions on the study of normalizedsolutions of autonomous problems. We also would like to point out [26–32] and thereferences therein for the studies on normalized solutions of problems with trappingpotentials.It is well-known that the number p = 2 + N plays an important role in studyingnormalized solutions which is called the L critical exponent or mass critical expo-nent in the literature. Since 2 ∗ = NN − > N , the nonlinearity of (1.5) growsfaster than | u | p − u at infinity and thus, it is well-known that E µ ( u ) is unboundedfrom below on S a , which makes one to find new constraints to prove the existence ofground states of E µ ( u ) on S a . The new constraint, which is introduced by Bartschand Soave in [9] for the general problem (1.4) and is widely used nowadays instudying normalized solutions, is the following L -Pohozaev manifold: P a,µ = { u ∈ S a | k∇ u k = µγ q k u k qq + k u k ∗ ∗ } , where γ q = N ( q − q . (1.6)By the Pohozaev identity of (1.5), P a,µ contains all nontrivial solutions of (1.5),thus, we have the following definition of ground states of (1.5). Definition 1.1.
We say ( u , λ ) is a ground state of (1.5) if u is a critical pointof E µ | S a ( u ) with E µ | S a ( u ) = inf u ∈P a,µ E µ ( u ) . The L -Pohozaev manifold P a,µ is quite related to the fibering mapsΨ( u, s ) = e s k∇ u k − µe qγ q s q k u k qq − e ∗ s ∗ k u k ∗ ∗ , which is introduced by Jeanjean in [33] for the general problem (1.4) and is wellstudied by Soave in [49]. According to the fibering maps Ψ( u, s ), P a,µ can benaturally divided into the following three parts: P a,µ + = { u ∈ S a | k∇ u k > µqγ q k u k qq + 2 ∗ k u k ∗ ∗ } , P a,µ = { u ∈ S a | k∇ u k = µqγ q k u k qq + 2 ∗ k u k ∗ ∗ } , P a,µ − = { u ∈ S a | k∇ u k < µqγ q k u k qq + 2 ∗ k u k ∗ ∗ } . Let m ± a,µ = inf u ∈P a,µ ± E µ ( u ) , (1.7)then Soave proved the following results in [49, Theorems 1.1 and 1.4]:(1) For 2 < q < N , there exists α N,q > µa q − qγ q <α N,q then m + a,µ = inf u ∈P a,µ + E µ ( u ) = inf u ∈P a,µ E µ ( u ) < u a,µ, + which is real valued, positive, radially symmetricand radially decreasing. Moreover, (1.5) has a ground state ( u a,µ, + , λ a,µ, + )with λ a,µ, + <
0, and m + a,µ → k∇ u a,µ, + k → µ → J. WEI AND Y.WU (2) For 2 + N ≤ q < ∗ , there exists α N,q > µa q − qγ q < α N,q then m − a,µ = inf u ∈P a,µ − E µ ( u ) = inf u ∈P a,µ E µ ( u ) ∈ (0 , N S N ) and it can beattained by some u a,µ, − which is real valued, positive, radially symmetricand radially decreasing, where S is the optimal constant in the Sobolevembedding, that is, k u k ∗ ≤ S − k∇ u k for all u ∈ D , ( R N ) . (1.8)Moreover, (1.5) has a ground state ( u a,µ, − , λ a,µ, − ) with λ a,µ, − <
0, and m − a,µ → N S N and k∇ u a,µ, − k → S N as µ → L -subcritical case 2 < q < N , since E µ ( u ) | S a is unbounded frombelow, it could be naturally to expect that E µ ( u ) | S a has a second critical pointof mountain-pass type, which is also positive, real valued and radially symmetric.This natural expectation has been pointed out by Soave in [49, Remark 1.1] whichcan be summarized to be the following question:( Q ) Does E µ ( u ) | S a has a critical point of mountain-pass type in the L -subcritical case < q < N ?Remark 1.1. In preparing this paper, we notice that in the very recent work [34],the question ( Q ) has been solved for N ≥ . Thus, it only need to consider thecase N = 3 for the question ( Q ) . Besides, since Soave only considered the case that µa q − qγ q > µ > µa q − qγ q > Q ) Does E µ ( u ) | S a have a ground state if µ > and µa q − qγ q > large? In [49], Soave conjectures that the answer of ( Q ) is negative in general.Finally, in these results, the asymptotic behavior is only for k∇ u a,µ, − k in thecases of 2 + N ≤ q < ∗ . Thus, it is also natural to ask if it is possible tocharacterize the asymptotic behavior of u a,µ, − , and not only of k∇ u a,µ, − k . In [49,Remark 8.1], Soave pointed out that in dimensions N = 3 ,
4, it could be provedthat k∇ u a,µ, − k → S N , but u a,µ, − ⇀ H while, in dimensions H ≥
5, both u a,µ, − ⇀ u a,µ, − ⇀ e u = 0 could happen. He also conjectures that the weaklimit of { u a,µ, − } will be the Aubin-Talanti babbles in the higher dimensions N ≥ Q ) Can we capture the precisely asymptotic behavior of u a,µ, − as µ → ? In this paper, we are interested in these questions and we shall give some answersto them, which will give more information on the ground states of (1.5). Our firstresult, which is devoted to the existence and nonexistence of ground states, can bestated as follows.
Theorem 1.1.
Let N ≥ , < q < ∗ and a, µ > . (1) If N = 3 and < q < N , then for µa q − qγ q < α N,q , m − a,µ can beattained by some u a,µ, − which is real valued, positive, radially symmetricand radially decreasing, and thus, (1.5) has a second solution u a,µ, − withsome λ a,µ, − < . ORMALIZED SOLUTIONS 5 (2) If q = 2 + N , then m − a,µ can not be attained for µa q − qγ q ≥ α N,q and thus, (1.5) has no ground states for µa q − qγ q ≥ α N,q . (3) If N < q < ∗ , then for all µ > , and m − a,µ can be attained bysome u a,µ, − which is real valued, positive, radially symmetric and radiallydecreasing, and thus, (1.5) has a ground state u a,µ, − with some λ a,µ, − < for all µ > . Remark 1.2. ( a ) (1) of Theorem 1.1, which together the results of [34],gives a completely positive answer to the question ( Q ) . ( b ) As pointed out in the very recent work [34], the crucial point in studying ( Q ) is to obtain a good energy estimate of m − a,µ for < q < N suchthat the compactness of minimizing sequence or ( P S ) sequence at the energylevel m − a,µ still holds. As for other concave-convex problems (cf. [6]) andobserved in [34], the threshold of such compactness should be m + a,µ + N S N .Since m − a,µ is a mountain-pass level, the classical idea, which can be tracedback to [16], is to use the ground state u a,µ, + and the Aubin-Talanti babblesto construct a good path, whose energy can be well controlled from above tomake sure that it is smaller than the threshold m + a,µ + N S N . This strategyis already used in [34] to prove the existence of critical points of E µ ( u ) | S a of mountain-pass type for N ≥ and < q < N . Unlike [34] inwhich nonradial test function composing of u a,µ, + and a bubble at ∞ isused, here we directly use the radial superposition of u a,µ, + and the Aubin-Talenti bubble. This test function seems to be more natural and it worksfor all dimensions. ( c ) (2) and (3) of Theorem 1.1 give partial answers to ( Q ) and they areproved by observing the non-increasing of m − a,µ and suitable choices of testfunctions. These two conclusions imply that the L -critical and supercriticalperturbations have quite different influence on (1.5) . Moreover, it seemsthat the critical mass of ground states also exists for (1.5) in the L -criticalcase. Our next result will be devoted to the precisely asymptotic behaviors of thesolutions found in [49, Theorem 1.1], [34, Theorem 1.6] and Theorem 1.1 as µ →
0. To state this result, let us first introduce some necessary notations. By [55,Theorem B], the Gagliardo-Nirenberg inequality, k u k q ≤ C N,q k u k − γ q k∇ u k γ q for all u ∈ H ( R N ) , (1.9)has a minimizer φ , which satisfies − ∆ φ + ν φ = σ φ q − in R N ,φ (0) = max x ∈ R N φ ( x ) ,φ ( x ) > R N ,φ ( x ) → | x | → + ∞ , (1.10)where ν = N ( q − (1 − ( q − N − ), σ = N ( q − and C N,q is the best constant inthe Gagliardo-Nirenberg inequality. On the other hand, the Aubin-Talanti babbles, U ε ( x ) = [ N ( N − N − (cid:18) εε + | x | (cid:19) N − , (1.11) J. WEI AND Y.WU is the only solutions to the following equation: − ∆ u = u ∗ − in R N ,u (0) = max x ∈ R N u ( x ) ,u ( x ) > R N ,u ( x ) → | x | → + ∞ . Now, our second result can be stated as follows.
Theorem 1.2.
Let N ≥ , < q < ∗ and a, µ > such that µ > is sufficientlysmall. Let e u µ be the minimizer of E µ ( u ) | S a in P a,µ + and b u µ be the minimizer of E µ ( u ) | S a in P a,µ − . Then (1) For < q < N , e w a,µ ( x ) = s N µ e u µ ( s µ x ) → ν q − a φ ( √ ν a x ) strongly in H ( R N ) as µ → , where φ is the unique solution of (1.10) , ν a = (cid:18) a k φ k (cid:19) q − − N ( q − . (1.12) and s µ ∼ µ − qγq is the unique solution of the following system: ( s µ k∇ ψ ν a , k − µγ q k ψ ν a , k qq s qγ q µ − k ψ ν a , k ∗ ∗ s ∗ µ = 0 , s µ k∇ ψ ν a , k − µqγ q k ψ ν a , k qq s qγ q µ − ∗ k ψ ν a , k ∗ ∗ s ∗ µ > , (1.13) where ψ ν a , ( x ) = ν q − a φ ( √ ν a x ) . Moreover, up to translations and rota-tions, e u µ is the unique ground state of (1.5) for µ > sufficiently small. (2) For N ≥ , b u µ → U ε strongly in H ( R N ) as µ → , where U ε is theAubin-Talanti babble satisfying k U ε k = a . Moreover, up to translationsand rotations, b u µ is the unique minimizer of E µ ( u ) | S a in P a,µ − for µ > sufficiently small. (3) For N = 3 , , b w a,µ ( x ) = ε N − µ b u µ ( ε µ x ) → U ε strongly in D , ( R N ) forsome ε > as µ → up to a subsequence, where ε µ satisfies µ ∼ ε − qµ e − ε − µ , N = 4 , < q < ,ε q − µ , N = 3 , < q < ,ε µ ln( ε µ ) , N = 3 , q = 3 ,ε − q µ , N = 3 , < q < . Remark 1.3. (1)
The precise asymptotic behaviors of e u µ and b u µ stated in (1) and (2) of Theorem 1.2 are captured by comparing the energy values andnorms by full using the variational formulas of e u µ and b u µ , and minimizersof the Gagliardo-Nirenberg inequality and the Aubin-Talanti bubbles. Inthis argument, the unique determination of minimizers of the Gagliardo–Nirenberg inequality (1.9) for < q < N and Aubin-Talanti bubbles for N ≥ in S a , respectively, is crucial. Moreover, (2) of Theorem 1.2 alsogives a positive answer to Soave’s conjecture on ( Q ) . ORMALIZED SOLUTIONS 7 (2)
For the local uniqueness, the standard strategy is to assume the contraryand obtain a contradiction by full using the non-degeneracy of minimizersof the Gagliardo–Nirenberg inequality and Aubin-Talanti bubbles in passingto the limit (cf. [21,27]), which is powerful in studying problems with poten-tials. Since (1.5) is autonomous, we can use a different method, based onthe precisely asymptotic behaviors of e u µ and the implicit function theorem,to prove the local uniqueness of e u µ in a more direct way. It is worth point-ing out that our method is also based on the non-degenerate of minimizersof the Gagliardo–Nirenberg inequality. For b u µ , we remark that since thelinear operator of the limit equation is different from that of (1.5) , our di-rect methods, based on implicit function theorem, is invalid. Thus, we willstill use the standard method, that is to assume the contrary and obtain acontradiction by full using the non-degeneracy of Aubin-Talanti bubbles. (3) Since we loss the L -integrability of the Aubin-Talanti babbles { U ε } for N = 3 , , the asymptotic behavior of b u µ as µ → for N = 3 , is muchweaker than that of N ≥ in the sense that, the convergence is only forsubsequences, which also leads us to loss the local uniqueness of b u µ for µ > sufficiently small in these two cases. We also remark that since weloss the L -integrability of the Aubin-Talanti babbles { U ε } for N = 3 , ,the asymptotic behavior of b u µ can not be obtained by merely using varia-tional arguments to compare the energy values and norms as that for (2) ofTheorem 1.2. Thus, to capture the precisely asymptotic behavior of b u µ , wedrive some uniformly pointwise estimates of b u µ by the maximum principle(cf. [20]) and some ODE technique used in [7] (see also [24, 38]). Withthese additional estimates, we obtain the precisely asymptotic behavior of b u µ for N = 3 , . It is worth pointing out that, in the case N = 3 and < q < , since the nonlinearity decays too slow at infinity, we need tofurther employ the bootstrapping argument to drive the desired estimates. Our final result is devoted to the asymptotic behavior of the minimizers of E µ ( u ) | S a in P a,µ − as µ close to its upper-bound in the cases of 2 + N ≤ q < ∗ .It can be stated as follows. Theorem 1.3.
Assume N ≥ , N ≤ q < ∗ and µ, a > . Let b u µ be theminimizer of E µ ( u ) | S a in P a,µ − , found in [49, Theorem 1.1] for q = 2 + N with < µa q − qγ q < α N,q and found in Theorem 1.1 for N < q < ∗ with all µ > .Then (1) For q = 2 + N , b v µ = ( a k φ k ) N − s N µ b u µ ( a k φ k s µ x ) → ( ν ′ a ) q − φ ( p ν ′ a x ) strongly in H ( R N ) as µ → α N,q,a up to a subsequence, where α N,q,a = a qγ q − q α N,q for some ν ′ a > and s µ = (1 − µα N,q,a ) − N − . (2) For N < q < ∗ , b v µ = s N µ b u µ ( s µ x ) → ν q − a φ ( √ ν a x ) strongly in H ( R N ) as µ → + ∞ , where s µ = µ qγq − . Moreover, up to translations androtations, b u µ is also the unique ground state of (1.5) for µ > sufficientlylarge. Remark 1.4. (1)
The ideas in proving Theorem 1.3 are similar to that of The-orem 1.2. However, in the L -critical case q = 2 + N , the convergence of b u µ is much weaker than that in the L -supcritical case N < q < ∗ J. WEI AND Y.WU in the sense that, it only holds for subsequences. The main reason is thatin the L -critical case q = 2 + N , we have k ϕ k = const. for all ϕ beinga minimizer of the Gagliardo–Nirenberg inequality (1.9) . Thus, the pre-cise mass k b u µ k = a is invalid in determining a unique minimizer of theGagliardo–Nirenberg inequality (1.9) in the case q = 2 + N . Moreover, un-like the studies for problems with homogeneous nonlinearities (cf. [26, 27]),combining nonlinearities ( L -critical and L -supercritical) of (1.5) makesthe asymptotic behavior of b u µ to be more complicated, which also make usloss the local uniqueness of b u µ for µ > close to its upper bound in thiscase. Indeed, as µ goes to its upper bound in the L -critical case, com-paring with the studies for problems with homogeneous nonlinearities, theSobolev critical term of (1.5) is an additionally inhomogenous perturbationin passing to the limit, which makes the oscillations occurring. Notations.
Throughout this paper, C and C ′ are indiscriminately used to denotevarious absolutely positive constants. a ∼ b means that C ′ b ≤ a ≤ Cb and a . b means that a ≤ Cb . 2. Asymptotic behavior of u a,µ, + By [49, Theorem 1.1], m + a,µ can always be attained by some u a,µ, + for 2 < q < N and µa q − qγ q < α N,q , where m + a,µ is given by (1.7) and u a,µ, + is real valued,positive, radially symmetric and radially decreasing. Our goal in this section is togive an asymptotic behavior of u a,µ, + as µ →
0, which is more precisely than thatin [49, Theorem 1.4], and capture the precisely decaying rate of u a,µ, + as µ → u a,µ, + is a solution of (1.5) for some λ a,µ, + < u µ, + = u a,µ, + and λ µ, + = λ a,µ, + , sincewe will fix a > Lemma 2.1.
Let < q < N . Then − λ µ, + ∼ k∇ u µ, + k ∼ µ − qγq as µ → .Proof. Since u µ, + ∈ P a,µ + , we have k∇ u µ, + k = µγ q k u µ, + k qq + k u µ, + k ∗ ∗ (2.1)and 2 k∇ u µ, + k > µqγ q k u µ, + k qq + 2 ∗ k u µ, + k ∗ ∗ . It follows from the Gagliardo–Nirenberg inequality that k∇ u µ, + k . µ k u µ, + k qq . µ k∇ u µ, + k qγ q , which together with qγ q < < q < N , implies k∇ u µ, + k . µ − qγq . (2.2)Thus, by (2.1) and (2.2), we also have µ k u µ, + k qq . µ − qγq . (2.3)Let us define V ε ( x ) = U ε ( x ) ϕ ( R − ε x ) (2.4)where U ε ( x ) is the Aubin-Talanti babbles given by (1.11) and ϕ ∈ C ∞ ( R N ) is aradial cut-off function with ϕ ≡ B , ϕ ≡ B c , and R ε is chosen such that ORMALIZED SOLUTIONS 9 V ε ∈ S a . More precisely, for N ≥
5, we choose ε = ε and R ε = + ∞ such that V ε = U ε ∈ S a while for N = 3 ,
4, we choose ε > a = Z R N ( U ε ( x ) ϕ ( R − ε x )) ∼ ε Z R ε ε − r − N ∼ ( ε ln( R ε ε − ) , for N = 4 ,εR ε , for N = 3 , (2.5)which implies R ε ε − → + ∞ as ε →
0. Then it is well-known (cf. [44, (4.2)–(4.5)]or [53, Chapter III]) that k∇ V ε k = S N + O (( R ε ε − ) − N ) , k V ε k ∗ ∗ = S N + O (( R ε ε − ) − N ) (2.6)for ε > k∇ V ε k ∼ S N ∼ k V ε k ∗ ∗ (2.7)for ε > ε = ε and R ε = + ∞ for N = 5, andfix ε > R ε as that in (2.5) for N = 3 , N ≥
3. By [49, Lemma 4.2], there exists t ( µ ) > V ε ) t ( µ ) ∈ P a,µ + for µ > V ε ) t ( µ ) = [ t ( µ )] N V ε ( t ( µ ) x ) . Then [ t ( µ )] k∇ V ε k = µγ q k V ε k qq [ t ( µ )] qγ q + k V ε k ∗ ∗ [ t ( µ )] ∗ and 2[ t ( µ )] k∇ V ε k > µqγ q k V ε k qq [ t ( µ )] qγ q + 2 ∗ k V ε k ∗ ∗ [ t ( µ )] ∗ . Since qγ q < < q < N , by(2 ∗ − t ( µ )] k∇ V ε k < µ (2 ∗ − qγ q ) γ q k V ε k qq [ t ( µ )] qγ q , it is easy to see that t ( µ ) → µ → N ≥
3. It follows that[ t ( µ )] ∼ µ [ t ( µ )] qγ q as µ → , which implies t ( µ ) ∼ µ − qγq as µ →
0. Thus, by qγ q < < q < N oncemore, E µ (( V ε ) t ( µ ) ) = ( 12 − qγ q ) k∇ V ε k [ t ( µ )] + ( 1 qγ q − ∗ ) k V ε k ∗ ∗ [ t ( µ )] ∗ ∼ − µ − qγq . Therefore, by E µ (( V ε ) t ( µ ) ) ≥ m + a,µ and m + a,µ & − µ k u µ, + k qq , we have µ k u µ, + k qq & µ − qγq , which together with (2.3), implies µ k u µ, + k qq ∼ µ − qγq . By the regularity of u µ, + and the Pohozaev identity, λ µ, + ∼ − µ k u µ, + k qq , and by(2.2) and u µ, + ∈ P a,µ + , k∇ u µ, + k ∼ µ k u µ, + k qq . Therefore, − λ µ, + ∼ k∇ u µ, + k ∼ µ − qγq as µ →
0. It completes the proof. (cid:3)
By the well-known uniqueness result (cf. [39]) and the scaling invariance of (1.10), φ ( x ) = (cid:18) ν σ (cid:19) q − w ( √ ν x ) , where w is the unique solution of the following equation: − ∆ u + u = u q − in R N ,u (0) = max x ∈ R N u ( x ) ,u ( x ) > R N ,u ( x ) → | x | → + ∞ , (2.8)A direct calculation also shows that ψ ν,σ ( x ) = ( νσ ) q − φ ( r νσ x ) (2.9)for ν, σ > ν a be given by (1.12), then for q = 2 + N , ψ ν a , ∈ S a and ψ ν a , is a minimizer of theGagliardo–Nirenberg inequality, that is, k ψ ν a , k qq = C qN,q a q − qγ q k∇ ψ ν a , k qγ q . (2.10)For the sake of simplicity, we re-denote ψ a = ψ ν a , . Proposition 2.1.
Let < q < N . Then w µ, + → ψ a strongly in H ( R N ) as µ → , where w µ, + = s − N µ u µ, + ( s − µ x ) with s µ being the unique solution of (1.13) .Moreover, up to translations and rotations, u µ, + is the unique ground state of (1.5) for µ > sufficiently small.Proof. Since ψ a ∈ S a , by [49, Lemma 4.2], there exists a unique s µ > ψ a ) s µ ∈ P a,µ + for µ > ψ a ) s µ = s N µ ψ a ( s µ x ). That is, s µ k∇ ψ a k = µγ q k ψ a k qq s qγ q µ + k ψ a k ∗ ∗ s ∗ µ (2.11)and 2 s µ k∇ ψ a k > µqγ q k ψ a k qq s qγ q µ + 2 ∗ k ψ a k ∗ ∗ s ∗ µ . (2.12)As that in the proof of Lemma 2.1, we have k∇ ( ψ a ) s µ k − qγ q < C qN,q γ q µa q − qγ q ∗ − qγ q ∗ − . (2.13)Since u µ, + ∈ P a,µ + , we also have k∇ u µ, + k − qγ q < C qN,q γ q µa q − qγ q ∗ − qγ q ∗ − . (2.14)Now, using ( ψ a ) s µ as a test function of m + a,µ and by (2.10), m + a,µ ≤ E µ (( ψ a ) s µ ) = 1 N k∇ ( ψ a ) s µ k − µa q − qγ q C qN,q q (1 − qγ q ∗ ) k∇ ( ψ a ) s µ k qγ q . By the Gagliardo–Nirenberg inequality (1.9), m + a,µ = E µ ( u µ, + ) ≥ N k∇ u µ, + k − µa q − qγ q C qN,q q (1 − qγ q ∗ ) k∇ u µ, + k qγ q . ORMALIZED SOLUTIONS 11
Let us consider the function f ( t ) = 1 N t − µa q − qγ q C qN,q q (1 − qγ q ∗ ) t qγ q . A direct calculation shows that f ( t ) is strictly decreasing in (0 , t ), where t = (cid:18) C qN,q γ q µa q − qγ q ∗ − qγ q ∗ − (cid:19) − qγq . Thus, by (2.13) and (2.14), k∇ u µ, + k ≥ k∇ ( ψ a ) s µ k . (2.15)By (2.11) and (2.12), we can use similar arguments as that used in the proof ofLemma 2.1 to show that s µ ∼ µ − qγq as µ →
0. It then follows from (2.9) and(2.11) that s µ = (1 + o µ (1)) (cid:18) µγ q k ψ a k qq k∇ ψ a k (cid:19) − qγq = (1 + o µ (1)) (cid:18) µγ q k φ k qq k∇ φ k (cid:19) − qγq . Since by the Pohozaev identity satisfied by φ , we have N k∇ φ k = ( q − σ q k φ k qq .By (1.6), s µ = [( σ + o µ (1)) µ ] − qγq . Let w µ, + = s − N µ u µ, + ( s − µ x ) . Since u µ, + satisfies (1.5), w µ, + satisfies the following equation: − ∆ w µ, + = λ µ, + s − µ w µ, + + µs − N ( q − µ w q − µ, + + s − N (2 ∗ − µ w ∗ − µ, + . (2.16)By Lemma 2.1 and Z R N w µ, + = Z R N u µ, + ≡ a , we have k∇ w µ, + k + k w µ, + k = s − µ k∇ u µ, + k + a ∼ . Therefore, { w µ, + } is bounded in H ( R N ). It follows that w µ, + ⇀ w ∗ weakly in H ( R N ) as µ → w µ, + is radial, by Struss’s radiallemma (cf. [12, Lemma A.IV, Theorem A.I’] or [44, Lemma 3.1]) and the Sobolevembedding theorem, w µ, + → w ∗ strongly in L q ( R N ) as µ → { λ µ, + µ − − qγq } is bounded. Thus, λ µ, + µ − − qγq → α ∗ as µ → qγ q < < q < N , s − N (2 ∗ − µ ∼ µ ∗− − qγq → µ →
0. Now, using (2.15) and (2.16), it is standard to show that w µ, + → w ∗ strongly in H ( R N ) as µ → w ∗ is the unique solutionof the following equation: − ∆ u + α ∗ u = σ u q − in R N ,u (0) = max x ∈ R N u ( x ) ,u ( x ) > R N ,u ( x ) → | x | → + ∞ , (2.17) by the well-known uniqueness result (cf. [39]) and the scaling invariance of (1.10), w ∗ ( x ) = ( α ∗ σ ) q − w ( √ α ∗ x ), where w is the unique solution of (2.8). It follows from k w µ, + k = a and the strong convergence of { w µ, + } in H ( R N ) that k w ∗ k = a ,which implies α ∗ = ν a ν where ν a is given by (1.12). Thus, w ∗ = ψ a . Since ψ a is unique, w µ, + → ψ a strongly in H ( R N ) as µ →
0. The system (1.13) directlycomes from (2.11) and (2.12). It remains to prove the local uniqueness of u µ, + for µ > ( F ( w, α, β, γ ) = ∆ w − αν w + βw q − + γw ∗ − , G ( w, α, β, γ ) = k w k − a , (2.18)where α, β, γ > F ( ψ a , ν a , σ ,
0) = 0 and G ( ψ a , ν a , σ ,
0) = 0. Let L ( ψ a , ν a , σ ,
0) = ∂ w F ( ψ a , ν a , σ , ∂ α F ( ψ a , ν a , σ , ∂ w G ( ψ a , ν a , σ , ∂ α G ( ψ a , ν a , σ , ! be the linearization of the system (2.18) at ( ψ a , ν a , σ ,
0) in H ( R N ) × R , that is, ∂ w F ( ψ a , ν a , σ ,
0) = ∆ − ν a ν + ( q − σ ψ q − a , ∂ α F ( ψ a , ν a , σ ,
0) = − ν ψ a and ∂ w G ( ψ a , ν a , σ ,
0) = 2 ψ a , ∂ α G ( ψ a , ν a , σ ,
0) = 0 . Then L ( ψ a , ν a , σ , φ, τ )] = 0 if and only if ∆ φ − ν a ν φ + ( q − σ ψ q − a φ − τ ν ψ a = 0 , Z R N ψ a φ = 0 . Let us consider the following system: ∆ φ − ν a ν φ + ( q − σ ψ q − a φ − τ ν ψ a = g, Z R N ψ a φ = b, (2.19)where ( g, b ) ∈ H rad ( R N ) × R with H rad ( R N ) = { u ∈ H ( R N ) | u is radial } . Then φ = φ g + τ ν φ a , where φ g and φ a satisfies∆ φ g − ν a ν φ g + ( q − σ ψ q − a φ g = g (2.20)and ∆ φ a − ν a ν φ a + ( q − σ ψ q − a φ a = ψ a , (2.21)respectively. By [54, (5.2) and (5.3)], φ a = q − ψ a + ( x · ∇ ψ a ) and Z R N φ a ψ a = ( 1 q − − N ) k ψ a k = 0since q = 2 + N . Thus, the unique solution of (2.19) is given by ( φ g + τ b,g ν φ a , τ b,g )where τ b,g = b − R R N φ g ψ a ν R R N φ a ψ a . ORMALIZED SOLUTIONS 13
Since q < ∗ , it is well-known that ψ a is nondegenerate (cf. [41, Theorem 2.12]and [45, Lemma 4.2]). Thus, by q < ∗ , (2.20) only has zero solution in H rad ( R N )for g = 0, which implies the linear operator L ( ψ a , ν a , σ ,
0) : H rad ( R N ) × R → H rad ( R N ) × R is bijective. Moreover, it is standard to show that | τ b,g | + k φ g + τ b,g φ a k H . | b | + k g k H . Now, by the implicit function theorem, there exists a unique C -curve ( w β,γ , α β,γ )in H rad ( R N ) × R for | β − σ | << | γ | << w σ , , α σ , ) = ( ψ a , ν a ),and F ( w β,γ , α β,γ , β, γ ) ≡ , G ( w β,γ , α β,γ , β, γ ) ≡ . We recall that w µ, + is radial and satisfies (2.16), and w µ, + → ψ a strongly in H ( R N )as µ → k w σ, + k = a , thus, by the uniqueness of s µ determined by (1.13), wemust have w µ, + = w β ( µ ) ,γ ( µ ) for β ( µ ) = µs − N ( q − µ and γ ( µ ) = s − N (2 ∗ − µ with µ > e u µ, + is another ground state of (1.5)with some e λ µ, + ∈ R for µ > e u µ, + = e iθ b u µ, + where θ is a constant and b u µ, + is real valued and positive. Since by thePohozaev identity, we always have e λ µ, + <
0. By applying the well-known Gidas-Ni-Nirenberg theorem (cf. [25]), e u µ, + must be radially symmetric. Now, by running thearguments as used above once more, we know that b w µ, + = s − N µ b u µ, + ( s − µ x ) → ψ a strongly in H ( R N ) as µ → + with k w σ, + k = a . It follows from the uniquenessof w β ( µ ) ,γ ( µ ) that b u µ, + = u µ, + for µ > u µ, + is theunique ground state of (1.5) for µ > (cid:3) Existence and nonexistence of u a,µ, − In this section, we shall mainly study the question ( Q ). Since in the very recentwork [34], the question ( Q ) has been solved for N ≥
4. we only consider the case N = 3 and prove that m − a,µ can also be attained by some u a,µ, − for 2 < q < N in the case N = 3 under some additional assumptions, where m − a,µ is also givenby (1.7) and u a,µ, − is also real valued, positive, radially symmetric and radiallydecreasing. The crucial point in this study is the following energy estimates. Lemma 3.1.
Let N = 3 , < q < N and µ, a > . Then for µa q − qγ q < α N,q , m − a,µ = inf u ∈P a,µ − E µ ( u ) < m + a,µ + 13 S . (3.1) Proof.
Since N = 3, we have U ε = 3 ( εε + | x | ) . Let W ε = χ ( x ) U ǫ where χ ( x ) is acut-off function such that χ ( x ) = 1 for | x | ≤ χ ( x ) = 0 for | x | >
2. By simplecomputations, we have that k∇ W ε k = S + O ( ε ) , k W ε k = S + O ( ε ) (3.2)and k W ε k pp ∼ ε − p , < p < ε ln 1 ε , p = 3; ε p , ≤ p < . (3.3) Now, we define c W ε,t = u µ, + + tW ε and W ε,t = s c W ε,t ( sx ). Then it is well-knownthat k∇ W ε,t k = k∇ c W ε,t k , k W ε,t k = k c W ε,t k , (3.4)and k W ε,t k = s − k c W ε,t k , k W ε,t k qq = s qγ q − q k c W ε,t k qq . (3.5)We choose s = k c W ε,t k a , then W ε,t ∈ S a . By [49, Lemma 4.2], there exist τ ε,t > W ε,t ) τ ε,t ∈ P a,µ − , where ( W ε,t ) τ ε,t = τ ε,t W ε,t ( τ ε,t x ). Thus, k∇ W ε,t k τ − qγ q ε,t = µγ q k W ε,t k qq + k W ε,t k ∗ ∗ τ ∗ − qγ q ε,t . (3.6)Since u µ, + ∈ P a,µ + , by [49, Lemma 4.2], τ ε, >
1. By (3.2) and (3.6), we also knowthat τ ε,t → t → + ∞ uniformly for ε > τ ε,t is uniqueby [49, Lemma 4.2], it is standard to show that τ ε,t is continuous for t , which impliesthat there exists t ε > τ ε,t ε = 1. It follows that m − µ,a ≤ sup t ≥ E µ ( W ε,t ) . (3.7)Recall that u µ, + ∈ S a and W ε are positive, by (3.2), (3.4) and (3.5), there exists t > E µ ( W ε,t ) = ( 12 k∇ c W ε,t k − µq s qγ q − q k c W ε,t k qq − k c W ε,t k ) < m + µ,a + 13 S − σ ′ (3.8)for t < t and t > t with σ ′ >
0. Since u µ, + is radial solution of (1.5) andexponentially decays to zero as r → + ∞ , Z R u µ, + W ε ∼ ε Z ε ( 11 + r ) r ∼ ε and Z R u µ, + W ε ∼ ε Z ε ( 11 + r ) r ∼ ε . (3.9)Thus, by (3.3), s = k c W ε,t k a = 1 + 2 ta Z R u µ, + W ε + t k W ε k = 1 + O ( ε )for t − ≤ t ≤ t . Since it is easy to see that f ( t ) = (1 + t ) q − − t q − qt − qt q − ≥ t ≥ q ≥
3, by (3.4), (3.5) and the fact that u µ, + is a solutionof (1.5) for some λ µ, + < E µ ( W ε,t ) = 12 k∇ c W ε,t k − µq s qγ q − q k c W ε,t k qq − k c W ε,t k ≤ m + µ,a + E µ ( tW ε ) − Z R ( tW ε ) u µ, + + t ( λ µ, + Z R u µ, + W ε + µa ( γ q − k c W ε,t k qq Z R u µ, + W ε ) + O ( ε )= m + µ,a + E µ ( tW ε ) − Z R ( tW ε ) u µ, + + O ( ε ) ORMALIZED SOLUTIONS 15 for t − ≤ t ≤ t , where we have used the fact that λ µ, + a = λ µ, + k u µ, + k = µ ( γ q − k u µ, + k qq which comes from the Pohozaev identity satisfied by u µ, + . Now,for t − ≤ t ≤ t , by (3.2), (3.3) and (3.9), E µ ( W ε,t ) ≤ m + µ,a + 13 S + O ( ε ) − Cε < m + µ,a + 13 S by taking ε > t ≥ E µ ( W ε,t ) < m + µ,a + 13 S . (3.10)The conclusion then follows from (3.7). (cid:3) Remark 3.1.
It is worth pointing our that the above argument also works for N ≥ . In these cases, we have k∇ W ε k = S N + O ( ε N − ) , k W ε k = S N + O ( ε N ) and k W ε k qq ∼ ε N − ( N − q , k W ε k ∼ ε ln 1 ε , N = 4 ,ε , N ≥ . Moreover, similar to (3.9) , Z R N u pµ, + W ε ∼ ε N − for all p ≥ . It follows that E µ ( W ε,t ) = 12 k∇ c W ε,t k − µq s qγ q − q k c W ε,t k qq − k c W ε,t k ≤ m + µ,a + E µ ( tW ε )+ t ( λ µ, + Z R N u µ, + W ε + µa ( γ q − k c W ε,t k qq Z R N u µ, + W ε ) + O ( ε ln 1 ε )= m + µ,a + E µ ( tW ε ) + O ( ε N − ) ≤ m + µ,a + 1 N S N − Cε N − ( N − q + O ( ε ln 1 ε ) < m + µ,a + 1 N S N for t − ≤ t ≤ t by taking ε > sufficiently small since N ≥ and q > . Ourproof is slightly simpler than that of [34] since our test function is radial and we donot need other variational formulas of m − µ,a . For every c > µc q − qγ q < α N,q , let u ∈ P c,µ ± , then v b = bc u ∈ S b forall b >
0. By [49, Lemma 4.2], there exists τ ± ( b ) > v b ) τ ± ( b ) = ( τ ± ( b )) N v b ( τ ± ( b ) x ) ∈ P b,µ ± , where b > µb q − qγ q < α N,q . Clearly, τ ± ( c ) = 1. Lemma 3.2.
Let < q < N . For every c > such that µc q − qγ q < α N,q , τ ′± ( c ) exist and τ ′± ( c ) = µqγ q k u k qq + 2 ∗ k u k ∗ ∗ − k∇ u k c (2 k∇ u k − µqγ q k u k qq − ∗ k u k ∗ ∗ ) . (3.11) Moreover, E µ (( v b ) τ ± ( b ) ) < E µ ( u ) for all b > c such that µb q − qγ q < α N,q .Proof.
The proof is mainly inspired by [19]. Since ( v b ) τ ± ( b ) ∈ P b,µ ± , we have( bc τ ( b )) k∇ u k = ( bc ) q ( τ ( b )) qγ q µγ q k u k qq + ( bc τ ( b )) ∗ k u k ∗ ∗ . Now, if we define the functionΦ( b, τ ) = ( bτc ) k∇ u k − ( bc ) q τ qγ q µγ q k u k qq − ( bτc ) ∗ k u k ∗ ∗ , then Φ( b, τ ( b )) ≡ b > µb q − qγ q < α N,q . Since u ∈ P c,µ ± , ∂ τ Φ( c,
1) = 2 k∇ u k − µqγ q k u k qq − ∗ k u k ∗ ∗ = 0 . It follows from the implicit function theorem that τ ′± ( c ) exist and (3.11) holds. By(1.6) and q < ∗ , 1 − γ q >
0. Thus, by u ∈ P c,µ ± ,1 + cτ ′ ( c ) = 1 + µqγ q k u k qq + 2 ∗ k u k ∗ ∗ − k∇ u k k∇ u k − µqγ q k u k qq − ∗ k u k ∗ ∗ = µqγ q (1 − γ q ) k u k qq k∇ u k − µqγ q k u k qq − ∗ k u k ∗ ∗ . Since ( v b ) τ ± ( b ) ∈ P b,µ ± and u ∈ P c,µ ± , E µ (( v b ) τ ± ( b ) ) = ( 12 − qγ q ) k∇ ( v b ) τ ( b ) k + ( 1 qγ q − ∗ ) k ( v b ) τ ( b ) k ∗ ∗ = ( bc τ ( b )) ( 12 − qγ q ) k∇ u k + ( bc τ ( b )) ∗ ( 1 qγ q − ∗ ) k u k ∗ ∗ = ( 12 − qγ q ) k∇ u k + ( 1 qγ q − ∗ ) k u k ∗ ∗ + o ( b − c )+ 1 + cτ ′ ( c ) c (2( 12 − qγ q ) k∇ u k + 2 ∗ ( 1 qγ q − ∗ ) k u k ∗ ∗ )( b − c )= E µ ( u ) − µ (1 − γ q ) k u k qq c ( b − c )+ o ( b − c ) . Therefore, d E µ (( v b ) τ ± ( b ) ) db | b = c = − µ (1 − γ q ) k u k qq c < . Since c >
0, which satisfies µc q − qγ q < α N,q , is arbitrary and ( v b ) τ ± ( b ) ∈ P b,µ ± , wehave E µ (( v b ) τ ± ( b ) ) < E µ ( u ) for all b > c such that µb q − qγ q < α N,q . (cid:3) With Lemma 3.2 in hands, we can obtain the following.
Proposition 3.1.
Let < q < N and µa q − qγ q < α N,q . If m − a,µ < m + a,µ + N S N then m − a,µ = inf u ∈P a,µ − E µ ( u ) can be attained by some u a,µ, − which is real valued, positive, radially symmetricand decreasing in r = | x | . Moreover, (1.5) has a second solution u a,µ, − with some λ a,µ, − < which is real valued, positive, radially symmetric and radially decreasing. ORMALIZED SOLUTIONS 17
Proof.
Let { u n } ⊂ P a,µ − be a minimizing sequence. Then by taking | u n | and adapt-ing the Schwarz symmetrization to | u n | if necessary, we can obtain a new minimizingsequence, say { u n } again, such that u n are all real valued, nonnegative, radiallysymmetric and decreasing in r = | x | . Since { u n } ⊂ P a,µ − , we have E µ ( u n ) = µq ( qγ q − k u n k qq + 1 N k u n k ∗ ∗ . (3.12)Thus, by the H¨older and Young inequalities and { u n } ⊂ P a,µ − again, we known that { u n } is bounded in H ( R N ) and thus, u n ⇀ u weakly in H ( R N ) as n → ∞ upto a subsequence. Since u n are all radial, by Struss’s radial lemma (cf. [12, LemmaA.IV, Theorem A.I’] or [44, Lemma 3.1]) and the Sobolev embedding theorem, u n → u strongly in L q ( R N ) as n → ∞ up to a subsequence. Without loss ofgenerality, we assume that u n ⇀ u weakly in H ( R N ) and u n → u strongly in L q ( R N ) as n → ∞ . We claim that u = 0. If not, then u n → L q ( R N )as n → ∞ . It follows from { u n } ⊂ P a,µ − that k∇ u n k = k u n k ∗ ∗ + o n (1) , which together with the Sobolev inequality (1.8), implies that either u n → D , ( R N ) as n → ∞ or k∇ u n k = k u n k ∗ ∗ + o n (1) ≥ S N + o n (1).Hence, by (3.12), either m − a,µ = 0 or m − a,µ ≥ N S N , which contradicts E µ ( u ) & u ∈ P c,µ − and Lemma 3.1. We remark that E µ ( u ) & u ∈ P c,µ − comes fromsimilar arguments as used for [50, Lemma 5.7]. Therefore, we must have u = 0.Let v n = u n − u . Then there are two cases:( i ) v n → H ( R N ) as n → ∞ up to a subsequence.( ii ) k∇ v n k + k v n k & i ), u ∈ P a,µ − and m − a,µ is attained by u which is real valued, radiallysymmetric, nonnegative and decreasing in r = | x | . By [49, Proposition 1.5], u isa solution of (1.5) with some λ ∈ R which appears as a Lagrange multiplier. Bymultiplying (1.5) with u and integrating by parts, and using u ∈ P a,µ − , we have λ a = µ ( γ q − k u k qq < , which implies λ <
0. Now, by the maximum principle and classical elliptic es-timates, we know that u is positive. It remains to consider the case ( ii ). Let k u k = t , then by the Fatou lemma, 0 < t ≤ a . There are two subcases:( ii ) k v n k ∗ → n → ∞ up to a subsequence.( ii ) k v n k ∗ ∗ & ii ), by [49, Lemma 4.2], there exists s > u ) s ∈P t ,µ − . By [49, Lemma 4.2] once more, { u n } ⊂ P a,µ − and u n → u strongly in L ∗ ( R N ) ∩ L q ( R N ) as n → ∞ up to a subsequence, m − a,µ + o n (1) = E µ ( u n ) ≥ E µ (( u n ) s ) ≥ E µ (( u ) s ) + o n (1) . By Lemma 3.2, we have m − t ,µ ≥ m − a,µ . Thus, E µ (( u ) s ) = m − t ,µ and m − t ,µ = m − a,µ .If t < a then by taking ( u ) s as the test function in the proof of Lemma 3.2, weknow that m − t ,µ > m − a,µ , which is a contradiction. Thus, in the subcase ( ii ),we must have t = a and so that m − a,µ is attained by ( u ) s which is real valued,radially symmetric, nonnegative and decreasing in r = | x | . As above, we can show that ( u ) s is positive and ( u ) s is a solution of (1.5) with some λ ′ <
0. It remainsto consider the subcase ( ii ). Let s n = (cid:18) k∇ v n k k v n k ∗ ∗ (cid:19) ∗− . Then in the subcase ( ii ), s n . k∇ ( v n ) s n k = k ( v n ) s n k ∗ ∗ ≥ N S N . Since 0 < t ≤ a , by [49, Lemma 4.2], there exists τ > u ) τ ∈ P t ,µ − .We claim that s n ≥ τ up to a subsequence. Suppose the contrary that s n < τ for all n . Then by [49, Lemma 4.2] once more, the Brezis-Lieb lemma (cf. [56,Lemma 1.32]), Lemma 3.2, the fact that u n → u strongly in L q ( R N ) as n → ∞ and the boundedness of { s n } , m − a,µ + o n (1) = E µ ( u n ) ≥ E µ (( u n ) s n )= E µ (( u ) s n ) + E (( v n ) s n ) + o n (1) ≥ m + t ,µ + 1 N S N + o n (1) ≥ m + a,µ + 1 N S N + o n (1) , which is impossible. Thus, we must have s n ≥ τ up to a subsequence. Without lossof generality, we may assume that s n ≥ τ for all n ∈ N . Again, by [49, Lemma 4.2],the Brezis-Lieb lemma (cf. [56, Lemma 1.32]) and the fact that u n → u stronglyin L q ( R N ) as n → ∞ , m − a,µ + o n (1) = E µ ( u n ) ≥ E µ (( u n ) τ ) = E µ (( u ) τ ) + E (( v n ) τ ) + o n (1) . Since s n ≥ τ , by [49, Proposition 2.2], E (( v n ) τ ) ≥
0, which, together withLemma 3.2, implies that t = a and m − a,µ is attained by ( u ) τ . Clearly, ( u ) τ is real valued, radially symmetric, nonnegative and decreasing in r = | x | . Asabove, we can show that ( u ) τ is positive and ( u ) τ is a solution of (1.5) withsome λ ′′ <
0. Therefore, we have proved that m − a,µ can always be attained by some u a,µ, − which is real valued, radially symmetric, positive and decreasing in r = | x | .By [49, Proposition 1.5], (1.5) has a second solution u a,µ, − which is real valued,radially symmetric, positive and decreasing in r = | x | . (cid:3) Our next goal in this section is to prove the existence and nonexistence of groundstates for µa q − qγ q ≥ α N,q in the L -critical and supercritical cases, which givespartial answers to the question ( Q ). In these two cases, 2 + N ≤ q < ∗ , whichimplies qγ q ≥ . We recall that the constant α N,q is given by [49, Theorem 1.1]. For q = 2 + N ,by [49, (5,1)], α N,q = C − qN,q (1 + 2 N ) = 1 C qN,q γ q , (3.13)where C N,q is the optimal constant in the Gagliardo–Nirenberg inequality (1.9).
ORMALIZED SOLUTIONS 19
Lemma 3.3.
Let N ≥ and N ≤ q < ∗ . Then m − a,µ is strictly decreasing for < µ < a qγ q − q α N,q and is nonincreasing for µ ≥ a qγ q − q α N,q , where m − a,µ is givenby (1.7) . Moreover, < m − a,µ < N S N for all µ > in the case of N < q < ∗ while, m − a,µ = 0 for µ ≥ a qγ q − q α N,q in the case of q = 2 + N .Proof. Modified the proof of [49, Lemma 8.2] in a trivial way (or by Lemma 3.2and [49, Theorem 1.1]), we can show that m − a,µ is strictly decreasing for 0 < µ u ( t ) is strictly increasing in (0 , t u ), isstrictly decreasing in ( t u , + ∞ ) and( u ) t u = t N u u ( t u x ) ∈ P a,µ − . Moreover, by [49, Lemma 6.2], we have m − a,µ > µ > L -supercriticalcase 2 + N < q < ∗ . It follows that we can always choose v ε ∈ P a,µ − such that E µ ( v ε ) < m − a,µ + ε in the L -supercritical case 2 + N < q < ∗ . Then by similararguments as used for [49, Lemma 8.2] (or by Lemma 3.2), we have m − a,µ ′ < m − a,µ + ε for all µ ′ > µ. Since ε > µ ≥ a qγ q − q α N,q are arbitrary, m − a,µ is nonincreasing for µ ≥ a qγ q − q α N,q in the L -supercritical case 2 + N < q < ∗ . It follows from [49,Lemma 6.4] that m − a,µ < N S N for all µ > L -critical case q = 2 + N , sincesup u ∈S a k∇ u k k u k q = + ∞ . For all µ >
0, we can always choose u ∈ S a such that k∇ u k k u k q > µγ q . Indeed, ifsup u ∈S a k∇ u k k u k q .
1, then by the Gagliardo-Nirenberg inequality, k∇ u k . k u k q . k∇ u k γ q for all u ∈ S a , which implies sup u ∈S a k∇ u k . . It is impossible since in any ball B R (0), the eigenvalue problem − ∆ u = λu , withDirichlet boundary conditions, has a sequence of eigenvalues λ j → + ∞ as j → ∞ .We note that qγ q = 2 in the L -critical case q = 2 + N . Thus,Ψ ′ u ( t ) = ( k∇ u k − µγ q k u k qq ) t − t ∗ − k u k ∗ ∗ = 0has a unique solution t u > u ∈ S a such that k∇ u k k u k q > µγ q . Moreover, Ψ u ( t ) isstrictly increasing in (0 , t u ), is strictly decreasing in ( t u , + ∞ ) and( u ) t u = t N u u ( t u x ) ∈ P a,µ = P a,µ − . Thus, P a,µ = P a,µ − = ∅ and Ψ u ( t u ) = max t ≥ Ψ u ( t ) for all µ > u ∈S a such that k∇ u k k u k q > µγ q . Now, as in the L -supercritical case 2 + N < q < ∗ , by similar arguments as used for [49, Lemma 8.2], we can show that m − a,µ isnonincreasing for µ ≥ a qγ q − q α N,q in the L -critical case q = 2 + N . It remainsto prove that m − a,µ = 0 for µ ≥ a qγ q − q α N,q in the case of q = 2 + N . Let { ϕ n } be the minimizing sequence of the Gagliardo-Nirenberg inequality (1.9). Then byscaling at N n k ϕ n k ϕ n ( t n x ) if necessary, we may assume that k ϕ n k = a , k ϕ n k qq = 1 and k∇ ϕ n k = C − γq N,q a γq − γq + o n (1). Let us consider the following function: h ϕ n ( µ, t ) = t ( k∇ ϕ n k − µγ q k ϕ n k qq ) − t ∗ k ϕ n k ∗ ∗ = ( C − γq N,q a γq − γq + o n (1) − µγ q ) t − t ∗ k ψ k ∗ ∗ = γ q ( α N,q a qγ q − q + o n (1) − µ ) t − t ∗ k ψ k ∗ ∗ , where we have used (3.13). By [49, Lemma 5.1], there exists a unique t n ( µ ) > h ϕ n ( µ, t n ( µ )) = 0 for 0 < µ < a qγ q − q α N,q . Thus, ( ϕ n ) t n ( µ ) ∈ P a,µ for 0 < µ < a qγ q − q α N,q , where ( ϕ n ) t n ( µ ) = [ t n ( µ )] N ϕ n ( t n ( µ ) x ). Moreover, since k ϕ n k qq = 1, by the H¨older inequality, k ϕ n k ∗ &
1. It follows that t ( µ ) → µ → a qγ q − q α N,q , which implies E µ (( ψ ) t ( µ ) ) = 1 N k ϕ n k ∗ ∗ [ t n ( µ )] ∗ = o n (1)as µ → a qγ q − q α N,q in the L -critical case q = 2 + N . Thus, we must have m − a,µ ≤ µ = a qγ q − q α N,q . By the monotone property of m − a,µ stated in Lemma 3.2, m − a,µ ≤ µ ≥ a qγ q − q α N,q . Recall that we always have E µ ( u ) = 1 N k u k ∗ ∗ ≥ u ∈ P a,µ , (3.14)thus, we must have m − a,µ = 0 for µ ≥ a qγ q − q α N,q . (cid:3) With Lemma 3.3 in hands, we can obtain the following.
Proposition 3.2.
Let N ≥ and N ≤ q < ∗ . (1) If N < q < ∗ , then m − a,µ is attained by same u a,µ, − which is realvalued, positive, radially symmetric and decreasing in r = | x | for all µ > ,and thus, u a,µ, − is a solution of (1.5) for all µ > with some λ a,µ, − < . (2) If q = 2 + N , then m − a,µ can not be attained and (1.5) has no groundstates for all µ ≥ a qγ q − q α N,q .Proof. (1) By Lemma 3.3, 0 < m − a,µ < N S N for all µ > N N < q < ∗ . Bysimilar arguments as used for [50, Lemma 6.2], we know that P a,µ = P a,µ − = ∅ is anatural constraint in S a for all µ > N < q < ∗ . Thus, u a,µ, − is a solution of (1.5) for all µ > λ a,µ, − in the case of 2 + N < q < ∗ .As that in the proof of Proposition 3.1, we can show that λ a,µ, − < u a,µ, − ispositive.(2) Suppose the contrary that m − a,µ is attained by some u a,µ, − for µ ≥ a qγ q − q α N,q ,then by Lemma 3.3 and (3.14), k u a,µ, − k ∗ ∗ = 0. It is impossible since u a,µ, − ∈ S a . ORMALIZED SOLUTIONS 21
Thus, m − a,µ can not be attained for µ ≥ a qγ q − q α N,q . It follows that (1.5) has noground state for all µ ≥ a qγ q − q α N,q . (cid:3) The asymptotic behavior of u a,µ, − In this section, we shall mainly study the question ( Q ) and give a preciselydescription of the asymptotic behavior of u a,µ, − as µ → + . Since we consider µ → + now, the assumptions of [49, Theorem 1.1], [34, Theorem 1.6] and Proposition 3.1always hold and thus, u a,µ, − , which is a minimizer of E| S a ( u ) on P a,µ − , exists for all N ≥
3, 2 < q < ∗ for µ > Proposition 4.1.
Let N ≥ , < q < ∗ and u a,µ, − is a critical point of E| S a ( u ) of mountain pass type. If N ≥ then u a,µ, − → U ε strongly in H ( R N ) as µ → + ,where U ε is the Aubin-Talanti babble such that U ε ∈ S a . Moreover, if N ≥ , thenup to translations and rotations, u a,µ, − is the unique minimizer of E| S a ( u ) on P a,µ − for µ > sufficiently small.Proof. By [49, Theorem 1.4], m + a,µ → µ → + . Moreover, by [49, Theorem 1.4]again, we know that m − a,µ → N S N as µ → + , and k∇ u a,µ, − k , k u a,µ, − k ∗ ∗ → S N as µ → + for 2 + N ≤ q < ∗ . On the other hand, by [49, Lemma 4.2] and similar argumentsas used for [50, Lemma 5.7], we also have m − a,µ & µ > < q < N . Thus, by adapting similar arguments as used in the proofof [49, Theorem 1.1] for the case of 2 + N ≤ q < ∗ to the case of 2 < q < N ,we can also show that m − a,µ → N S N as µ → + , and k∇ u a,µ, − k , k u a,µ, − k ∗ ∗ → S N as µ → + for 2 < q < N (see also [34, Theorem 1.7]). It follows that, up to a subsequence, { u a,µ, − } is a minimizing sequence of the following minimizing problem: S = inf u ∈ D , ( R N ) \{ } k∇ u k k u k ∗ . (4.1)Since N ≥ U ε ∈ L ( R N ) for all ε >
0. We then choose ε > U ε ∈ S a . By [49, Lemma 4.2], there exists t ( µ ) > U ε ) t ( µ ) ∈ P a,µ − for µ > t ( µ )] S N = µγ q k U ε k qq [ t ( µ )] qγ q + [ t ( µ )] ∗ S N . Clearly, by the implicit function theorem, t ( µ ) is of class C for | µ | << t (0) = 1. It follows from S N (1 − [ t ( µ )] ∗ − ) = µγ q k U ε k qq [ t ( µ )] qγ q − that t ( µ ) = 1 − γ q k U ε k qq (2 ∗ − S N µ + o ( µ ) , (4.2)which implies m − a,µ ≤ E µ (( U ε ) t ( µ ) )= 1 N S N − µγ q k U ε k qq ∗ − µq (1 − qγ q ∗ ) k U ε k qq + o ( µ )= 1 N S N − µ k U ε k qq q + o ( µ ) (4.3) for N ≥
5. Since we have m − a,µ = N k∇ u a,µ, − k − µq (1 − qγ q ∗ ) k u a,µ, − k qq , by (4.3),1 N k∇ u a,µ, − k − µq (1 − qγ q ∗ ) k u a,µ, − k qq ≤ N S N − µ k U ε k qq q + o ( µ ) . (4.4)On the other hand, by (4.1) and u a,µ, − ∈ P a,µ − , S ≤ k∇ u a,µ, − k k u a,µ, − k ∗ = k∇ u a,µ, − k ( k∇ u a,µ, − k − µγ q k u a,µ, − k qq ) ∗ = ( k∇ u a,µ, − k − µγ q k u a,µ, − k qq ) N + µγ q k u a,µ, − k qq S N − + o ( µ k u a,µ, − k qq ) . It follows that k∇ u a,µ, − k ≥ S N − N − µγ q k u a,µ, − k qq + o ( µ k u a,µ, − k qq ) . (4.5)Combining (4.4) and (4.5), we have k u a,µ, − k qq ≥ k U ε k qq + o (1) . (4.6)Since { u a,µ, − } is bounded in H ( R N ), u a,µ, − ⇀ u , − weakly in H ( R N ) as µ → + up to a subsequence. Since u a,µ, − is radial and decreasing for r = | x | ,sup y ∈ R N Z B ( y ) | u a,µ, − | dx = Z B (0) | u a,µ, − | dx. Thus, by (4.6), Lions’ lemma [56, Lemma 1.21] and the Sobolev embedding theorem, u , − = 0. Note that it is standard to show that u , − is a weak solution of thefollowing equation, − ∆ U = U ∗ − , in R N , thus, we must have k∇ u , − k ≥ S N . It follows from k∇ u a,µ, − k → S N as µ → + that u a,µ, − → u , − strongly in D , ( R N ) as µ → + up to a subsequence, whichimplies u , − = U ε for some ε >
0. Since k U ε k = a , by the Fatou lemma and(4.6), k U ε k qq ≥ k U ε k qq and k U ε k ≤ k U ε k . Hence, we must have ε = ε and thus, k u , − k = k U ε k = a , which implies u a,µ, − → U ε strongly in H ( R N ) as µ → + up to a subsequence. Since U ε isthe unique Aubin-Talanti babble in S a , we have u a,µ, − → U ε strongly in H ( R N )as µ → + . Moreover, since u a,µ, − is a solution of (1.5), by the Pohozaev identityand u a,µ, − ∈ P a,µ , we have − λ a,µ, − k u a,µ, − k = (1 − γ q ) µ k u a,µ, − k qq , (4.7)which implies λ a,µ, − → µ → + . It remains to prove that u a,µ, − is the uniqueminimizer of E| S a ( u ) on P a,µ − for µ > u a,µ, − . (cid:18)
11 + r (cid:19) N − (4.8)for all r ≥ µ > u a,µ, − is a positiveand radially decreasing solution of (1.5), by Struss’s radial lemma (cf. [12, Lemma ORMALIZED SOLUTIONS 23
A.IV, Theorem A.I’] or [44, Lemma 3.1]), u a,µ, − . r − N − for r ≥
1. Thus, by(4.7), u a,µ, − satisfies − u ′′ a,µ, − − N − r u ′ a,µ, − . u N − a,µ, − u a,µ, − . r − (2+ δ ) u a,µ, − for r & δ >
0. By bootstrapping we obtain the desired decaying estimate (4.8).Now, let us consider the following system: ( F ( w, α, µ ) = ∆ w − αw + µw q − + w ∗ − , G ( w, α, µ ) = k w k − a , (4.10)where α, µ > F ( U ε , ,
0) = 0 and G ( U ε , ,
0) = 0. Let L ( U ε , ,
0) = ∂ w F ( U ε , , ∂ α F ( U ε , , ∂ w G ( U ε , , ∂ α G ( U ε , , ! be the linearization of the system (4.10) at ( U ε , ,
0) in H ( R N ) × R , that is, ∂ w F ( U ε , ,
0) = ∆ + (2 ∗ − U ∗ − ε , ∂ α F ( U ε , ,
0) = − U ε and ∂ w G ( U ε , ,
0) = 2 U ε , ∂ α G ( U ε , ,
0) = 0 . Then L ( U ε , , φ, τ )] = 0 if and only if ∆ φ + (2 ∗ − U ∗ − ε φ − τ U ε = 0 , Z R N U ε φ = 0 . (4.11)We claim that in H rad ( R N ) × R , L ( U ε , , φ, τ )] = 0 if and only if ( φ, τ ) = (0 , L ( U ε , , φ, τ )] = 0 for some ( φ, τ ) ∈ H rad ( R N ) × R . Since it is well-known(cf. [14]) that W = N − U ε + U ′ ε r is the unique radial solution of the followingequation ∆ φ + (2 ∗ − U ∗ − ε φ = 0in H rad ( R N ), by multiplying the first equation of (4.11) with W and integratingby parts, we have 0 = τ Z R N W U ε = − τ Z R N U ε . It follows that τ = 0 and thus, φ = CW for some constant C ∈ R . By the secondequation of (4.11), 0 = Z R N U ε φ = − C Z R N U ε , which implies that φ = 0. Thus, the kernel of the linearization of the system (4.10)at ( U ε , ,
0) in H rad ( R N ) × R is trivial, which implies that the linear operator L ( U ε , ,
0) : H rad ( R N ) × R → H rad ( R N ) × R is injective. On the other hand,by similar arguments as used for Proposition 2.1, we know that all minimizers of E µ ( u ) | S a on P a,µ − are real valued, positive, radially symmetric and radially decreas-ing up to translations and rotations. Now, suppose that there are at least twominimizers of E µ ( u ) | S a on P a,µ − , say u ∗ µ and u ∗∗ µ , then without loss of generality, wemay assume that they are all real valued, positive, radially symmetric and radially decreasing. The corresponding Lagrange multipliers are λ ∗ µ and λ ∗∗ µ , respectively.Let w µ = u ∗ µ − u ∗∗ µ k u ∗ µ − u ∗∗ µ k H + | λ ∗ µ − λ ∗∗ µ | and ς µ = λ ∗ µ − λ ∗∗ µ k u ∗ µ − u ∗∗ µ k H + | λ ∗ µ − λ ∗∗ µ | , where k · k H is the usual norm in H ( R N ). It is easy to see that { w µ } is boundedin H ( R N ) and { ς µ } is bounded. Moreover, by (1.5), we also have − ∆ w µ − λ ∗ µ w µ − ς µ u ∗∗ µ = µ ( q − (cid:18) u ∗ µ + θ µ ( u ∗ µ − u ∗∗ µ ) (cid:19) q − w µ +(2 ∗ − (cid:18) u ∗ µ + θ ′ µ ( u ∗ µ − u ∗∗ µ ) (cid:19) ∗ − w µ , (4.12)where θ µ , θ ′ µ ∈ (0 , u ∗ µ and u ∗∗ µ belong to S a , we also have2 Z R N u ∗ µ w µ = −k u ∗ µ − u ∗∗ µ k k w µ k . Since the linear operator L ( U ε , ,
0) : H rad ( R N ) × R → H rad ( R N ) × R is injective,it is standard to prove that ( w µ , ς µ ) ⇀ (0 ,
0) weakly in D , ( R N ) × R as µ → + .Now, by multiplying (4.12) with w µ and integrating by parts, we can use fact that u ∗∗ µ , u ∗ µ → U ε strongly in H ( R N ) as µ → + ∞ to show that ( w µ , ς µ ) → (0 , D , ( R N ) × R as µ → + . Moreover, by (4.7), we also have ς µ ∼ µ Z R N (cid:18) u ∗ µ + θ µ ( u ∗ µ − u ∗∗ µ ) (cid:19) q − w µ = o ( µ ) . By (4.7), (4.8) and (4.12), − ∆ w µ − λ ∗ µ w µ . µr N − for r & | λ ∗ µ | . By (4.7), we also have − ∆( r − N ) − λ ∗ µ r − N = − λ ∗ µ r − N & µr N − for r & | λ ∗ µ | . Since w µ is radial and { w µ } is bounded in H ( R N ), by [12, Lemma A.2], | w µ | . r − N − for r & . (4.13)Thus, by the maximum principle, | w µ | . r − N for r & | λ ∗ µ | . (4.14)For 1 . r . | λ ∗ µ | , by (4.8), (4.12) and (4.13), − ∆ w µ . r ( 1 r N − + 1 r N − ) . r − N . r − α + N in the case of N ≥
5, where α = . Recall that for N ≥ r − α + N is also asuperharmornic function. Thus, by the maximum principle, | w µ | . r − α + N for 1 . r . | λ ∗ µ | . (4.15) ORMALIZED SOLUTIONS 25
Note that by w µ → D , ( R N ) as µ → + and k w µ k H = 1, we knowthat k w µ k = 1 + o µ (1). Thus, by w µ → D , ( R N ) as µ → + , theSobolev embedding theorem and (4.14) and (4.15),1 ∼ Z R N | w µ | . o µ (1) + Z | λ ∗ µ | r r − α + Z + ∞ | λ ∗ µ | r − N = o µ (1) + 12 r − , which is a contradiction by taking r > u a,µ, − is the unique minimizer of E| S a ( u ) on P a,µ − for µ > N ≥ (cid:3) For N = 3 ,
4, The Aubin-Talanti babbles U ε L ( R N ). Thus, we need tomodify the arguments for Proposition 4.1 to give a precise description of u a,µ, − as µ → + in these two cases. As in the proof of Proposition 4.1, { u a,µ, − } is also aminimizing sequence of the minimizing problem (4.1) in the cases N = 3 ,
4. Since u a,µ, − is radially symmetric for r = | x | , by Lions’ result (cf. [56, Theorem 1.41]),up to subsequence, there exists σ µ > ε ∗ > v a,µ, − ( x ) = σ N − µ u a,µ, − ( σ µ x ) → U ε ∗ strongly in D , ( R N ) as µ → + . (4.16)We also remark that since U ε ∗ L ( R N ) for N = 3 , k v a,µ, − k = a σ µ , by theFatou lemma, we have σ µ → µ → + . Lemma 4.1.
Let N = 3 , and < q < ∗ . Then ∼ µσ N − N − qµ − λ a,µ, − , NN − < q < ∗ ,µσ µ − λ a,µ, − ln (cid:18) p − λ a,µ, − σ µ (cid:19) , N = 3 , q = 3 ,µσ − q µ (cid:18)p − λ a,µ, − σ µ (cid:19) q − − λ a,µ, − , N = 3 , < q < . Proof.
By the equation (1.5), we know that v a,µ, − satisfies − ∆ v a,µ, − − λ a,µ, − σ µ v a,µ, − = µσ N − N − qµ v q − a,µ, − + v ∗ − a,µ, − in R N . (4.17)It follows from (4.7) that − λ a,µ, − σ µ k v a,µ, − k = (1 − γ q ) µσ N − N − qµ k v a,µ, − k qq . (4.18)Recall that k u a,µ, − k = σ µ k v a,µ, − k = a and k∇ u a,µ, − k → S N as µ → + , by(4.7) and the H¨older inequality, λ a,µ, − → µ → + . Clearly, µσ N − N − qµ → µ → + . (4.19)By the H¨older inequality once more, | λ a,µ, − | σ µ k v a,µ, − k . µσ N − N − qµ k v a,µ, − k N − N − q . Since q > N − N − q <
2. It follows from k v a,µ, − k ∼ σ − µ → + ∞ as µ → + that | λ a,µ, − | σ µ = o ( µσ N − N − qµ ) as µ → + . (4.20) Recall that v a,µ, − → U ε ∗ strongly in D , ( R N ) as µ → + up to a subsequence, by(4.19)-(4.20), adapting the Moser iteration (cf. [53, B.3 Lemma]) and the L p theoryof elliptic equations to (4.17) and the Sobolev embedding theorem, v a,µ, − → U ε ∗ strongly in L ∞ ( R N ) as µ → + up to a subsequence (4.21)In what follows, we follow the ideas in [7] (see also [24, 38]) to drive a uniformlyupper bound of v a,µ, − . We define e v a,µ, − = 1 v a,µ, − (0) v a,µ, − ( q v a,µ, − (0) x ) . Since e v a,µ, − is radial, e v a,µ, − satisfies − e v ′′ a,µ, − − N − r e v ′ a,µ, − = f ( e v a,µ, − ) in R N . (4.22)where f ( e v a,µ, − ) = λ a,µ, − σ µ v a,µ, − (0) e v a,µ, − + µσ N − N − qµ [ v a,µ, − (0)] q − e v q − a,µ, − +[ v a,µ, − (0)] ∗ − e v ∗ − a,µ, − . Let H ( r ) = r N ( e v ′ a,µ, − ) + ( N − r N − e v a,µ, − e v ′ a,µ, − + N − N r N e v a,µ, − f ( e v a,µ, − ) . Then by direct calculations and using (4.20)-(4.22), H ′ ( r ) = r N e v ′ a,µ, − N (4 | λ a,µ, − | σ µ − ( N − ∗ − q ) µσ N − N − qµ v q − a,µ, − ) v a,µ, − = µσ N − N − qµ r N e v ′ a,µ, − N ( o µ (1) − ( N − ∗ − q ) U q − ε ∗ ) v a,µ, − . Since v a,µ, − > e v ′ a,µ, − < v a,µ, − exponentially decays to zero as r → + ∞ ,there exists r µ > H ′ ( r ) > < r < r µ and H ′ ( r ) < r > r µ . Thus, H ( r ) > H (0) = 0 for all r >
0. LetΨ( r ) = − e v ′ a,µ, − r e v NN − a,µ, − . Then by direct calculations and using (4.22),Ψ ′ ( r ) = NN − r − (1+ N ) e v − N − N − a,µ, − H ( r ) > r >
0. It follows from (4.22) once more thatΨ( r ) > Ψ(0) = − e v ′′ a,µ, − (0) = d µ N where d µ = λ a,µ, − σ µ v a,µ, − (0) + µσ N − N − qµ [ v a,µ, − (0)] q − + [ v a,µ, − (0)] ∗ − . Let Z µ ( r ) = 1(1 + d µ N ( N − r ) N − . ORMALIZED SOLUTIONS 27
Then it is easy to check that − Z ′ µ ( r )[ Z µ ( r )] NN − = d µ N r . It follows that e v ′ a,µ, − e v NN − a,µ, − ≤ Z ′ µ ( r )[ Z µ ( r )] NN − for all r > , which together with (4.21), implies v a,µ, − . r ) N − for all r > µ > NN − < q < ∗ , k v a,µ, − k qq . Z + ∞ r ) ( N − q r N − dr . . By the Fatou lemma, k v a,µ, − k qq ≥ k U ε ∗ k qq + o µ (1) &
1. Thus, by (4.18), µσ N − N − qµ − λ a,µ, − ∼ NN − < q < ∗ . For N = 3 and q = 3, we need to drive the uniformly exponential decay of v a,µ, − at infinitely both from below and above to obtain the conclusions. LetΦ = r − e − √ | λ a,µ, − | σ µ r . Then it is easy to check (cf. [44]) that − ∆Φ − λ a,µ, − σ µ Φ ≤ r ≥ N = 3. Since v a,µ, − → U ε ∗ strongly in L ∞ ( R N ) as µ → + up to a subsequence,by the maximum principle, v a,µ, − & r − e − √ | λ a,µ, − | σ µ r for r ≥ N = 3. On the other hand, letΥ = r − e − √ | λ a,µ, − | σ µ r . Then it is also easy to check that − ∆Υ − λ a,µ, − σ µ Υ ≥ r ≥
1. Since µσ − q µ → µ → + , by (4.23), for r & | λ a,µ, − | σ µ , we have − ∆ v a,µ, − − λ a,µ, − σ µ v a,µ, − ≤ N = 3. Thus, by themaximum principle and (4.21) once more, v a,µ, − . r − e − √ | λ a,µ, − | σ µ r for r & | λ a,µ, − | σ µ . (4.25)For q = 3 and N = 3, by (4.23) and (4.25), k v a,µ, − k . Z | λa,µ, −| σ µ r − + Z + ∞ | λa,µ, −| σ µ e − √ | λ a,µ, − | σ µ r . ln (cid:18) p | λ a,µ, − | σ µ (cid:19) . By (4.24), for q = 3 and N = 3, we also have k v a,µ, − k & Z √ | λa,µ, −| σµ r − & ln (cid:18) p | λ a,µ, − | σ µ (cid:19) . Thus, the conclusion for q = 3 and N = 3 then follows (4.18). For N = 3 and2 < q <
3, we need to construct a newly upper bound of v a,µ, − by adapting ideasin [20]. By (4.23), we have − ∆ v a,µ, − − λ a,µ, − σ µ v a,µ, − . µσ − q µ r − q for r & p | λ a,µ, − | σ µ . On the other hand, let φ µ ∼ µσ − q µ | λ a,µ, − | σ µ r − q , then by direct calculations, − ∆ φ µ − λ a,µ, − σ µ φ µ & µσ − q µ r − q for r & p | λ a,µ, − | σ µ . By (4.23) and the maximum principle, v a,µ, − . µσ − q µ | λ a,µ, − | σ µ r − q for r & p | λ a,µ, − | σ µ . (4.26)Now, using (4.26) as a new barrier and by (4.23), we know that − ∆ v a,µ, − − λ a,µ, − σ µ v a,µ, − . µσ − q µ (cid:18) µσ − q µ | λ a,µ, − | σ µ r − q (cid:19) q − for r & √ | λ a,µ, − | σ µ . Thus, by similar comparisons, we have v a,µ, − . (cid:18) µσ − q µ | λ a,µ, − | σ µ (cid:19) q r − ( q − for r & p | λ a,µ, − | σ µ . By iterating the above arguments n times for a sufficiently large n such that q ( q − n − >
0, we have v a,µ, − . (cid:18) µσ − q µ | λ a,µ, − | σ µ (cid:19) s n r − ( q − n for r & p | λ a,µ, − | σ µ , (4.27)where s n = s n − ( q −
1) + 1 which implies s n = ( q − n +1 − q − . By (4.24), we have k v a,µ, − k qq & Z √ | λa,µ, −| σµ r − q e − q √ | λ a,µ, − | σ µ r & (cid:18) | λ a,µ, − | σ µ (cid:19) − q (4.28)and k v a,µ, − k & Z √ | λa,µ, −| σµ e − √ | λ a,µ, − | σ µ r & (cid:18) | λ a,µ, − | σ µ (cid:19) . It follows from (4.18) that | λ a,µ, − | & µσ − q µ (cid:18) | λ a,µ, − | σ µ (cid:19) − q and (cid:18) | λ a,µ, − | σ µ (cid:19) | λ a,µ, − | . µ (4.29) ORMALIZED SOLUTIONS 29 which implies µσ − q µ | λ a,µ, − | σ µ . | λ a,µ, − | − q σ − qµ . σ − q ) µ . Now, by (4.27) and (4.29), k v a,µ, − k qq . Z √ | λa,µ, −| σµ r − q + ( σ µ ) − q − n +1 − Z + ∞ √ | λa,µ, −| σµ r − q ( q − n . (cid:18) | λ a,µ, − | σ µ (cid:19) − q + σ q ( q − n − − q − n +1 − µ . (cid:18) | λ a,µ, − | σ µ (cid:19) − q + σ ( q − n (2+ o n (1)) µ = (cid:18) | λ a,µ, − | σ µ (cid:19) − q . (4.30)The conclusion for N = 3 and 2 < q < (cid:3) With Lemma 4.1 in hands, we can obtain the following.
Proposition 4.2.
Let N = 3 , and < q < ∗ . Then w µ, − = ε N − µ u a,µ, − ( ε µ x ) → U ε ∗ strongly in D , ( R N ) as µ → + up to a subsequence for some ε ∗ > , where ε µ > satisfies µ ∼ ε − qµ e − ε − µ , N = 4 , < q < ∗ ,ε q − µ , N = 3 , < q < ∗ ,ε µ ln( ε µ ) , N = 3 , q = 3 ,ε − q µ , N = 3 , < q < . (4.31) Proof.
Let { V ε } be the family given by (2.4). Since 2 ≥ NN − for N ≥
4, By [44,(4.2)–(4.5)], k V ε k qq ∼ ε N − N − q , N = 3 , , NN − < q < ∗ ,ε ln( R ε ε − ) , N = 3 , q = 3 ,ε − q ( R ε ε − ) − q , N = 3 , < q < ε > t µ,ε > k∇ V ε k = µγ q k V ε k qq t qγ q − µ,ε + k V ε k ∗ ∗ t ∗ − µ,ε and 2 k∇ V ε k < µqγ q k V ε k qq t qγ q − µ,ε + 2 ∗ k V ε k ∗ ∗ t ∗ − µ,ε . Thus, { t µ,ε } is uniformly bounded and bounded from below away from 0 for all ε, µ > S N (1 − t ∗ − µ,ε µ ) = µγ q k V ε k qq t qγ q − µ,ε + O (( R ε ε − ) − N )) . Then we can use similar arguments as used for (4.2) to show that t µ,ε = 1 − (1 + o (1)) µγ q k V ε k qq + O (( R ε ε − ) − N ))(2 ∗ − S N and thus by similar arguments as used for (4.6), k u a,µ, − k qq ≥ (1 + o (1))( k V ε k qq − Cµ − ( R ε ε − ) − N ))) , which together with (2.5) and (4.32), implies k u a,µ, − k qq & ε − q − Cµ − e − ε − , N = 4 , < q < ∗ ,ε − q − Cµ − ε , N = 3 , < q < ∗ ,ε ln( 1 ε ) − Cµ − ε , N = 3 , q = 3 ,ε q − − Cµ − ε , N = 3 , < q < . By choosing ε µ such that the right hand sides of the above estimate take themaximum, we have (4.31) and k u a,µ, − k qq & ε N − N − qµ , N = 3 , , NN − < q < ∗ ,ε µ ln( 1 ε µ ) , N = 3 , q = 3 ,ε q − µ , N = 3 , < q < . (4.33)We define w µ, − = ε N − µ u a,µ, − ( ε µ x ), then k w µ, − k ∗ ∗ , k∇ w µ, − k ∼ k w µ, − k qq & , N = 3 , , NN − < q < ∗ , ln( 1 ε µ ) , N = 3 , q = 3 ,ε q − µ , N = 3 , < q < . (4.34)It is easy to see that σ N − N − qµ k v a,µ, − k qq = k u a,µ, − k qq = ε N − N − qµ k w µ, − k qq . (4.35)Then by Lemma 4.1, (4.18) and (4.34), we have σ N − N − qµ & ε N − N − qµ (4.36)for NN − < q < ∗ and N = 3 ,
4. On the other hand, we know that w µ, − ( x ) = (cid:18) ε µ σ µ (cid:19) N − v a,µ, − (cid:18) ε µ σ µ x (cid:19) (4.37)and e w µ, − satisfies − ∆ e w µ, − = g ( e w µ, − ) in R N . ORMALIZED SOLUTIONS 31 where e w µ, − = 1 w µ, − (0) w µ, − ([ w µ, − (0)] s x )with s ∈ R and g ( e w µ, − ) = λ a,µ, − ε µ [ w µ, − (0)] s e w µ, − + µε N − N − qµ [ w µ, − (0)] s + q − e w q − µ, − +[ w µ, − (0)] s +2 ∗ − e w ∗ − µ, − . By similar arguments as used for (4.23), we have w µ, − . w µ, − (0)(1 + b µ r ) N − for all r > , (4.38)where b µ = [ w µ, − (0)] s − ( λ a,µ, − ε µ w µ, − (0) + µε N − N − qµ [ w µ, − (0)] q − +[ w µ, − (0)] ∗ − ) . We recall that µ, σ µ , λ a,µ, − → µ → + , and by (4.7), we have λ a,µ, − . µ .Thus, by Lemma 4.1, (4.21) and (4.37), b µ ∼ (cid:18) ε µ σ µ (cid:19) ( N − s +2 . (4.39)Now, take s = − k w µ, − k qq . (cid:18) ε µ σ µ (cid:19) N − q − N (4 − N ) for NN − < q < ∗ and N = 3 ,
4, which together with (4.35), implies that σ µ . ε µ for NN − < q < ∗ and N = 3 ,
4. It follows from (4.36) that σ µ ∼ ε µ for NN − < q < ∗ and N = 3 ,
4. For the case N = 3 and q = 3, by k w µ, − k ∼ ε − µ , Struss’sradial lemma (cf. [12, Lemma A.IV, Theorem A.I’] or [44, Lemma 3.1]) and similararguments as used for (4.25), w µ, − . ε − µ r − e − √ | λ a,µ, − | ε µ r for r & p | λ a,µ, − | ε µ . (4.40)It follows from (4.38) and (4.39) that k w µ, − k qq . (cid:18) ε µ σ µ (cid:19) − ( s +2) ln( 1 p | λ a,µ, − | σ µ ) . By Lemma 4.1, taking s = − ε µ & σ µ . By Lemma 4.1, taking s = 2 and (4.35), we have ε µ . σ µ . Thus, for N = 3 and q = 3, we also have ε µ ∼ σ µ . For the case N = 3 and 2 < q <
3, by (4.7), (4.33), Lemma 4.1 and µ ∼ ε − q µ , σ q − q µ & ε q − q µ (cid:18) µε − q µ (cid:19) − q − q ∼ ε q − q µ which implies σ µ & ε µ . Thus, by (4.21), we can adapt the maximum principle asthat in the proof of Proposition 4.2 to show that w a,µ, − & r − e − √ | λ a,µ, − | ε µ r for r ≥ . (4.41)By (4.41), we can see that the estimates for (4.29) works for ε µ and thus, we have | λ a,µ, − | & µε − q µ (cid:18) | λ a,µ, − | ε µ (cid:19) − q , (cid:18) | λ a,µ, − | ε µ (cid:19) | λ a,µ, − | . µ and µε − q µ | λ a,µ, − | ε µ . | λ a,µ, − | − q ε − qµ . ε − q ) µ . Now, we can follow similar arguments as used in the proof of Lemma 4.1 to showthat k w µ, − k qq . (cid:18) | λ a,µ, − | ε µ (cid:19) − q , which, together with Lemma 4.1 and (4.35), implies that σ µ . ε µ . Thus, we alsohave σ µ ∼ ε µ as µ → + in the case of N = 3 and 2 < q < (cid:3) We are ready to give the proofs of Theorem 1.1 and 1.2.
Proof of Theorem 1.1:
It follows immediately from Lemma 3.1, Propositions 3.1and 3.2. ✷ Proof of Theorem 1.2:
It follows immediately from Propositions 2.1, 4.1 and4.2. ✷ We close this section by
Proof of Theorem 1.3: (1) Since the proof is similar to that of Proposition 2.1,we only sketch it. In the case of q = 2 + N , we have k ϕ k = k φ k for all minimizersof the Gagliardo–Nirenberg inequality (1.9), where φ is the unique solution of (1.10).Thus, we choose ϕ = a k φ k φ ∈ S a as a test function of m − a,µ . By using similararguments as used in the proof of Proposition 2.1 and direct calculations, m − a,µ ≤ N (1 − µα N,q,a ) ∗ ∗− (cid:18) k∇ φ k k φ k ∗ (cid:19) N . It follows from u a,µ, − ∈ P a,µ , the Gagliardo–Nirenberg and Sobolev inequalitiesthat S N ≤ k∇ u a,µ, − k (1 − µα N,q,a ) ∗− ≤ (cid:18) k∇ φ k k φ k ∗ (cid:19) N , (4.42)which, together with u a,µ, − ∈ P a,µ once more and the Pohozaev identity satisfiedby u a,µ, − , implies that − λ µ, − = 1 − γ q a µ k u a,µ, − k qq ≥ (1 + o µ (1)) 1 − γ q a γ q S N (1 − µα N,q,a ) ∗− and − λ µ, − = 1 − γ q a µ k u a,µ, − k qq ≤ (1 − µα N,q,a ) ∗− (cid:18) k∇ φ k k φ k ∗ (cid:19) N . ORMALIZED SOLUTIONS 33
Thus, { v a,µ, − } is bounded in H ( R N ), where v a,µ, − = ( a k φ k ) N − s N µ u a,µ, − ( a k φ k s µ x )and s µ = (1 − µα N,q,a ) − N − . Clearly, v a,µ, − satisfies − ∆ v a,µ, − = λ µ, − a k φ k s µ v a,µ, − + µ ( a k φ k ) N v q − a,µ, − + s − N (2 ∗ − µ v ∗ − a,µ, − By (3.13) and [55, (I.3)], we know that α N,q,a ( a k φ k ) N = 1 for q = 2 + N . Onthe other hand, since v a,µ, − is radial, it is standard to show that v a,µ, − → ψ ν ′ a , strongly in H ( R N ) as µ → α − N,q,a up to a subsequence for some ν ′ a > N < q < ∗ , q − − N = 0. Thus, we can choose ν a >
0, as that in (2.10), such that k ψ ν a , k = a . Again, we use ψ ν a , ∈ S a as a test function of m − a,µ . By using similar arguments as used in the proof ofProposition 2.1 and direct calculations, m − a,µ . µ − qγq − as µ → + ∞ . It followsthat u a,µ, − → D , ( R N ) ∩ L q ( R N ) as µ → + ∞ . This, together with u a,µ, − ∈ P a,µ and the Gagliardo–Nirenberg and Sobolev inequalities, implies k∇ u a,µ, − k ≥ (1 + o µ (1))( µγ q a q − qγ q C qN,q ) − qγq − . On the other hand, for the test function ψ ν a , , it satisfies k∇ ψ ν a , k = µγ q k ψ ν a , k qq t qγ q − µ + k ψ ν a , k ∗ ∗ t ∗ − µ , where ( ψ ν a , ) t µ ∈ P a,µ . It follows that t µ k∇ ψ ν a , k ≤ (cid:18) µa q − qγ q γ q C qN,q (cid:19) qγq − . Thus, E µ (( ψ ν a , ) t µ ) = (1 + o µ (1))( 12 − qγ q ) k∇ ( ψ ν a , ) t µ k ≤ (1 + o µ (1))( 12 − qγ q ) (cid:18) µa q − qγ q γ q C qN,q (cid:19) qγq − . Note that E µ (( ψ ν a , ) t µ ) ≥ m − a,µ and m − a,µ = E µ ( u a,µ, − ) = (1 + o µ (1))( 12 − qγ q ) k∇ u a,µ, − k as µ → + ∞ , we must have k∇ u a,µ, − k = (1 + o µ (1))( µγ q a q − qγ q C qN,q ) − qγq − . (4.43)As in (1), { v a,µ, − } is bounded in H ( R N ), where v a,µ, − = s N µ u a,µ, − ( s µ x ) and s µ = µ qγq − . Again, v a,µ, − satisfies − ∆ v a,µ, − = λ µ, − s µ v a,µ, − + v q − a,µ, − + s − N (2 ∗ − µ v ∗ − a,µ, − . Using (4.43), the Pohozaev identity satisfied by u a,µ, − and u a,µ, − ∈ P a,µ once more,we have − λ µ, − = (1 + o µ (1)) 1 − γ q a ( µγ q a q − qγ q C qN,q ) − qγq − . Now, by similar arguments as used in (1), v a,µ, − → ψ ν ′ a , strongly in H ( R N ) as µ → + ∞ up to a subsequence for some ν ′ a >
0. Since in the cases of 2+ N < q < ∗ , q − − N = 0. By k v a,µ, − k = a , we must have ν ′ a = ν a . By the uniqueness of ψ ν a , in S a , we know that v a,µ, − → ψ ν a , strongly in H ( R N ) as µ → + ∞ . Using theuniqueness of ψ ν a , in S a and the nondegenerate of ψ ν a , , we can prove the localuniqueness of u a,µ, − for µ > u a,µ, + in the proof of Proposition 2.1. ✷ Acknowledgements
The research of J. Wei is partially supported by NSERC of Canada and theresearch of Y. Wu is supported by NSFC (No. 11701554, No. 11771319, No.11971339).
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