Normalized solutions for a Schrödinger equation with critical growth in \mathbb{R}^{N}
aa r X i v : . [ m a t h . A P ] F e b Normalized solutions for a Schr¨odinger equation withcritical growth in R N Claudianor O. Alves, ∗ Chao Ji †‡ and Olimpio H. Miyagaki § February 16, 2021
Abstract
In this paper we study the existence of normalized solutions to the followingnonlinear Schr¨odinger equation with critical growth − ∆ u + λu = f ( u ) , in R N ,u > , Z R N | u | dx = a , where a > λ < f has an exponential critical growth when N = 2, and f ( t ) = µ | u | q − u + | u | ∗ − u with q ∈ (2 + N , ∗ ), µ > ∗ = NN − when N ≥ N ≥ N = 2. Keywords:
Normalized solutions, Nonlinear Schr¨odinger equation, Variational methods,Critical exponents.
This paper concerns the existence of normalized solutions to the following nonlinearSchr¨odinger equation with critical growth − ∆ u + λu = f ( u ) , in R N ,u > , Z R N | u | dx = a , (1.1)where a > λ < f has an exponential critical growth when N = 2, and f ( t ) = µ | u | q − u + | u | ∗ − u with q ∈ (2 + N , ∗ ), µ > ∗ = NN − when N ≥ ∗ C.O. Alves was partially supported by CNPq/Brazil grant 304804/2017-7. † Corresponding author ‡ C. Ji was partially supported by Natural Science Foundation of Shanghai(20ZR1413900,18ZR1409100). § O. H. Miyagaki was supported by FAPESP/Brazil grant 2019/24901-3 and CNPq/Brazil grant307061/2018-3. i ∂ψ∂t + △ ψ + h ( | ψ | ) ψ = 0 in R N . A stationary wave solution is a solution of the form ψ ( t, x ) = e iλt u ( x ), where λ ∈ R and u : R N → R is a time-independent that must solve the elliptic problem − ∆ u + λu = g ( u ) , in R N , (1.2)where g ( u ) = h ( | u | ) u . For some values of λ the existence of nontrivial solutions for (1.2)are obtained as the critical points of the action functional J λ : H ( R N ) → R given by J λ ( u ) = 12 Z R N ( |∇ u | + λ | u | ) dx − Z R N G ( u ) dx, where G ( t ) = R t g ( s ) ds . In this case the particular attention is devoted to the least actionsolutions , namely, the minimizing solutions of J λ among all non-trivial solutions.Another important way to find the nontrivial solutions for (1.2) is to search for solutionswith prescribed mass , and in this case λ ∈ R is part of the unknown. This approach seems tobe particularly meaningful from the physical point of view, because there is a conservationof mass.The present paper has been motivated by a seminal paper due to Jeanjean [26] thatstudied the existence of normalized solutions for a large class of Schr¨odinger equations of thetype − ∆ u + λu = g ( u ) , in R N ,u > , Z R N | u | dx = a , (1.3)with N ≥
2, where function g : R → R is an odd continuous function with subcriticalgrowth that satisfies some technical conditions. One of these conditions is the following: ∃ ( α, β ) ∈ R × R satisfying N +4 N < α ≤ β < NN − , for N ≥ , N +4 N < α ≤ β, for N = 1 , , such that αG ( s ) ≤ g ( s ) s ≤ βG ( s ) with G ( s ) = Z s g ( t ) dt. As an example of a function g that satisfies the above condition is g ( s ) = | s | q − s with q ∈ (2 + N , ∗ ) when N ≥ q > N = 2. In order to overcome the loss of compactnessof the Sobolev embedding in whole R N , the author worked on the space H rad ( R N ) to get somecompactness results. However the most important and interesting point, in our opinion, isthe fact that Jeanjean did not work directly with the energy functional I : H ( R N ) → R associated with the problem (1.3) given by 2 ( u ) = 12 Z R N |∇ u | dx − Z R N G ( u ) dx. In his approach, he considered the functional e I : H ( R N ) × R → R given by e I ( u, s ) = e s Z R N |∇ u | dx − e Ns Z R N G ( e Ns u ( x )) dx. After a careful analysis, it was proved that I and e I satisfy the mountain pass geometry onthe manifold S ( a ) = { u ∈ H , ( R N ) : | u | = a } , and their mountain pass levels are equal, which we denote by γ ( a ). Moreover, using theproperties of e I ( u, s ), it was obtained a ( P S ) sequence ( u n ) to I associated with the mountainpass level γ ( a ) which is bounded in H rad ( R N ). Finally, after some estimates, the author wasable to prove that the weak limit of ( u n ), denoted by u , is nontrivial, u ∈ S ( a ) and it verifies − ∆ u − g ( u ) = λ a u in R N , for some λ a <
0. An example of a nonlinearity explored in [26] we cite( g ) g ( t ) = µ | t | q − t t ∈ R , where µ > q ∈ (2 , ∗ ). We recall that the study of the normalized problem despitebeing more convenient in the application, this bring some difficulties such as Nehari manifoldmethod can not be applied because the constants µ and λ a are unknow in the problem; itis necessary to prove that the weak limit belongs to the constrained manifold; and also itbrings some difficult to apply some usual approach for get the boundedness of the PalaisSmale sequence.We recall that the number ¯ q := 2 + N is called in the literature as L − critical exponent,which come from Gagliardo Nirenberg inequality, (see [15, Theorem 1.3.7, page 9]. If g isof the form ( g ) with q ∈ (2 , ¯ q ), we say that the problem is L − subcritical, while in thecase q ∈ (¯ q, ∗ ) the problem is L − supercritical. Associated with the L − supercritical, wewould like to cite [10], where the authors studied a problem involving vanishing potential.In the purely L − critical case, that is, q = 2 + N , related problems were studied in [16, 32].In [36], Soave studied the normalized solutions for the nonlinear Schr¨odinger equation(1.1) with combined power nonlinearities of the type( f ) f ( t ) = µ | t | q − t + | t | p − t, t ∈ R , where 2 < q ≤ N ≤ p < ∗ , p = q and µ ∈ R . He showed that interplay between subcritical, critical and supercritical nonlinearities stronglyaffects the geometry of the functional and existence and properties of ground states.Recently some authors have considered the problem (1.1) with f of the form ( f ) but with p = 2 ∗ , which implies that f has a critical growth in the Sobolev sense. In [27], the existence3f a ground state normalized solution is obtained as minimizer of the constrained functionalassuming that q ∈ (2 , N ) . While in [29] a multiplicity result is established, where thesecond solution is not a ground state. For the general case q ∈ (2 , ∗ ), we would like tomention Soave [37], where the existence result is obtained by imposing that µa (1 − γ p ) q < α ,where α is a specific constant that depends on N and q and γ p = N ( p − p . We have seen thatthe results in the paper are deeply dependent on the assumptions about a and µ , becauseby the Pohozaev identidy the problem (1.1) does not have any solution if λ = − a ≥ µ > . Still related to the case q ∈ (2+ N , ∗ ), we would like to refer [2, 3] where the existenceof least action solutions was proved with µ > N ≥ , and for N = 3 by supposing atechnical on the constants λ and µ .We recall that elliptic problems involving critical Sobolev exponent were studied by manyresearchers after appeared the pioneering paper by Brezis and Nirenberg[13] , which havehad many progresses in several directions. We would like to mention the excellent book [42],for a review on this subject. In our setting, since if λ = 0, the problem (1.1) does not haveany solution for any µ , then taking λ as a Lagrange multiplier, it is able to get solution, bycombining the arguments made in [13] with the concentration compactness principle. Forthe reader interested in normalized solutions for the Schr¨odinger equations, we would alsolike to refer [9], [11], [12], [17], [25], [28], [31], [34], [38], [39], [41] and references therein.Our main result for the Sobolev critical case is the following: Theorem 1.1.
Assume that f is of the form ( f ) with p = 2 ∗ and q ∈ (2+ N , ∗ ) . Then, thereexists µ ∗ = µ ∗ ( a ) > such that the problem (1.1) admits a couple ( u a , λ a ) ∈ H ( R N ) × R ofweak solutions such that R R N | u | dx = a and λ a < for all µ ≥ µ ∗ . The above theorem complements the results found in [37] for the L − supercritical case,because in that paper µ ∈ (0 , a − (1 − γ p ) q α ) for some α >
0, then µ cannot be large enough,while in our paper µ can be large enough, because µ ∈ [ µ ∗ ( a ) , + ∞ ). Here, we used adifferent approach from that explored in [37], because we work directly with the mountainpass geometry and concentration-compactness principle due to Lions [30], while in [37], Soaveemployed minimization technique and properties of the Pohozaev manifold.Motivated by the research made in the critical Sobolev case, in this paper we also studythe exponential critical growth for N = 2, which is a novelty for this type of problem. Tothe best our knowledge we have not found any reference involving normalizing problem withthe exponential critical growth. We recall that in R , the natural growth restriction on thefunction f is given by the inequality of Trudinger and Moser [33, 40]. More precisely, we saythat a function f has an exponential critical growth if there is α > | s |→∞ | f ( s ) | e αs = 0 ∀ α > α and lim | s |→∞ | f ( s ) | e αs = + ∞ ∀ α < α . We would like to mention that problems involving exponential critical growth have receiveda special attention at last years, see for example, [4, 5, 7, 8, 14, 18, 19, 20, 21] for semilinearelliptic equations, and [1, 6, 22, 23] for quasilinear equations.In this case, we assume that f is a continuous function that satisfies the followingconditions:( f ) lim t → | f ( t ) || t | τ = 0 as t →
0, for some τ > f ) lim | t |→ + ∞ | f ( t ) | e αt = ( , for α > π, + ∞ , for 0 < α < π ;( f ) there exists a positive constant θ > < θF ( t ) ≤ tf ( t ) , ∀ t = 0 , where F ( t ) = Z t f ( s ) ds ;( f ) there exist constants p > µ > f ( t ) ≥ µ sgn ( t ) | t | p − t for all t = 0 , where sign : R \ { } → R is given by sgn ( t ) = (cid:26) , if t > − , if t < . Our main result is as follows:
Theorem 1.2.
Assume that f satisfies ( f ) − ( f ) . If a ∈ (0 , , then there exists µ ∗ = µ ∗ ( a ) > such that the problem (1.1) admits a couple ( u a , λ a ) ∈ H ( R ) × R ofweak solutions with R R | u | dx = a and λ a < for all µ ≥ µ ∗ . In the proof of Theorem 1.1 and Theorem 1.2 we borrow the ideas developed in Jeanjean[26]. The main difficulty in the proof of these theorems is associated with the fact thatwe are working with critical nonlinearities in whole R N . As above mentioned, in the proofof Theorem 1.1, the concentration-compactness principle due to Lions [30] is crucial in ourarguments, while in the proof of Theorem 1.2, the Trundiger-Moser inequality developedby Cao [14] plays an important role in a lot of estimates. Moreover, in the proofs of thesetheorems we will work on the space H rad ( R N ), because it has very nice compact embeedings.Moreover, by Palais’ principle of symmetric criticality, see [35], it is well known that thesolutions in H rad ( R N ) are in fact solutions in whole H ( R N ). Notation:
From now on in this paper, otherwise mentioned, we use the following notations: • B r ( u ) is an open ball centered at u with radius r > B r = B r (0). • C, C , C , ... denote any positive constant, whose value is not relevant. • | | p denotes the usual norm of the Lebesgue space L p ( R N ), for p ∈ [1 , + ∞ ], k k denotesthe usual norm of the Sobolev space H ( R N ). • o n (1) denotes a real sequence with o n (1) → n → + ∞ .5 Normalized solutions: The Sobolev critical case for N ≥ In order to follow the same strategy of [26], we need the following definitions to introduceour variational procedure.(1) S ( a ) = { u ∈ H ( R N ) : | u | = a } is the sphere of radius a > | | .(2) J : H ( R N ) → R with J ( u ) = 12 Z R N |∇ u | dx − Z R N F ( u ) dx, where F ( t ) = µq | t | q + 12 ∗ | t | ∗ , t ∈ R . Hereafter, H = H ( R N ) × R is equipped with the scalar product h· , ·i H = h· , ·i H ( R N ) + h· , ·i R and corresponding norm k · k H = ( k · k H + | · | R ) / . In this section, f denotes the function f ( t ) = µ | t | q − t + | t | ∗ − t with t ∈ R , and so, F ( t ) = R t f ( s ) ds .(3) H : H ( R N ) × R → H ( R N ) with H ( u, s )( x ) = e Ns u ( e s x ) . (4) ˜ J : H ( R N ) × R → R with˜ J ( u, s ) = e s Z R N |∇ u | dx − e Ns Z R N F ( e Ns u ( x )) dx or ˜ J ( u, s ) = 12 Z R N |∇ v | dx − Z R N F ( v ( x )) dx = J ( v ) for v = H ( u, s )( x ) . Throughout this section, S denotes the following constant S = inf u ∈ D , ( R N ) u = 0 R R N |∇ u | dx (cid:0)R R N | u | ∗ dx (cid:1) ∗ , (2.1)where 2 ∗ = NN − for N ≥
3, and D , ( R N ) is the Banach space given by D ,p ( R N ) = (cid:8) u ∈ L ∗ ( R N ) : |∇ u | ∈ L ( R N ) (cid:9) endowed with the norm k u k D , ( R N ) = (cid:18)Z R N |∇ u | dx (cid:19) . It is well known that the embedding D , ( R N ) ֒ → L ∗ ( R N ) is continuous.6 .1 The minimax approach We will prove that ˜ J on S ( a ) × R possesses a kind of mountain-pass geometrical structure. Lemma 2.1.
Let u ∈ S ( a ) be arbitrary but fixed. Then we have:(i) |∇H ( u, s ) | → and J ( H ( u, s )) → as s → −∞ ;(ii) |∇H ( u, s ) | → + ∞ and J ( H ( u, s )) → −∞ as s → + ∞ .Proof. By a straightforward calculation, it follows that Z R N |H ( u, s )( x ) | dx = a , Z R N |H ( u, s )( x ) | ξ dx = e ( ξ − Ns Z R N | u ( x ) | ξ dx, ∀ ξ ≥ , (2.2)and Z R N |∇H ( u, s )( x ) | dx = e s Z R N |∇ u | dx. (2.3)From the above equalities, fixing ξ >
2, we have |∇H ( u, s ) | → |H ( u, s ) | ξ → s → −∞ . (2.4)Hence, Z R N | F ( H ( u, s )) | dx ≤ Z R N C |H ( u, s ) | q dx + C Z R N |H ( u, s ) | ∗ dx → s → −∞ , from where it follows that J ( H ( u, s )) → s → −∞ , showing ( i ).In order to show ( ii ), note that by (2.3), |∇H ( u, s ) | → + ∞ as s → + ∞ . On the other hand, J ( H ( u, s )) ≤ |∇H ( u, s ) | − µp Z R N |H ( u, s ) | p dx = e s Z R N |∇ u | dx − µe ( q − Ns q Z R N | u ( x ) | q dx. Since q > N , the last inequality yields J ( H ( u, s )) → −∞ as s → + ∞ . Lemma 2.2.
There exists K ( a ) > small enough such that < sup u ∈ A J ( u ) < inf u ∈ B J ( u ) with A = (cid:26) u ∈ S ( a ) , Z R N |∇ u | dx ≤ K ( a ) (cid:27) and B = (cid:26) u ∈ S ( a ) , Z R N |∇ u | dx = 2 K ( a ) (cid:27) . roof. We will need the following Gagliardo-Sobolev inequality: for any ξ ≥ | u | ξ ≤ C ( ξ, |∇ u | γ | u | − γ , where γ = N ( − ξ ). If we fix |∇ u | ≤ K ( a ) and |∇ v | = 2 K ( a ), we derive that Z R N F ( u ) dx ≤ C | u | qq + C | u | ∗ ∗ . Then, by the Gagliardo-Sobolev inequality, Z R N F ( v ) dx ≤ C ( |∇ v | ) N ( q − ) + C ( |∇ v | ) N ( ∗− ) . Since F ( u ) ≥ u ∈ H ( R N ), we have J ( v ) − J ( u ) = 12 Z R N |∇ v | dx − Z R N |∇ u | dx − Z R N F ( v ) dx + Z R N F ( u ) dx ≥ Z R N |∇ v | dx − Z R N |∇ u | dx − Z R N F ( v ) dx, and so, J ( v ) − J ( u ) ≥ K ( a ) − C K ( a ) N ( q − ) − C K ( a ) N ( ∗− ) . Thereby, fixing K ( a ) small enough of such way that,12 K ( a ) − C K ( a ) N ( q − ) − C K ( a ) N ( ∗− ) > , we get the desired result.As a byproduct of the last lemma is the following corollary. Corollary 2.1.
There exists K ( a ) > such that if u ∈ S ( a ) and |∇ u | ≤ K ( a ) , then J ( u ) > .Proof. Arguing as in the last lemma, J ( u ) ≥ |∇ u | − C |∇ u | N ( q − )2 − C |∇ u | N ( ∗− )2 > , for K ( a ) small enough.In what follows, we fix u ∈ S ( a ) and apply Lemma 2.1 to get two numbers s < s >
0, of such way that the functions u = H ( s , u ) and u = H ( s , u ) satisfy |∇ u | < K ( a )2 , |∇ u | > K ( a ) , J ( u ) > J ( u ) < . Now, following the ideas from Jeanjean [26], we fix the following mountain pass levelgiven by γ µ ( a ) = inf h ∈ Γ max t ∈ [0 , J ( h ( t ))8here Γ = { h ∈ C ([0 , , S ( a )) : h (0) = u and h (1) = u } . From Lemma 2.2, max t ∈ [0 , J ( h ( t )) > max { J ( u ) , J ( u ) } . Lemma 2.3.
There holds lim µ → + ∞ γ µ ( a ) = 0 .Proof. In what follows we set the path h ( t ) = H ((1 − t ) s + ts , u ) ∈ Γ. Then, γ λ ( a ) ≤ max t ∈ [0 , J ( h ( t )) ≤ max r ≥ (cid:26) r |∇ u | − µq r N ( q − | u | qq (cid:27) , and so, for some positive constant C , γ λ ( a ) ≤ C (cid:18) µ (cid:19) N ( q − − → µ → + ∞ . Here, we have used the fact that q > N .In what follows ( u n ) denotes the ( P S ) sequence associated with the level γ µ ( a ), which isobtained by making u n = H ( v n , s n ), where ( v n , s n ) is the ( P S ) sequence for ˜ J obtained by[26, Proposition 2.2 ], associated with the level γ µ ( a ) . More precisely, we have J ( u n ) → γ µ ( a ) as n → + ∞ , (2.5)and k J | ′ S ( a ) ( u n ) k → n → + ∞ . Setting the functional Ψ : H ( R N ) → R given byΨ( u ) = 12 Z R N | u | dx, it follows that S ( a ) = Ψ − ( { a / } ). Then, by Willem [42, Proposition 5.12], there exists( λ n ) ⊂ R such that || J ′ ( u n ) − λ n Ψ ′ ( u n ) || H − → n → + ∞ . Hence, − ∆ u n − f ( u n ) = λ n u n + o n (1) in ( H ( R N )) ∗ . (2.6)Moreover, another important limit involving the sequence ( u n ) is Q ( u n ) = Z R N |∇ u n | dx + N Z R N F ( u n ) dx − N Z R N f ( u n ) u n dx → n → + ∞ , (2.7)which is obtained using the limit below ∂ s ˜ J ( v n , s n ) → n → + ∞ , u n ) is a bounded sequence, andso, the number λ n must satisfy the equality below λ n = 1 | u n | (cid:26) |∇ u n | − Z R N f ( u n ) u n dx (cid:27) + o n (1) , or equivalently, λ n = 1 a (cid:26) |∇ u n | − Z R N f ( u n ) u n dx (cid:27) + o n (1) . (2.8) Lemma 2.4.
There exists
C > such that lim sup n → + ∞ Z R N F ( u n ) dx ≤ Cγ µ ( a ) and lim sup n → + ∞ Z R N f ( u n ) u n dx ≤ Cγ µ ( a ) . Proof.
From (2.5) and (2.7)
N J ( u n ) + Q ( u n ) = N γ µ ( a ) + o n (1) , then N + 22 Z R N |∇ u n | dx − N Z R N f ( u n ) u n dx = N γ µ ( a ) + o n (1) . Using again (2.5), we get N + 22 (cid:18) Z R N F ( u n ) dx + 2 γ µ ( a ) + o n (1) (cid:19) − N Z R N f ( u n ) u n dx = N γ µ ( a ) + o n (1) , that is, − ( N + 2) Z R N F ( u n ) dx + N Z R N f ( u n ) u n dx = 2 γ µ ( a ) + o n (1) . (2.9)Since q > N +2) N and F ( t ) = µq | t | q + ∗ | t | ∗ , ∀ t ∈ R N , we obtain qF ( t ) ≤ f ( t ) t, t ∈ R . (2.10)This together with (2.9) yields (cid:18) qN − ( N + 2) (cid:19) Z R N F ( u n ) dx ≤ γ µ ( a ) + o n (1) , and so, lim sup n → + ∞ Z R N F ( u n ) dx ≤ Cγ µ ( a ) . This inequality combined again with (2.9) ensures thatlim sup n → + ∞ Z R N f ( u n ) u n dx ≤ Cγ µ ( a ) . emma 2.5. lim sup n → + ∞ |∇ u n | ≤ Cγ µ ( a ) .Proof. First of all, let us recall that Z R N |∇ u n | dx = 2 γ µ ( a ) + Z R N F ( v n ) dx + o n (1) . Then, from Lemma 2.4, lim sup n → + ∞ |∇ u n | ≤ (2 + C ) γ µ ( a ) . Now, from (2.9), the sequence ( R R N F ( u n ) dx ) is bounded away from zero, otherwise wewould have Z R N F ( u n ) dx → n → + ∞ , which leads to Z R N f ( u n ) u n dx → n → + ∞ . These limits combined with (2.9) imply that γ µ ( a ) = 0, which is absurd. From this, in whatfollows we can assume that Z R N F ( u n ) dx → C > , as n → ∞ . (2.11) Lemma 2.6.
The sequence ( λ n ) is bounded with lim sup n → + ∞ λ n ≤ − ( N + 2)2 a lim inf n → + ∞ Z R N F ( u n ) dx and lim sup n → + ∞ | λ n | ≤ Ca γ µ ( a ) , for some C > .Proof. The boundedness of ( u n ) yields that ( λ n ) is bounded, because λ n a = λ n | u n | = |∇ u n | − Z R N f ( u n ) u n dx + o n (1) , (2.12)and so, | λ n | ≤ a (cid:18) |∇ u n | + Z R N f ( u n ) u n dx (cid:19) + o n (1) ≤ Ca γ µ ( a ) + o n (1) . This guarantees the boundedness of ( λ n ) and the second inequality is proved.11n order to prove the first inequality, we know by (2.7) that |∇ u n | = N Z R N f ( u n ) u n dx − N Z R N F ( u n ) dx + o n (1) . Inserting this equality in (2.12), we obtain λ n a = ( N − Z R N f ( u n ) u n dx − N Z R N F ( u n ) dx + o n (1) , showing the first inequality.In the sequel, we restrict our study to the space H rad ( R N ). Then, it is well known thatlim n → + ∞ Z R N | u n | q dx = Z R N | u | q dx, (2.13)where u n ⇀ u in H ( R N ), because q ∈ (2 + N , ∗ ). Lemma 2.7.
There exists µ ∗ > such that u = 0 for all µ ≥ µ ∗ > .Proof. Seeking for a contradiction, let us assume that u = 0. Then,lim n → + ∞ Z R N | u n | q dx = 0 , (2.14)and by Lemma 2.6, lim sup n → + ∞ λ n ≤ . The equality a λ n = |∇ u n | − Z R N f ( u n ) u n dx + o n (1)together with (2.14) leads to a λ n = |∇ u n | − | u n | ∗ ∗ + o n (1) . (2.15)In what follows, going to a subsequence, we assume that |∇ u n | − a λ n = L + o n (1) and | u n | ∗ ∗ = L + o n (1) . We claim that
L >
0, otherwise if L = 0, we must have a λ n = |∇ u n | + o n (1) . (2.16)From this, 0 ≥ lim sup n → + ∞ λ n = lim sup n → + ∞ |∇ u n | ≥ lim inf n → + ∞ |∇ u n | ≥ , then, |∇ u n | → , which is absurd, because γ µ ( a ) >
0. 12ince
L >
0, by definition of S in (2.1), S ≤ R R N |∇ u n | dx (cid:0)R R N | u n | ∗ dx (cid:1) ∗ = R R N |∇ u n | dx − a λ n (cid:0)R R N | u | ∗ dx (cid:1) ∗ + a λ n (cid:0)R R N | u n | ∗ dx (cid:1) ∗ . Taking the limsup as n → + ∞ , we obtain S ≤ LL ∗ , that is, L ≥ S N . On the other hand o n (1) + γ µ ( a ) − a λ n |∇ u n | − a λ n ) − µq | u n | qq − ∗ | u n | ∗ ∗ = 1 N L + o n (1) . Recalling that lim sup n → + ∞ | λ n | ≤ Ca γ µ ( a ), it follows that1 N S N ≤ Cγ µ ( a ) . Now, fixing µ ∗ large enough of a such way that Cγ µ ( a ) < N S N , ∀ µ ≥ µ ∗ , we get a new contradiction. This proves that u = 0 for µ > Lemma 2.8.
Increasing if necessary µ ∗ , we have u n → u in L ∗ ( R N ) for all µ ≥ µ ∗ .Proof. Using the concentration-compactness principle due to Lions [30], it follows that( i ) |∇ u n | → κ weakly- ∗ in the sense of measureand( ii ) | u n | ∗ → ν weakly- ∗ in the sense of measure,and for a most countable index set J , we have ( a ) ν = | u | ∗ + P j ∈ J ν j δ x j , ν j ≥ , ( b ) κ ≥ |∇ u | + P j ∈ J µ j δ x j , µ j ≥ , ( c ) Sν ∗ j ≤ κ j , ∀ j ∈ J. Since − ∆ u n − f ( u n ) = λ n u n + o n (1) in ( H ( R N )) ∗ , we derive that Z R N ∇ u n ∇ φ dx − λ n Z R N u n φ dx = µ Z R N | u n | q − v n φ dx + Z R N | u n | ∗ − v n φ dx, ∀ φ ∈ H ( R N ) . J is empty or otherwise J is nonempty but finite. Inthe case that J is nonempty but finite, we must have ν j ≥ S N , ∀ j ∈ J. However, by Lemma 2.5, lim sup n → + ∞ |∇ u n | ≤ Cγ µ ( a ) . Then, if µ ∗ > Cγ µ ( a ) < S N , we get a contradiction, and so, J = ∅ . From this, u n → u in L ∗ loc ( R N ) . (2.17) Claim 2.1.
For each
R > , we have u n → u in L ∗ loc ( R N \ B R (0)) . Indeed, as u n ∈ H rad ( R N ), we know that | u n ( x ) | ≤ k u n k| x | N − , a.e. in R N . Since ( u n ) is a bounded sequence in H ( R N ), we obtain | u n ( x ) | ≤ C | x | N − , a.e. in R N , and so, | u n ( x ) | ∗ ≤ C | x | N ( N − N − , a.e. in R N . Recalling that C | · | N ( N − N − ∈ L ( R N \ B R (0)) and u n ( x ) → u ( x ) a.e. in R N \ B R (0), theLebesgue’s Theorem gives u n → u in L ∗ ( R N \ B R (0)) , showing the Claim 2.1. Now, the Claim 2.1 combined with (2.17) ensures that u n → u in L ∗ ( R N ) . u = 0. Therefore, the inequalitylim sup n → + ∞ λ n ≤ − C lim inf n → + ∞ Z R N F ( u n ) dx together with (2.13) and Lemma 2.8 ensures thatlim sup n → + ∞ λ n ≤ − C lim inf n → + ∞ Z R N F ( u ) dx < . So, according to the Lemma 2.6, we can assume without loss of generality that λ n → λ a < n → + ∞ . Now, the equality (2.6) implies that − ∆ u − f ( u ) = λ a u, in R N . (2.18)Thus, |∇ u | − λ a | u | = Z R N f ( u ) u dx. On the other hand, |∇ u n | − λ n | u n | = Z R N f ( u n ) u n dx + o n (1) , or yet, |∇ u n | − λ a | u n | = Z R N f ( u n ) u n dx + o n (1) . Recalling that u n → u in L ∗ ( R N )and u n → u in L q ( R N ) , we obtain lim n → + ∞ Z R N f ( u n ) u n dx = Z R N f ( u ) u dx, from where it follows thatlim n → + ∞ ( |∇ u n | − λ a | u n | ) = |∇ u | − λ a | u | . Since λ a <
0, the last equality implies that u n → u in H ( R N ) , implying that | u | = a . This establishes the desired result.15 Normalized solutions: The exponential criticalgrowth case for N = 2 In this section we will deal with the case N = 2, where f has an exponential critical growthand a ∈ (0 , f ) and ( f ), we know that fixed q ≥ ζ > α > π , there exists a constant C >
0, which depends on q , α , ζ , suchthat | f ( t ) | ≤ ζ | t | τ + C | t | q − ( e αt −
1) for all t ∈ R (3.1)and, using ( f ), we have | F ( t ) | ≤ ζ | t | τ +1 + C | t | q ( e αt −
1) for all t ∈ R . (3.2)Moreover, it is easy to see that, by (3.1), | f ( t ) t | ≤ ζ | t | τ +1 + C | t | q ( e αt −
1) for all t ∈ R . (3.3)Finally, let us recall the following version of Trudinger-Moser inequality as stated e.g. in[14]. Lemma 3.1. If α > and u ∈ H ( R ) , then Z R ( e αu − dx < + ∞ . Moreover, if |∇ u | ≤ , | u | ≤ M < + ∞ , and < α < π , then there exists a positiveconstant C ( M, α ) , which depends only on M and α , such that Z R ( e αu − dx ≤ C ( M, α ) . Lemma 3.2.
Let ( u n ) be a sequence in H ( R ) with u n ∈ S ( a ) and lim sup n → + ∞ |∇ u n | < − a . Then, there exist t > , t close to 1, and C > satisfying Z R (cid:16) e π | u n | − (cid:17) t dx ≤ C, ∀ n ∈ N . Proof.
As lim sup n →∞ |∇ u n | < | u n | = a < , there exist m > n ∈ N verifying k u n k < m < , for any n ≥ n . Fix t >
1, with t close to 1, and β > t satisfying βm < . Then, there exists C = C ( β ) > Z R (cid:16) e π | u n | − (cid:17) t dx ≤ Z R (cid:16) e βmπ ( | un ||| un || ) − (cid:17) dx, for any n ≥ n , e t − s ≤ e ts − , for s > t ≥ . Hence, by Lemma 3.1, Z R (cid:16) e π | u n | − (cid:17) t dx ≤ C ∀ n ≥ n , for some positive constant C . Now, the lemma follows fixing C = max (cid:26) C , Z R (cid:16) e π | u | − (cid:17) t dx, ...., Z R (cid:16) e π | u n | − (cid:17) t dx (cid:27) . Corollary 3.1.
Let ( u n ) be a sequence in H ( R ) with u n ∈ S ( a ) and lim sup n → + ∞ |∇ u n | < − a . If u n ⇀ u in H ( R ) and u n ( x ) → u ( x ) a.e in R , then F ( u n ) → F ( u ) in L ( B R (0)) , for any R > . Proof.
By ( f ) − ( f ), for each β >
1, there is
C > | F ( t ) | ≤ C | t | τ +1 + C ( e βπ | t | − , ∀ t ∈ R , from where it follows that, | F ( u n ) | ≤ C | u n | τ +1 + C ( e βπ | u n | − , ∀ n ∈ N . (3.4)Setting h n ( x ) = C ( e βπ | u n ( x ) | − , we can argue as in the proof of Lemma 3.2, to find two numbers β, q >
1, with β and q closeto 1, such that h n ∈ L q ( R ) and sup n ∈ N | h n | q < + ∞ , which is an immediate consequence of Lemma 3.2. Therefore, for some subequence of ( u n ),still denoted by itself, we derive that h n ⇀ h = C ( e βπ | u | − , in L q ( R ) . (3.5) Claim 3.1. As h n , h ≥ , the last limit yields h n → h in L ( B R (0)) , ∀ R > . R >
0, we consider the characteristic function χ R associated with B R (0) ⊂ R , that is, χ R ( x ) = (cid:26) , x ∈ B R (0) , , x ∈ R N \ B R (0) , which belongs to L q ′ ( R ) with q + q ′ = 1. Thus, by the weak limit (3.5), Z R h n χ R dx → Z R hχ R dx, or equivalently, Z B R (0) h n dx → Z B R (0) h dx. Once h n , h ≥
0, we derive that | h n | L ( B R (0) → | h | L ( B R (0) . Moreover, we also have h n ( x ) → h ( x ) , a.e. in R . Now, the Lebesgue’s Theorem combined with the above limits gives h n → h in L ( B R (0)) . Using the last limit, there exists a subsequence of ( h n ), still denoted by itself, and anonnegative function g R ∈ L ( B R (0)), such that | h n ( x ) | ≤ g R ( x ) a.e. in B R (0) . Consequently, by (3.4), | F ( u n ) | ≤ C | u n | τ +1 + g R ( x ) a.e. in B R (0) . Since u n → u in L τ +1 ( B R ) , there exists a nonnegative function Q R ∈ L τ +1 ( B R ) such that | u n | τ +1 ≤ Q R . Observing that F ( u n ( x )) → F ( u ( x )) , a.e. in R , we can use again the Lebesgue’s Theorem to guarantee that F ( u n ) → F ( u ) in L ( B R (0)) . The next lemma is crucial in our argument18 emma 3.3.
Let ( u n ) ⊂ H rad ( R ) be a sequence with u n ∈ S ( a ) and lim sup n → + ∞ |∇ u n | < − a . Then, there are β, q close to 1, such that for all l > , | u n | l ( e βπ | u n ( x ) | − → | u | l ( e βπ | u ( x ) | − in L ( R N ) . Proof.
Arguing as in Corollary 3.1, there are β, q close to 1 such that the sequence h n ( x ) = C ( e βπ | u n ( x ) | − , is a bounded sequence in L q ( R N ). Therefore, for some subsequence of ( h n ), still denoted byitself, we derive that h n ⇀ h = C ( e βπ | u | −
1) in L q ( R ) . For q ′ = qq − , we know that the embedding H rad ( R N ) ֒ → L lq ′ ( R N ) is compact, then u n → u in L lq ′ ( R N ) , and so, | u n | l → | u | l in L q ′ ( R N ) . Thus, lim n → + ∞ Z R | u n | l h n ( x ) dx = Z R | u | l h ( x ) dx, that is, lim n → + ∞ Z R | u n | l ( e βπ | u n ( x ) | − dx = Z R | u | l ( e βπ | u ( x ) | − dx. Since | u n | l ( e βπ | u n ( x ) | − ≥ | u | l ( e βπ | u ( x ) | − ≥ , the last limit gives | u n | l ( e βπ | u n ( x ) | − → | u | l ( e βπ | u ( x ) | −
1) in L ( R ) . Corollary 3.2.
Let ( u n ) be a sequence in H rad ( R ) with u n ∈ S ( a ) and lim sup n → + ∞ |∇ u n | < − a . If u n ⇀ u in H ( R ) and u n ( x ) → u ( x ) a.e in R , then F ( u n ) → F ( u ) and f ( u n ) u n → f ( u ) u in L ( R ) . roof. By ( f ) − ( f ), | F ( t ) | ≤ C | t | τ +1 + C | u n | l ( e βπ | t | − ∀ t ∈ R , where β > l > | F ( u n ) | ≤ C | u n | τ +1 + C | u n | l ( e βπ | u n | − , ∀ n ∈ N . (3.6)By Lemma 3.3, | u n | l ( e βπ | u n ( x ) | − → | u | l ( e βπ | u ( x ) | −
1) in L ( R ) , and by the compact embedding H rad ( R N ) ֒ → L τ +1 ( R N ), u n → u in L τ +1 ( R ) . Now, we can use the Lebesgue’s Theorem to conclude that F ( u n ) → F ( u ) in L ( R ) . A similar argument works to show that f ( u n ) u n → f ( u ) u in L ( R ) . From now on, we will use the same notations introduced in Section 2 to apply ourvariational procedure, more precisely(1) S ( a ) = { u ∈ H ( R ) : | u | = a } is the sphere of radius a > | | .(2) J : H ( R ) → R with J ( u ) = 12 Z R |∇ u | dx − Z R F ( u ) dx. (3) H : H ( R ) × R → H ( R ) with H ( u, s )( x ) = e s u ( e s x ) . (4) ˜ J : H ( R ) × R → R with˜ J ( u, s ) = e s Z R |∇ u | dx − e s Z R F ( e s u ( x )) dx. .1 The minimax approach We will prove that ˜ J on S ( a ) × R possesses a kind of mountain-pass geometrical structure. Lemma 3.4.
Assume that ( f ) − ( f ) and ( f ) hold and let u ∈ S ( a ) be arbitrary but fixed.Then we have:(i) |∇H ( u, s ) | → and J ( H ( u, s )) → as s → −∞ ;(ii) |∇H ( u, s ) | → + ∞ and J ( H ( u, s )) → −∞ as s → + ∞ .Proof. By a straightforward calculation, it follows that Z R |H ( u, s )( x ) | dx = a , Z R |H ( u, s )( x ) | ξ dx = e ( ξ − s Z R | u ( x ) | ξ dx, ∀ ξ ≥ , (3.7)and Z R |∇H ( u, s )( x ) | dx = e s Z R |∇ u | dx. (3.8)From the above equalities, fixing ξ >
2, we have |∇H ( u, s ) | → |H ( u, s ) | ξ → s → −∞ . (3.9)Thus, there are s < m ∈ (0 ,
1) such that kH ( u, s ) k ≤ m, ∀ s ∈ ( −∞ , s ] . By ( f ) − ( f ), | F ( t ) | ≤ C | t | τ +1 + C | u n | l ( e βπ | t | − ∀ t ∈ R , where β is closed to 1 and l > | F ( H ( u, s )) | ≤ C |H ( u, s ) | τ +1 + C |H ( u, s ) | l ( e βπ |H ( u,s ) | − , ∀ s ∈ ( −∞ , s ] . Using the H¨older’s inequality together with Lemma 3.1, there exists C = C ( u, m ) > Z R | F ( H ( u, s )) | dx ≤ Z R C |H ( u, s ) | τ +1 dx + C Z R |H ( u, s ) | lq ′ dx, ∀ s ∈ ( −∞ , s ] , where q ′ = qq − , and q is close to 1 of way that lq ′ >
2. Now, by using (3.9), Z R | F ( H ( u, s )) | → s → −∞ , from where it follows that J ( H ( u, s )) → s → −∞ , showing ( i ).In order to show ( ii ), note that by (3.8), |∇H ( u, s ) | → + ∞ as s → + ∞ .
21n the other hand, by ( f ), J ( H ( u, s )) ≤ |∇H ( u, s ) | − µp Z R |H ( u, s ) | p dx = e s Z R |∇ u | dx − µe ( p − s p Z R | u ( x ) | p dx. Since p >
4, the last inequality yields J ( H ( u, s )) → −∞ as s → + ∞ . Lemma 3.5.
Assume that ( f ) − ( f ) hold. Then there exists K ( a ) > small enough suchthat < sup u ∈ A J ( u ) < inf u ∈ B J ( u ) with A = (cid:26) u ∈ S ( a ) , Z R |∇ u | dx ≤ K ( a ) (cid:27) and B = (cid:26) u ∈ S ( a ) , Z R |∇ u | dx = 2 K ( a ) (cid:27) . Proof.
We will need the following Gagliardo-Sobolev inequality: for any ξ ≥ | u | ξ ≤ C ( ξ, |∇ u | γ | u | − γ , where γ = 2( − ξ ). If we fix K ( a ) < − a , |∇ u | ≤ K ( a ) and |∇ v | = 2 K ( a ), the conditions( f ) − ( f ) combined with Lemma 3.1 ensure that Z R F ( u ) dx ≤ C | u | τ +1 τ +1 + C | u | lq ′ lq ′ where l > q ′ = qq − and q is close to 1. Then, by the Gagliardo-Sobolev inequality, Z R F ( v ) dx ≤ C |∇ v | τ − + C |∇ v | lq ′ − . From ( f ), F ( u ) ≥ u ∈ H ( R ), then J ( v ) − J ( u ) = 12 Z R |∇ v | dx − Z R |∇ u | dx − Z R F ( v ) dx + Z R F ( u ) dx ≥ Z R |∇ v | dx − Z R |∇ u | dx − Z R F ( v ) dx, and so, J ( v ) − J ( u ) ≥ K ( a ) − C K ( a ) τ − − C K ( a ) lq ′− . Since τ > q close to 1 of way that lq ′ >
4, decreasing K ( a ) if necessary,it follows that 12 K ( a ) − C K ( a ) τ − − C K ( a ) lq ′− > , showing the desired result. 22s a byproduct of the last lemma is the following corollary. Corollary 3.3.
There exists K ( a ) > small enough such that if u ∈ S ( a ) and |∇ u | ≤ K ( a ) ,then J ( u ) > .Proof. Arguing as in the last lemma, J ( u ) ≥ |∇ u | − C |∇ u | τ − − C |∇ u | lq ′ − > , for K ( a ) > u ∈ S ( a ) and apply Lemma 3.4 and Corollary 3.3 to get twonumbers s < s >
0, of such way that the functions u = H ( u , s ) and u = H ( u , s )satisfy |∇ u | < K ( a )2 , |∇ u | > K ( a ) , J ( u ) > J ( u ) < . Now, following the ideas from Jeanjean [26], we fix the following mountain pass levelgiven by γ µ ( a ) = inf h ∈ Γ max t ∈ [0 , J ( h ( t ))where Γ = { h ∈ C ([0 , , S ( a )) : h (0) = u and h (1) = u } . From Lemma 3.5, max t ∈ [0 , J ( h ( t )) > max { J ( u ) , J ( u ) } . Lemma 3.6.
There holds lim µ → + ∞ γ µ ( a ) = 0 .Proof. In what follow we set the path h ( t ) = H (cid:0) u , (1 − t ) s + ts (cid:1) ∈ Γ. Then, γ µ ( a ) ≤ max t ∈ [0 , J ( h ( t )) ≤ max r ≥ (cid:26) r |∇ u | − µp r p − | u | pp (cid:27) and so, γ µ ( a ) ≤ C (cid:18) µ (cid:19) p − → µ → + ∞ , for some C >
0. Here, we have used the fact that p > u n ) denotes the ( P S ) sequence associated withthe level γ µ ( a ), which satisfies: J ( u n ) → γ µ ( a ) , as n → + ∞ , (3.10) − ∆ u n − f ( u n ) = λ n u n + o n (1) , in ( H ( R )) ∗ , (3.11)23or some sequence ( λ n ) ⊂ R , and Q ( u n ) = Z R |∇ u n | dx + 2 Z R F ( u n ) dx − Z R f ( u n ) u n dx → , as n → + ∞ . (3.12)Moreover, ( u n ) is a bounded sequence, and so, the number λ n must satisfy the equality below λ n = 1 a (cid:26) |∇ u n | − Z R f ( u n ) u n dx (cid:27) + o n (1) . (3.13) Lemma 3.7.
There holds lim sup n → + ∞ Z R F ( u n ) dx ≤ θ − γ µ ( a ) . Proof.
Using the fact that J ( u n ) = γ µ ( a ) + o n (1) and Q ( u n ) = o n (1), it follows that2 J ( u n ) + Q ( u n ) = 2 γ µ ( a ) + o n (1) , and so, 2 |∇ u n | − Z R f ( u n ) u n dx = 2 γ µ ( a ) + o n (1) . (3.14)Using that J ( u n ) = γ µ ( a ) + o n (1), we get4 Z R F ( u n ) dx + 4 γ µ ( a ) + o n (1) − Z R f ( u n ) u n dx = 2 γ µ ( a ) + o n (1) . Hence, 2 γ µ ( a ) + o n (1) = Z R f ( u n ) u n dx − Z R F ( u n ) dx ≥ ( θ − Z R F ( u n ) dx. By ( f ), we know that θ >
4, consequentlylim sup n → + ∞ Z R F ( u n ) u n dx ≤ θ − γ µ ( a ) . Lemma 3.8.
The sequence ( u n ) satisfies lim sup n → + ∞ |∇ u n | ≤ θ − θ − γ µ ( a ) . Hence, there exists µ ∗ > such that lim sup n → + ∞ |∇ u n | < − a , for any µ ≥ µ ∗ . Proof.
Since J ( u n ) = γ µ ( a ) + o n (1), we have Z R |∇ u n | dx = 2 γ µ ( a ) + Z R F ( u n ) dx + o n (1) . Thereby, by Lemma 3.7, lim sup n → + ∞ |∇ u n | ≤ θ − θ − γ µ ( a ) . emma 3.9. Fix µ ≥ µ ∗ , where µ ∗ is given in Lemma 3.8. Then, ( λ n ) is a bounded sequencewith lim sup n → + ∞ | λ n | ≤ θ − a ( θ − γ µ ( a ) and lim sup n → + ∞ λ n = − a lim inf n → + ∞ Z R F ( u n ) dx. Proof.
The boundedness of ( u n ) yields that ( λ n ) is bounded, because λ n | u n | = |∇ u n | − Z R f ( u n ) u n dx + o n (1) , and as | u n | = a , we have λ n a = |∇ u n | − Z R f ( u n ) u n dx + o n (1) . Hence, | λ n | a ≤ |∇ u n | + Z R f ( u n ) u n dx + o n (1) . The limit (3.12) together with Lemmas 3.7 and 3.8 ensures that ( R R f ( u n ) u n dx ) isbounded with lim sup n → + ∞ Z R f ( u n ) u n dx ≤ θ − θ − γ µ ( a ) . This is enough to conclude that ( λ n ) is a bounded sequence withlim sup n → + ∞ | λ n | ≤ θ − a ( θ − γ µ ( a ) . In order to prove the second inequality, the equality λ n a = |∇ u n | − Z R f ( u n ) u n dx + o n (1)together with the limit (3.12) leads to λ n a = − Z R F ( u n ) dx + o n (1) , showing the desired result.Now, we restrict our study to the space H rad ( R ). For any µ ≥ µ ∗ , using Lemmas 3.3and 3.8, it follows that lim n → + ∞ Z R f ( u n ) u n dx = Z R f ( u ) u dx, and lim n → + ∞ Z R F ( u n ) dx = Z R F ( u ) dx, u n ⇀ u in H ( R ). The last limit implies that u = 0, because otherwise, Corollary3.2 gives lim n → + ∞ Z R F ( u n ) dx = lim n → + ∞ Z R f ( u n ) u n dx = 0 , and by Lemma 3.9, lim sup n → + ∞ λ n ≤ . Since ( u n ) is bounded in H ( R ) and lim sup n → + ∞ |∇ u n | < − a if µ ≥ µ ∗ , Corollary 3.2together with ( f ) − ( f ) and the equality below λ n | u n | = |∇ u n | − Z R f ( u n ) u n dx + o n (1) , lead to λ n a = |∇ u n | + o n (1) . (3.15)From this, 0 ≥ lim sup n → + ∞ λ n a = lim sup n → + ∞ |∇ u n | ≥ lim inf n → + ∞ |∇ u n | ≥ , then |∇ u n | → , which is absurd, because γ µ ( a ) > u = 0. Moreover, the equalitylim sup n → + ∞ λ n = − a lim inf n → + ∞ Z R F ( u n ) dx ensures that lim sup n → + ∞ λ n = − a lim inf n → + ∞ Z R F ( u ) dx < . From this, for some subsequence, still denoted by ( λ n ), we can assume that λ n → λ a < n → + ∞ . Now, the equality (3.11) implies that − ∆ u − f ( u ) = λ a u in ( H ( R )) ∗ . (3.16)Thus, |∇ u | − λ a | u | = Z R f ( u ) u dx. On the other hand, |∇ u n | − λ n | u n | = Z R f ( u n ) u n dx + o n (1) , or yet, |∇ u n | − λ a | u n | = Z R f ( u n ) u n dx + o n (1) . n → + ∞ Z R f ( u n ) u n dx = Z R f ( u ) u dx, we derive that lim n → + ∞ ( |∇ u n | − λ a | u n | ) = |∇ u | − λ a | u | . Since λ a <
0, the last limit implies that u n → u in H ( R ) , implying that | u | = a . This establishes the desired result. References [1] A. Adimurthi.
Existence of Positive solutions of the semilinear Dirichlet problem withcritical growth for the N -Laplacian . Ann. Sc. Norm. Super. Pisa 17 (1990), 393-413.[2] T. Akahori, S. Ibrahim, H. Kikuchi, and H. Nawa. Existence of a ground state andscattering for a nonlinear Schr¨odinger equation with critical growth.
Selecta Math. (N.S.)19(2)(2013), 545-609.[3] T. Akahori, S. Ibrahim, H. Kikuchi, and H. Nawa.
Global dynamics above the groundstate energy for the combined power type nonlinear Schr¨odinger equations with energycritical growth at low frequencies . arXiv.1510.08034, 2019.[4] C. O. Alves.
Multiplicity of solutions for a class of elliptic problem in R with Neumannconditions . J. Differential Equations 219 (2005), 20–39.[5] C. O. Alves, J. M. B. do ´O and O. H. Miyagaki. On nonlinear perturbations of a periodicelliptic problem in R involving critical growth , Nonlinear Anal. 56 (2004), 781–791.[6] C.O. Alves and G.M. Figueiredo. On multiplicity and concentration of positive solutionsfor a class of quasilinear problems with critical exponential growth in R N . J. DifferentialEquations 219 (2009), 1288-1311.[7] C.O. Alves and S.H.M. Soares.
Nodal solutions for singularly perturbed equations withcritical exponential growth.
J. Differential Equations 234 (2007), 464-484.[8] C. O. Alves, M. A. S. Souto and M. Montenegro.
Existence of a ground state solutionfor a nonlinear scalar field equation with critical growth.
Calc. Var. Partial DifferentialEquations 43(3-4)(2012), 537-554.[9] T. Bartsch, L. Jeanjean and N. Soave.
Normalized solutions for a system of coupledcubic Schr¨odinger equations on R . J. Math. Pures Appl. (9) 106(4)(2016), 583-614.[10] T. Bartsch, R. Molle, M. Rizzi and G.Verzini. Normalized solutions of mass supercriticalSchr¨odinger equations with potential. arXiv:2008.07431V1, 2020.2711] T. Bartsch and N. Soave.
Multiple normalized solutions for a competing system ofSchr¨odinger equations . Calc. Var. Partial Differential Equations 58(1) (2019), art 22,pp.24.[12] J. Bellazzini, L. Jeanjean and T. Luo.
Existence and instability of standing waves withprescribed norm for a class of Schr¨odinger-Poisson equations . Proc. Lond. Math. Soc.(3), 107(2)(2013), 303-339.[13] H. Brezis and L. Nirenberg.
Positive solutions of nonlinear elliptic equations involvingcritical Sobolev exponents,
Comm. Pure Appl. Math. 36 (1983), 437–477.[14] D. M. Cao.
Nontrivial solution of semilinear elliptic equation with critical exponent in R . Comm. Partial Differential Equation 17 (1992),407–435.[15] T. Cazenave, Semilinear Schr¨odinger equations,
Courant Lecture Notes in Mathematics (New York University, Courant Institute of Mathematical Sciences, New York;American Mathematical Society, Providence, RI, 2003, 323 pp. ISBN: 0-8218-3399-5.[16] X. Cheng, C.X. Miao and L.F. Zhao. Global well-posedness and scattering for nonlinearSchrodinger equations with combined nonlinearities in the radial case.
J. DifferentialEquations 261(6)(2016), 2881-2934.[17] S. Cingolani and L. Jeanjean.
Stationary waves with prescribed L -norm for the planarSchr¨odinger-Poisson system . SIAM J. Math. Anal. 51(4)(2019), 3533-3568.[18] D.G. de Figueiredo, O.H. Miyagaki and B. Ruf. Elliptic equations in R withnonlinearities in the critical growth range . Calc. Var. Partial Differential Equations 3(1995), 139-153.[19] D.G. de Figueiredo, Jo˜ao Marcos do ´O and B. Ruf. On an inequality by N. Trudingerand J. Moser and related elliptic equations . Comm. Pure Appl. Math. 55 (2002), 1-18.[20] J. M. B. do ´O and M.A.S. Souto.
On a class of nonlinear Schr¨odinger equations in R involving critical growth . J. Differential Equations 174 (2001), 289-311.[21] J. M. B. do ´O and B. Ruf. On a Schr¨odinger equation with periodic potential and criticalgrowth in R . Nonlinear Differential Equations Appl. 13 (2006), 167-192.[22] J. M. B. do ´O. Quasilinear elliptic equations with exponential nonlinearities . Comm.Appl. Nonlin. Anal. 2 (1995), 63-72.[23] J. M. B. do ´O, M. de Souza, E. de Medeiros and U. Severo.
An improvement for theTrudinger-Moser inequality and applications . J. Differential Equations 256 (2014), 1317-1349.[24] J. Garcia Azorero and I. Peral Alonso.
Multiplicity of solutions for elliptic problemswith critical exponent or with a nonsymmetric term.
Trans. Amer. Math. Soc. 2 (1991),877-895. 2825] T.X. Gou and L. Jeanjean.
Multiple positive normalized solutions for nonlinearSchr¨odinger systems.
Nonlinearity 31(5)(2018), 2319-2345.[26] L. Jeanjean.
Existence of solutions with prescribed norm for semilinear elliptic equations.
Nonlinear Anal. 28 (1997), 1633-1659.[27] L. Jeanjean, J. Jendrej, T. T. Le and N. Visciglia.
Orbital stability of ground states fora Sobolev critical Schr¨odinger equation . arXiv.2008.12084, 2020.[28] L. Jeanjean and S.S. Lu.
Nonradial normalized solutions for nonlinear scalar fieldequations . Nonlinearity 32(12)(2019), 4942- 4966.[29] L. Jeanjean and T. T. Le.
Multiple normalized solutions for a Sobolev criticalSchr¨odinger equations . arXiv:2011.02945v1, 2020.[30] P.L. Lions.
The concentration-compactness principle in the calculus of variations. Thelocally compact case, part 2 . Anal. Inst. H. Poincar´e, Sect. C 1 (1984), 223-283.[31] J. Mederski and J. Schino.
Least energy solutions to a cooperative system of Schr¨odingerequations with prescribed L -bounds: At least L -crtical growth. arXiv:2101.02611v1,2021[32] C. X. Miao, G. X. Xu and L. F. Zhao. The dynamics of the 3D radial NLS with thecombined terms.
Comm. Math. Phys. 318(3)(2013), 767-808.[33] J. Moser.
A sharp form of an inequality by N. Trudinger . Ind. Univ. Math. J. (20)(1971), 1077–1092.[34] B. Noris, H. Tavares and G. Verzini.
Normalized solutions for nonlinear Schr¨odingersystems on bounded domains.
Nonlinearity 32(3)(2019), 10441072.[35] R.S. Palais, The principle of symmetric criticality,
Comm. Math. Phys. (1979), 19-30.[36] N. Soave. Normalized ground states for the NLS equation with combined nonlinearities.
J. Differential Equations 269(9)(2020), 6941- 6987.[37] N. Soave.
Normalized ground states for the NLS equation with combined nonlinearities:the Sobolev critical case.
J. Funct. Anal. 279(6)(2020), 108610, 43.[38] A. Stefanov.
On the normalized ground states of second order PDE’s with mixed powernon-linearities . Comm. Math. Phys. 369(3)(2019), 929-971.[39] T. Tao, M. Visan and X. Zhang.
The nonlinear Schr¨odinger equation with combinedpower-type nonlinearities . Comm. Partial Differential Equations 32(7-9)(2007), 1281-1343.[40] N. S. Trudinger.
On imbedding into Orlicz spaces and some application . J. Math Mech.17 (1967), 473–484. 2941] W. Wang, Q. Li, J. Zhou and Y. Li.
Normalized solutions for p-Laplacian equationswith a L -supercritical growth . Ann. Funct. Anal. 12(1) (2020), doi:10.1007/s43034-020-00101-w.[42] M. Willem. Minimax Theorems,
Birkhauser, 1996.
Claudianor O. Alves
Unidade Acadˆemica de Matem´aticaUniversidade Federal de Campina GrandeCampina Grande, PB, CEP:58429-900, Brazil [email protected] and
Chao Ji
Department of MathematicsEast China University of Science and TechnologyShanghai 200237, PR China [email protected] and
Ol´ımpio Hiroshi, Miyagaki
Departamento de Matem´aticaUniversidade Federal de S˜ao CarlosS˜ao Carlos, SP, CEP:13565-905, Brazil [email protected]@ufscar.br