Semirelativistic Choquard equations with singular potentials and general nonlinearities arising from Hartree-Fock theory
SSEMIRELATIVISTIC CHOQUARD EQUATIONS WITHSINGULAR POTENTIALS AND GENERAL NONLINEARITIESARISING FROM HARTREE-FOCK THEORY
FEDERICO BERNINI, BARTOSZ BIEGANOWSKI, AND SIMONE SECCHI
Abstract.
We are interested in the general Choquard equation q − ∆ + m u − mu + V ( x ) u − µ | x | u = (cid:18)Z R N F ( y, u ( y )) | x − y | N − α dy (cid:19) f ( x, u ) − K ( x ) | u | q − u under suitable assumptions on the bounded potential V and on the nonlinearity f . Our analysis extends recent results by the second and third author on theproblem with µ = 0 and pure-power nonlinearity f ( x, u ) = | u | p − u . We showthat, under appropriate assumptions on the potential, whether the groundstate does exist or not. Finally, we study the asymptotic behaviour of groundstates as µ → + . C o n t e n t s
1. Introduction 12. Variational setting 73. Existence and boundedness of a Cerami sequence 104. Decomposition of bounded minimizing sequences 145. Existence and nonexistence of ground states 236. Compactness of ground states 24Acknowledgements 27References 281.
I n t ro d u c t i o n
In a model for an atom with n electrons and nuclear charge Z , the kinetic energyof the electrons is described by the expression p ( | p | c ) + ( mc ) − mc . Date : February 4, 2021.2020
Mathematics Subject Classification. a r X i v : . [ m a t h . A P ] F e b FEDERICO BERNINI, BARTOSZ BIEGANOWSKI, AND SIMONE SECCHI
This model takes into account some relativistic effects, and gives rise to a Hamil-tonian of the form H = n X j =1 (cid:26)q − α − ∆ j + α − − α − − Z | x j | (cid:27) + X ≤ i
0. In particular, they appear in models inastrophysics describing the evolution of many-body quantum systems, like bosonstars. The external potential accounts for gravitational fields from other stars. Itis also possible to describe the evolution of other type stars, like white dwarfs orneutron stars, using the time-dependent equation of the form (1.2). In particular, in[17] the collapse of white dwarfs has been studied via the analysis of existence andblow-up of solutions to the Hartee and Hartree-Fock equations. See also [14, 18–20]for more details on these physical models.In [16] Fefferman and de la Llave showed how a system governed by operator H canimplode: in fact, this is happen for a single quantized electron attracted to a single EMIRELATIVISTIC CHOQUARD EQUATIONS 3 nucles of charge Z fixed at the origin. In [28] Lieb and Yau studied the quantummechanical many-body problem where they consider the problem where electronsand fixed nuclei interact via Coulomb forces with a relativistic kinect energy: theyproved that stability of relatve matters occurs for suitable values Z and α . In[27], the same authors consider operator H with Z = 0, that is electrically neutralgravitating particles (e.g. fermions or bosons) and they showed that the groundstate of stars can be obtained as the limit G (the gravitation constant) goes to zeroand n (the number of particles) goes to infinity. We refer to [24–26] for furtherresults.The stationary Schrödinger equation with singular potential of the form − ∆u + V ( x ) u − µ | x | u = f ( x, u ) , x ∈ R N (1.3)has been studied in [22]. In their paper, Guo and Mederski show that (1.3) admitsa ground state solution for sufficiently small µ > H ( R N ). They were able to study the strongly indefinite case, i.e. the situationwhere the infimum of the spectrum σ ( − ∆ + V ( x )) lies below 0, since the nonlinearpart I ( u ) = R R N F ( x, u ) dx of the variational functional J : H ( R N ) → R definedby J ( u ) = 12 Z R N |∇ u | + V ( x ) u dx − µ Z R N u | x | dx − Z R N F ( x, u ) dx is nonnegative. Later, this result has been extended in [2] in the fractional settingfor positive potentials V , but with sign-changing nonlinearities. The approach in[2] is based on the Nehari manifold technique, combined with recent results on thefractional Hardy inequality.Turning to our work, we study the Choquard equation (1.1) where the classicallaplacian − ∆ is replaced by the nonlocal operator √− ∆ + m , which is known inthe literature as the semirelativistic Schrödinger operator. The second difficultyis that the nonlinear part is nonlocal and has additional power-type term, whichmakes the right hand side to change sign. As a consequence we will study onlypositive potentials V .The problem (1.1) with µ = 0 and K ≡ f (see [32]) and also with general nonlinearity (see [1, 33]).The case µ = 0 with K f has been studied bythe second and third author in [5]. See also [8–11, 13, 23, 31, 35, 36] and referencestherein.We consider the following relation between numbers p , q , α and the dimension N . FEDERICO BERNINI, BARTOSZ BIEGANOWSKI, AND SIMONE SECCHI (N) N ≥
2, ( N − p − N < α < N , 2 < q < min { p, N/ ( N − } and p > p < NN − , so that the growth parameter p is smaller thanthe critical Sobolev exponent for the space H / ( R N ). In this sense we consider anonlinearity f with subcritical growth, see assumption (F1) below.The following are our assumptions on the potential function V :(V1) V = V p + V l , where V p ∈ L ∞ ( R N ) is Z N -periodic and V l ∈ L ∞ ( R N ) ∩ L N ( R N ) satisfies lim | x |→ + ∞ V l ( x ) = 0 . (V2) ess inf x ∈ R N V ( x ) > m .Conditions (V1) and (V2) ensure that the operator √− ∆ + m + V ( x ) − m ispositive definite. In particular, the quadratic form associated to this operatorgenerates a norm in H / ( R N ) which is equivalent to the standard one. Similarassumptions were considered in [5] in the case µ = 0 in the presence of the purepower nonlinearity f ( x, u ) = | u | p − u . However, for µ > √− ∆ + m + V ( x ) − m − µ | x | is positive definite, and we can show it onlyfor small values of µ . Moreover, the potential V is not necessarily Z N -periodic, andthe application of Lions’ concentration-compactness principle is not straightforward.With respect to the nonlinearity f we assume:(F1) f : R N × R → R is a Carathéodory function , Z N -periodic in x ∈ R N andthere is C > | f ( x, u ) | ≤ C (cid:0) | u | αN + | u | p − (cid:1) . (F2) f ( x, u ) = o ( u ) as u → x .(F3) F ( x, u ) / | u | q → + ∞ as | u | → + ∞ , uniformly with respect to x , where F ( x, u ) = R u f ( x, s ) ds and F ( x, u ) ≥ u ∈ R and a.e. x ∈ R N .(F4) The function u f ( x, u ) / | u | q − is non-decreasing on each half-line ( −∞ , , + ∞ ).Finally,(K) K ∈ L ∞ ( R N ) is Z N -periodic and non-negative. Remark . Assumptions (F3) and (F4) imply that0 ≤ q F ( x, u ) ≤ f ( x, u ) u (1.4)for almost every x ∈ R N and u ∈ R , which is a weaker variant of the well-knownAmbrosetti-Rabinowitz condition. It is also classical to check that conditions (F1),(F2) and (F3) imply that for any ε > C ε > F ( x, u ) ≤ ε | u | + C ε | u | p (1.5) We say that f : R N × R → R is a Carathéodory function if f ( · , u ) is measurable for every u ∈ R and f ( x, · ) is continuous for a.e. x ∈ R N . EMIRELATIVISTIC CHOQUARD EQUATIONS 5
Example . One can easily check that the pure power nonlinearity f ( x, u ) = | u | p − u satisfies (F1)–(F4) as soon as (N) holds true. Example . Consider f ( x, u ) = L ( x ) u log(1 + | u | p − ), with Z N -periodic L ∈ L ∞ ( R N ), inf x ∈ R N L ( x ) >
0. It is clear that (F1) and (F2) are satisfied. Note thatit follows from (N) that q < NN − ≤ . Hence, to get (F3) we use the L’Hôpital’s rulelim | u |→ + ∞ F ( x, u ) | u | q/ = 2 L ( x ) q lim | u |→ + ∞ u log(1 + | u | p − ) | u | q − u = 2 L ( x ) q lim | u |→ + ∞ log(1 + | u | p − ) | u | q − = 2 L ( x ) q lim | u |→ + ∞ | u | − q log(1 + | u | p − ) = + ∞ . To get the inequality F ( x, u ) ≥ u ≥ F ( x, u ) = L ( x ) Z u s ln(1 + s p − ) ds ≥ . Note that f is odd in u , and therefore F is even in u . Hence F ( x, u ) = F ( x, − u ) ≥ u <
0. To obtain (F4) we note that f ( x, u ) / | u | is clearly non-decreasing on(0 , + ∞ ). Moreover f ( x, u ) ≥ , + ∞ ). Hence f ( x, u ) | u | q − = f ( x, u ) | u | | u | − q is non-decreasing on (0 , + ∞ ). We proceed similarly on ( −∞ , Example . Suppose that ˜ f satisfy (F1)–(F4) and, for simplicity, does not dependon x ∈ R N . It is clear from (F1), (F2), (F4) that ˜ f ( u ) > , + ∞ ). Take M > u ≥ f ( x, u ) = L ( x ) ˜ f ( u ) if u < ,L ( x ) ˜ f (1) u q − if 1 ≤ u ≤ M ,L ( x ) M q − ˜ f (1)˜ f ( M ) ˜ f ( u ) if u > M and f ( x, u ) = f ( x, − u ) for u <
0, where L ∈ L ∞ ( R N ) is Z N -periodic and inf x ∈ R N L ( x ) >
0. Then we can easily check that (F1)–(F4) are satisfied. Moreover, on [1 , M ] thefunction f is sublinear, since q − ≤ √− ∆ + m by means of the Fourier trans-form, given by the symbol p | ξ | + m , i.e. for any rapidly decaying function u wedefine that √− ∆ + m u is defined by F (cid:16)p − ∆ + m u (cid:17) = p | ξ | + m F ( u ) , where F is the Fourier transform on L ( R N ).Our first result shows that equation (1.1) possesses a least-energy solution as longas the parameter µ is sufficiently small. FEDERICO BERNINI, BARTOSZ BIEGANOWSKI, AND SIMONE SECCHI
F i g u r e 1 .
Plot of µ ∗ ( N ) as N ranges from 2 to 10. Theorem 1.5.
Suppose that (N), (V1), (V2), (F1)–(F4), (K) are satisfied. Thereexists µ ∗ > such that for all µ ∈ (0 , µ ∗ ) and any V l satisfying V l ( x ) < µ | x | for a.e. x ∈ R N \ { } , (1.6) there is a ground state solution u ∈ H / ( R N ) of (1.1) . The constant µ ∗ = µ ∗ ( N ) := 2 Γ (cid:0) N +14 (cid:1) Γ (cid:0) N − (cid:1) , where Γ denotes the Euler Γ -function, depends only on the dimension of the space N , but is independent of the potential V or of the nonlinearity f .Remark . We emphasize the fact that we do not require V l ( x ) < x ∈ R N ,and indeed V l may be positive in some neighborhood of the origin. Example . For N = 2 the constant µ ∗ is equal to µ ∗ (2) = 2 Γ (cid:0) (cid:1) Γ (cid:0) (cid:1) = 2 (cid:18) √ πΓ ( ) (cid:19) Γ (cid:0) (cid:1) = 4 π Γ (cid:0) (cid:1) ≈ . , while for N = 3 it equals µ ∗ (3) = 2 Γ (cid:0) (cid:1) Γ (cid:0) (cid:1) = 2 π ≈ . . See Figure (1).Our second result describes a counterpart to Theorem 1.5 when µ < Theorem 1.8.
Suppose that (N), (V1), (F1)–(F4), (K) and(V2’) ess inf x ∈ R N V p ( x ) > m are satisfied. If µ < and V l ( x ) > µ | x | for a.e. x ∈ R N \ { } , (1.7) there are no ground states of (1.1) . Lastly, we focus on the parameter µ , and we prove a compactness result for groundstates as µ → + . EMIRELATIVISTIC CHOQUARD EQUATIONS 7
Theorem 1.9.
Suppose that (N), (V1), (V2), (F1)–(F4), (K) are satisfied and V l ≡ . Let { µ n } ⊂ (0 , µ ∗ ) be a sequence such that µ n → + . Then for anychoice of ground states u n ∈ H / ( R N ) of (1.1) with µ = µ n there is a sequence oftranslations { z n } ⊂ Z N , such that u n ( · − z n ) * u , up to a subsequence, in H / ( R N ) , where u ∈ H / ( R N ) is a ground state solution to (1.1) with µ = 0 . Moreover c n → c , where c n is the energy of u n and c is the energy of u , where the energy isdefined by (2.1) below. Va r i at i o n a l s e t t i n g
Here and in the sequel | · | k denotes the usual norm in L k ( R N ), where k ∈ [1 , ∞ ].The energy functional, associated with (1.1), E : H / ( R N ) → R is given by(2.1) E ( u ) := 12 Z R N p | ξ | + m | ˆ u ( ξ ) | dξ + 12 Z R N ( V ( x ) − m ) | u ( x ) | dx − µ Z R N | u ( x ) | | x | dx − Z R N × R N F ( x, u ( x )) F ( y, u ( y )) | x − y | N − α dx dy + 1 q Z R N K ( x ) | u ( x ) | q dx. Lemma 2.1.
The quadratic form u Q ( u ) := Z R N p | ξ | + m | ˆ u ( ξ ) | dξ + Z R N ( V ( x ) − m ) | u ( x ) | dx is positive-definite and generates a norm on H / ( R N ) that is equivalent to thestandard one. In particular there exist two positive constants < C ( N, m ) ≤ C ( N, m, | V | ∞ ) such that C ( N, m ) (cid:0) [ u ] + | u | (cid:1) ≤ Q ( u ) ≤ C ( N, m, | V | ∞ ) (cid:0) [ u ] + | u | (cid:1) , where [ u ] := Z R N × R N | u ( x ) − u ( y ) | | x − y | N +1 dx dy is the Gagliardo semi-norm in H / ( R N ) .Proof. We note that, by the Plancherel’s theorem, Q ( u ) = Z R N p | ξ | + m | ˆ u ( ξ ) | dξ + Z R N ( V ( x ) − m ) u dx ≤ Z R N ( | ξ | + m ) | ˆ u ( ξ ) | dξ + Z R N ( | V | ∞ − m ) u dx = Z R N | ξ || ˆ u ( ξ ) | dξ + Z R N | V | ∞ u dx = 12 C (cid:18) N, (cid:19) [ u ] + Z R N | V | ∞ u dx ≤ max (cid:26) C (cid:18) N, (cid:19) , | V | ∞ (cid:27) (cid:0) [ u ] + | u | (cid:1) , FEDERICO BERNINI, BARTOSZ BIEGANOWSKI, AND SIMONE SECCHI where (see [21]) C (cid:18) N, (cid:19) = (cid:18)Z R N − cos ζ | ζ | N +1 dζ (cid:19) − . On the other hand we have Q ( u ) = Z R N p | ξ | + m | ˆ u ( ξ ) | dξ + Z R N ( V ( x ) − m ) u dx ≥ Z R N | ξ || ˆ u ( ξ ) | dξ + Z R N (cid:18) ess inf R N V − m (cid:19) u dx = 12 C (cid:18) N, (cid:19) [ u ] + (cid:18) ess inf R N V − m (cid:19) | u | ≥ min (cid:26) C (cid:18) N, (cid:19) , ess inf R N V − m (cid:27) (cid:0) [ u ] + | u | (cid:1) . (2.2)Recall that, see [21], C (cid:18) N, (cid:19) − = Z R N − | η | ) N +12 dη · Z R − cos tt dt. By contour integration it follows that R R − cos tt dt = π , so C (cid:18) N, (cid:19) − = π Z R N − | η | ) N +12 dη. For N = 2 we have C (cid:18) , (cid:19) − = π Z R | η | ) dη = 2 π. (2.3)For N ≥
3, using polar coordinates we see that C (cid:18) N, (cid:19) − = π Z R N − | η | ) N +12 dη = πω N − Z + ∞ r N − (1 + r ) N +12 dr = πω N − r N − ( N − r + 1) N − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r =+ ∞ r =0 = πω N − N − . Recalling that ω N − = 2 π N − /Γ (cid:0) N − (cid:1) we finally get that C (cid:18) N, (cid:19) − = πN − π N − Γ (cid:0) N − (cid:1) = π N +12 N − Γ (cid:0) N − (cid:1) = π N +12 Γ (cid:0) N +12 (cid:1) , (2.4)where Γ is the Euler Γ -function. (cid:3) We recall the following variant of the fractional Hardy inequality, adapted to oursetting.
Lemma 2.2 ([15, Theorem 1.1]) . There exists a constant C N, / , / > , dependingonly on N ≥ , such that for every u ∈ H / ( R N ) there holds [ u ] ≥ C N, / , / Z R N | u ( x ) | | x | dx, EMIRELATIVISTIC CHOQUARD EQUATIONS 9 where C N, / , / = 2 π N/ Γ (cid:0) N +14 (cid:1) (cid:12)(cid:12) Γ (cid:0) − (cid:1)(cid:12)(cid:12) Γ (cid:0) N − (cid:1) Γ (cid:0) N +12 (cid:1) is the sharp constant of the inequality (see [15]). Lemma 2.3.
There is µ ∗ > such that for any < µ < µ ∗ the quadratic form Q µ : u Q ( u ) − µ Z R N | u ( x ) | | x | dx is positive-definite and generates a norm on H / ( R N ) that is equivalent to thestandard one. The constant µ ∗ = µ ∗ ( N ) = 2 Γ ( N +14 ) Γ ( N − ) depends only on the dimension N .Proof. Note that Q µ ( u ) ≤ Q ( u ) for any u ∈ H / ( R N ). On the other hand, recalling(2.2) and Lemma 2.2, Q µ ( u ) = Q ( u ) − µ Z R N | u ( x ) | | x | dx ≥ Z R N | ξ || ˆ u ( ξ ) | dξ + (cid:18) ess inf R N V − m (cid:19) | u | − µ Z R N | u ( x ) | | x | dx = 12 C (cid:18) N, (cid:19) [ u ] + (cid:18) ess inf R N V − m (cid:19) | u | − µ Z R N | u ( x ) | | x | dx ≥ C (cid:18) N, (cid:19) [ u ] + (cid:18) ess inf R N V − m (cid:19) | u | − µC N, / , / [ u ] = (cid:18) C (cid:18) N, (cid:19) − µC N, / , / (cid:19) [ u ] + (cid:18) ess inf R N V − m (cid:19) | u | ≥ min (cid:26) C (cid:18) N, (cid:19) − µC N, / , / , ess inf R N V − m (cid:27) (cid:0) [ u ] + | u | (cid:1) . Recalling (2.3) and (2.4), the conclusion for N ≥ µ < C N, / , / · C (cid:18) N, (cid:19) = 12 · π N/ Γ (cid:0) N +14 (cid:1) Γ (cid:0) N − (cid:1) (cid:12)(cid:12) Γ (cid:0) − (cid:1)(cid:12)(cid:12) Γ (cid:0) N +12 (cid:1) N − π Γ (cid:0) N − (cid:1) π N − = ( N − Γ (cid:0) N +14 (cid:1) Γ (cid:0) N − (cid:1) Γ (cid:0) N − (cid:1) Γ (cid:0) N +12 (cid:1) = 2 Γ (cid:0) N +14 (cid:1) Γ (cid:0) N − (cid:1) , where we have used the fact that Γ (cid:18) − (cid:19) = − √ π,N − Γ (cid:18) N − (cid:19) = Γ (cid:18) N + 12 (cid:19) . When N = 2, we conclude under the constraint µ < C (cid:18) , (cid:19) C , / , / = 12 Γ (cid:0) (cid:1) Γ (cid:0) (cid:1) √ πΓ (cid:0) (cid:1) = 2 Γ (cid:0) (cid:1) Γ (cid:0) (cid:1) . (cid:3) Thanks to Lemma 2.3 we can introduce the norm k u k µ := p Q µ ( u ) on H / ( R N )for 0 < µ < µ ∗ . In the rest of the paper we will use h· , ·i for the scalar productcorresponding to Q and h· , ·i µ for the scalar product which corresponds to Q µ .Moreover we define D ( u ) := Z R N × R N F ( x, u ( x )) F ( y, u ( y )) | x − y | N − α dx dy, that is well-defined on H / ( R N ) by (F1) and (N). Thus we can rewrite our func-tional in the form E ( u ) = 12 k u k µ − D ( u ) + 1 q Z R N K ( x ) | u ( x ) | q dx. It is standard to check that E is of C -class and its critical points are weak solutionsto (1.1).3. E x i s t e n c e a n d b o u n d e d n e s s o f a C e r a m i s e q u e n c e
Suppose that ( E, k · k ) is a Hilbert space and E : E → R is a nonlinear functional ofthe general form E ( u ) = 12 k u k − I ( u ) , where I is of C class and I (0) = 0. We introduce the following set N := { u ∈ E \ { } | E ( u )( u ) = 0 } , which is known as the Nehari manifold . It is obvious that any nontrivial criticalpoint of E belongs to N . The following theorem follows from [4, 30], see also theabstract setting in [3]. Theorem 3.1 ([3, Theorem 5.1]) . Suppose that(J1) there is r > such that inf k u k = r E ( u ) > (J2) I ( t n u n ) t n → + ∞ for t n → + ∞ and u n → u = 0 ;(J3) for all t > and u ∈ N there holds t − I ( u )( u ) − I ( tu ) + I ( u ) ≤ . Then
N 6 = ∅ , Γ = ∅ and c := inf N E = inf γ ∈ Γ sup t ∈ [0 , E ( γ ( t )) = inf u ∈ E \{ } sup t ≥ E ( tu ) > , EMIRELATIVISTIC CHOQUARD EQUATIONS 11 where Γ := { γ ∈ C ([0 , , E ) | γ (0) = 0 , k γ (1) k > r, E ( γ (1)) < } . Moreover there is a Cerami sequence for E at level c , i.e. a sequence { u n } n ⊂ E such that E ( u n ) → c, (1 + k u n k ) E ( u n ) → . Observe that assumptions (J1)–(J3) do not require that I ( u ) ≥ u ∈ E . In fact, I may change its sign, which is possible in our situation. Moreover the theorem givesequivalent min-max-type characterizations of the level c (which, in our situation,will be a critical value). Lemma 3.2.
There is
C > such that D ( u ) ≤ C (cid:0) k u k µ + k u k p +2 µ + k u k pµ (cid:1) for all u ∈ H / ( R N ) .Proof. In the proof, C denotes a generic, positive constant which may vary fromline to line. Fix any ε >
0. Using (1.5) we have D ( u ) ≤ Z R N × R N | F ( x, u ( x )) || F ( y, u ( y )) || x − y | N − α dx dy ≤ Z R N × R N (cid:0) ε | u ( x ) | + C ε | u ( x ) | p (cid:1) (cid:0) ε | u ( y ) | + C ε | u ( y ) | p (cid:1) | x − y | N − α dx dy. Applying the Hardy-Littlewood-Sobolev inequality we obtain D ( u ) ≤ C (cid:12)(cid:12) ε | u | + C ε | u | p (cid:12)(cid:12) r (cid:12)(cid:12) ε | u | + C ε | u | p (cid:12)(cid:12) r = C (cid:12)(cid:12) ε | u | + C ε | u | p (cid:12)(cid:12) r , where r = 2 NN + α . By Minkowski inequality we obtain D ( u ) ≤ C (cid:0) ε (cid:12)(cid:12) u (cid:12)(cid:12) r + C ε | u p | r (cid:1) = C (cid:16) ε | u | r + C ε | u | ppr (cid:17) = C (cid:16) ε | u | r + 2 εC ε | u | r | u | ppr + C ε | u | ppr (cid:17) . From (N) there follows that pr < NN − , and by Sobolev embeddings D ( u ) ≤ C (cid:0) ε k u k µ + 2 εC ε k u k p +2 µ + C ε k u k pµ (cid:1) , and therefore D ( u ) ≤ C (cid:0) k u k µ + k u k p +2 µ + k u k pµ (cid:1) . (cid:3) Put I ( u ) := 12 D ( u ) − q Z R N K ( x ) | u ( x ) | q dx. Lemma 3.3.
Suppose (N), (F1)–(F4) and ( K ) are satisfied. Then (J1)–(J3) aresatisfied on E = H / ( R N ) with the norm k · k µ . Proof. (J1) Since I ( u ) = D ( u ) − q R R N K ( x ) | u ( x ) | q dx , by Lemma 3.2 we have I ( u ) ≤ D ( u ) ≤ C (cid:0) k u k µ + k u k p +2 µ + k u k pµ (cid:1) = C k u k µ (cid:0) k u k µ + k u k pµ + k u k p − µ (cid:1) . Note that for r > I ( u ) ≤ C k u k µ (cid:0) r + r p + r p − (cid:1) for k u k µ ≤ r . Put A ( r ) := C (cid:0) r + r p + r p − (cid:1) and then I ( u ) ≤ A ( r ) k u k µ . Note that A : [0 , + ∞ ) → [0 , + ∞ ) is a continuous function with A (0) = 0 andlim r → + ∞ A ( r ) = + ∞ . Hence we can take r > A ( r ) = , that is I ( u ) ≤ k u k µ . Hence, for k u k µ = r E ( u ) = 12 k u k µ − I ( u ) ≥ k u k µ = 14 r > . (J2) Since t n → + ∞ , we may assume that t n ≥ q >
2. Then I ( t n u n ) t n ≥ I ( t n u n ) t qn = 12 Z R N × R N F ( x, t n u n ( x )) F ( y, t n u n ( y )) t qn | x − y | N − α dx dy − q Z R N K ( x ) | u n ( x ) | q dx → + ∞ by (F3) and Fatou’s lemma.(J3) Let u ∈ N and define ϕ ( t ) = t − I ( u ) u − I ( tu ) + I ( u )for t ≥
0; we remark that ϕ (1) = 0. Moreover, k u k µ = I ( u ) u >
0, which isequivalent to Z R N × R N F ( x, u ( x )) f ( y, u ( y )) u ( y ) | x − y | N − α dx dy > Z R N K ( x ) | u ( x ) | q dx. (3.1)We compute dϕ ( t ) dt = t I ( u ) u − I ( tu ) u = t Z R N × R N F ( x, u ( x )) f ( y, u ( y )) u ( y ) | x − y | N − α dx dy − t Z R N K ( x ) | u ( x ) | q dx − Z R N × R N F ( x, tu ( x )) f ( y, tu ( y )) u ( y ) | x − y | N − α dx dy + Z R N K ( x ) | tu ( x ) | q dx = Z R N × R N (cid:20) F ( x, u ( x )) f ( y, u ( y )) tu ( y ) | x − y | N − α − F ( x, tu ( x )) f ( y, tu ( y )) u ( y ) | x − y | N − α (cid:21) dx dy + (cid:0) t q − − t (cid:1) Z R N K ( x ) | u ( x ) | q dx. EMIRELATIVISTIC CHOQUARD EQUATIONS 13
For almost every fixed x, y ∈ R N we define the map ψ : (0 , + ∞ ) → R as ψ ( t ) := ψ ( x,y ) ( t ) := F ( x, tu ( x )) f ( y, tu ( y )) u ( y ) t q − . (3.2)If t < dϕ ( t ) dt = I ( u )( tu ) − I ( tu )( u ) ≥ Z R N × R N (cid:20) F ( x, u ( x )) f ( y, u ( y )) tu ( y ) | x − y | N − α − F ( x, tu ( x )) f ( y, tu ( y )) u ( y ) | x − y | N − α + t q − F ( x, u ( x )) f ( y, u ( y )) u ( y ) | x − y | N − α − F ( x, u ( x )) f ( y, u ( y )) tu ( y ) | x − y | N − α (cid:21) dx dy = Z R N × R N (cid:20) t q − F ( x, u ( x )) f ( y, u ( y )) u ( y ) | x − y | N − α − F ( x, tu ( x )) f ( y, tu ( y )) u ( y ) | x − y | N − α (cid:21) dx dy = t q − Z R N × R N ψ ( x,y ) (1) − ψ ( x,y ) ( t ) | x − y | N − α dx dy. We claim that ψ ( t ) ≥ ψ is non-decreasing on (0 , t <
1. From (F3) and (1.4) there follows that ψ ( t ) ≥
0. We rewrite (3.2)in the form ψ ( t ) = F ( x, tu ( x )) t q f ( y, tu ( y )) u ( y ) t q − Again, taking (1.4) into account ddt (cid:20) F ( x, tu ( x )) t q (cid:21) = f ( x, tu ( x )) tu ( x ) − q F ( x, tu ( x )) t q +1 ≥ , so t F ( x,tu ( x )) t q is non-decreasing on (0 , + ∞ ). Moreover f ( y, tu ( y )) u ( y ) t q − = f ( y, tu ( y )) u ( y ) | tu ( y ) | q − | u ( y ) | q − = f ( y, tu ( y )) | tu ( y ) | q − | u ( y ) | q − u ( y )so that t f ( y, tu ( y )) u ( y ) t q − is non-decreasing by (F4), if u ( y ) = 0. Hence ψ is non-decreasing as a product ofnon-negative, non-decreasing functions. Hence dϕ ( t ) dt = I ( u )( tu ) − I ( tu )( u ) ≥ t q − Z R N × R N ψ ( x,y ) (1) − ψ ( x,y ) ( t ) | x − y | N − α dx dy ≥ , that is ϕ ( t ) ≤ ϕ (1) = 0 for t ∈ (0 , t ∈ (1 , + ∞ ) then dϕ ( t ) dt = I ( u )( tu ) − I ( tu )( u ) ≤ , therefore ϕ ( t ) ≤ ϕ (1) = 0 for t ∈ (1 , + ∞ ). (cid:3) Lemma 3.4.
Any Cerami sequence { u n } n for E is bounded. Proof.
By the properties of Cerami’s sequences, we may writelim sup n → + ∞ E ( u n ) = lim sup n → + ∞ (cid:18) E ( u n ) − q E ( u n ) u n (cid:19) = lim sup n → + ∞ (cid:20)(cid:18) − q (cid:19) k u n k µ + 1 q Z R N × R N F ( x, u n ( x )) (cid:2) f ( y, u n ( y )) u n ( y ) − q F ( y, u n ( y )) (cid:3) | x − y | N − α dx dy ≥ lim sup n → + ∞ (cid:18) − q (cid:19) k u n k µ by (1.4). Since lim sup n → + ∞ E ( u n ) is finite, the proof is complete. (cid:3) D e c o m p o s i t i o n o f b o u n d e d m i n i m i z i n g s e q u e n c e s
To ease notation we set I α ( x ) := 1 | x | N − α , x ∈ R N \ { } . Lemma 4.1.
Suppose that { u n } n ⊂ H / ( R N ) is a bounded sequence such that u n * u in H / ( R N ) . Then D ( u n − u ) − D ( u n ) + D ( u ) → as n → + ∞ . The proof is similar to the proof of [7, Lemma 2.2] in the space H ( R N ), so we omitit. Lemma 4.2.
Suppose that { u n } n ⊂ H / ( R N ) and there are ‘ ≥ , { z kn } n ⊂ Z N and w k ∈ H / ( R N ) for k = 1 , . . . , ‘ such that u n * u , u n ( · − z kn ) * w k in x ∈ H / ( R N ) and (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) u n − u − ‘ X k =1 w k ( · − z kn ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) → . (4.1) Then D ( u n ) → D ( u ) + ‘ X k =1 D ( w k ) . Proof.
For any m ∈ { , , . . . , ‘ } we introduce a nm := u n − u − m X k =1 w k (cid:0) · − z kn (cid:1) . From Lemma 4.1 we have that D ( a n ) − D ( u n ) + D ( u ) → . Taking a n (cid:0) · + z n (cid:1) as u n and w as u in Lemma 4.1 we obtain D (cid:0) a n (cid:0) · + z n (cid:1) − w (cid:1) − D (cid:0) a n (cid:0) · + z n (cid:1)(cid:1) + D ( w ) → EMIRELATIVISTIC CHOQUARD EQUATIONS 15 or equivalently D ( a n ) − D ( a n ) + D ( w ) → . (4.2)Similarly D ( a n ) − D ( a n ) + D ( w ) → . (4.3)Combining (4.2) and (4.3) gives D ( a n ) + D ( w ) + D ( w ) − D ( a n ) → . Iterating the same reasoning we obtain D ( a n‘ ) + ‘ X k =1 D ( w k ) − D ( a n ) → . Taking into account that a n = u n − u we see that D ( a n‘ ) + ‘ X k =1 D ( w k ) − D ( u n − u ) → . (4.4)In view of (4.1) we obtain a n‘ →
0, so that D ( a n‘ ) → D ( u n − u ) → ‘ X k =1 D ( w k ) . Taking again Lemma 4.1 into account we obtain D ( u n ) → D ( u ) + ‘ X k =1 D ( w k ) . (cid:3) Lemma 4.3. D : H / ( R N ) → (cid:0) H / ( R N ) (cid:1) ∗ = H − / ( R N ) is weak-to-weak* con-tinuous, i.e. if { u n } n is bounded and u n * u in H / ( R N ) and ϕ ∈ H / ( R N ) then D ( u n )( ϕ ) → D ( u )( ϕ ) . Proof.
Observe that D ( u n )( ϕ ) = Z R N ( I α ∗ ( F ( · , u n ( · ))) ( x ) f ( x, u n ( x )) ϕ ( x ) dx, which (for brevity) we will write shortly as D ( u n )( ϕ ) = Z R N ( I α ∗ ( F ◦ u n ))( f ◦ u n ) ϕ dx. Since { u n } n is bounded in H / ( R N ), it is bounded also in L ( R N ) ∩ L NN − ( R N ).From (1.5) there follows that { F ( · , u n ( · )) } n is bounded in L NN + α ( R N ), since (N)implies that p · NN + α < NN − . The weak convergence u n * u in H / ( R N ) impliesthat u n ( x ) → u ( x ) for a.e. x ∈ R N . Hence F ( x, u n ( x )) → F ( x, u ( x )) for a.e. x ∈ R N , and therefore F ( · , u n ( · )) * F ( · , u ( · )) in L NN + α ( R N ). From the Hardy-Littlewood-Sobolev inequality we obtain that I α ∗ ( F ◦ u n ) * I α ∗ ( F ◦ u ) in L NN − α ( R N ) . Moreover, from (F1), f ◦ u n → f ◦ u in L NN + α loc ( R N ). Hence, for any ϕ ∈ C ∞ ( R N )there holds( I α ∗ ( F ◦ u n ))( f ◦ u n ) ϕ → ( I α ∗ ( F ◦ u ))( f ◦ u ) ϕ in L ( R N ) , i.e. D ( u n )( ϕ ) → D ( u )( ϕ ). (cid:3) Corollary 4.4. E : H / ( R N ) → (cid:0) H / ( R N ) (cid:1) ∗ = H − / ( R N ) is weak-to-weak*continuous.Proof. Indeed, take u n * u in H / ( R N ) and ϕ ∈ C ∞ ( R N ), we have that E ( u n )( ϕ ) = h u n , ϕ i µ − D ( u n )( ϕ ) + Z R N K ( x ) | u n ( x ) | q − u n ( x ) ϕ ( x ) dx. We may assume that, up to a subsequence, u n ( x ) → u ( x ) for a.e. x ∈ R N .Obviously h u n , ϕ i → h u , ϕ i µ . Moreover, from Lemma 4.3, D ( u n )( ϕ ) → D ( u )( ϕ ).Note that for any measurable subset E ⊂ supp ϕ , (cid:12)(cid:12)(cid:12)(cid:12)Z E K ( x ) | u n ( x ) | q − u n ( x ) ϕ ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ | K | ∞ | u n | q − q | ϕχ E | q , so that in view of the Vitali convergence theorem Z R N K ( x ) | u n ( x ) | q − u n ( x ) ϕ ( x ) dx → Z R N K ( x ) | u ( x ) | q − u ( x ) ϕ ( x ) dx and we conclude. (cid:3) Define E per ( u ) := E ( u ) − Z R N V l ( x ) u dx + µ Z R N u | x | dx. (4.5)Note that E per ( u ( · − z )) = E per ( u ) for any z ∈ Z N . Theorem 4.5.
Let { u n } n be a bounded Palais-Smale sequence. Then (up to asubsequence) there is an integer ‘ ≥ and sequences ( z kn ) ⊂ Z N , w k ∈ H / ( R N ) , k = 1 , . . . , ‘ such that(i) u n * u and E ( u ) = 0 ;(ii) | z kn | → + ∞ and | z kn − z k n | → + ∞ for k = k ;(iii) w k = 0 and E per ( w k ) = 0 for ≤ k ≤ ‘ ;(iv) u n − u − P ‘k =1 w k ( · − z kn ) → ;(v) E ( u n ) → E ( u ) + P ‘k =1 E per ( w k ) .Proof. Step 1:
Up to a subsequence, u n * u and E ( u ) = 0 . Since { u n } n is bounded, u n * u up to a subsequence. Taking into account that EMIRELATIVISTIC CHOQUARD EQUATIONS 17 E ( u ) → E ( u ) = 0. Step 2:
Let v n := u n − u . Suppose that lim n → + ∞ sup z ∈ R N Z B ( z, | v n ( x ) | dx = 0 . Then u n → u and the statement follows with ‘ = 0 . Note that E ( u n )( v n ) = k v n k µ + h u , u n − u i µ − D ( u n )( v n ) + Z R N K ( x ) | u n | q − u n v n dx. Taking into account that h u , u n − u i µ − D ( u )( v n ) + Z R N K ( x ) | u | q − u v n dx = 0we have k v n k µ = E ( u n )( v n ) − h u , u n − u i µ + 12 D ( u n )( v n ) − Z R N K ( x ) | u n | q − u n v n dx = E ( u n )( v n ) − D ( u )( v n ) + Z R N K ( x ) | u | q − u v n dx + 12 D ( u n )( v n ) − Z R N K ( x ) | u n | q − u n v n dx. From the boundedness of { v n } n there follows that E ( u n )( v n ) → . From Hölder’s inequality and Lions’ Concentration-Compactness Principle [29] weobtain (cid:12)(cid:12)(cid:12)(cid:12)Z R N K ( x ) | u | q − u v n dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ | K | ∞ | u | q − q | v n | q → , (cid:12)(cid:12)(cid:12)(cid:12)Z R N K ( x ) | u n | q − u n v n dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ | K | ∞ | u n | q − q | v n | q → . Moreover, from the Hardy-Littlewood-Sobolev inequality there follows that D ( u n )( v n ) = D ( u n )( u n − u ) → D ( u )( v n ) = D ( u )( u n − u ) → . Thus k v n k µ → u n → u . Hence E ( u n ) → E ( u ) and the statement of thetheorem holds true for ‘ = 0. Step 3:
Suppose that there is a sequence { z n } n ⊂ Z N such that lim inf n → + ∞ Z B ( z n , √ N ) | v n | dx > . Then there is w ∈ H / ( R N ) such that (up to a subsequence):(i) | z n | → + ∞ ;(ii) u n ( · + z n ) * w = 0 ;(iii) E per ( w ) = 0 . Statements (i) and (ii) are standard, so we will show only (iii). Put v n := u n ( · + z n ).Similarly as in Step 1 we see that E per ( v n )( ϕ ) → E per ( w )( ϕ )for any ϕ ∈ C ∞ ( R N ). Moreover o (1) = E ( u n )( ϕ ( · − z n )) = E per ( v n )( ϕ ) + Z R N V l ( x + z n ) v n ϕ dx − µ Z R N u n ϕ ( · − z n ) | x | dx = E per ( w )( ϕ ) + Z supp ϕ V l ( x + z n ) v n ϕ dx − µ Z R N u n ϕ ( · − z n ) | x | dx + o (1)and E per ( w )( ϕ ) = − Z supp ϕ V l ( x + z n ) v n ϕ dx + µ Z R N u n ϕ ( · − z n ) | x | dx + o (1) . Lemma 2.2 implies that { u n } n is bounded in L ( R N ; | x | − dx ), and from [2, Lemma2.5] we obtain that (cid:12)(cid:12)(cid:12)(cid:12)Z R N u n ϕ ( · − z n ) | x | dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:18)Z R N | u n | | x | dx (cid:19) / (cid:18)Z R N | ϕ ( x − z n ) | | x | dx (cid:19) / → . Hence it is sufficient to show that R supp ϕ V l ( x + z n ) v n ϕ dx →
0. Fix any measurableset E ⊂ supp ϕ . From the Hölder inequality we obtain that Z E | V l ( x + z n ) v n ϕ | dx ≤ | V l | ∞ | v n | | ϕχ E | . Then, the boundedness of { v n } n in L ( R N ) implies that the family { V l ( · + z n ) v n ϕ } n is uniformly integrable on supp ϕ and from Vitali’s convergence theorem we derive Z supp ϕ V l ( x + z n ) v n ϕ dx → , and the proof of Step 3 is completed. Step 4:
Suppose that there are m ≥ , { z kn } n ⊂ Z N , w k ∈ H / ( R N ) for k ∈{ , , . . . , m } such that | z kn | → + ∞ , | z kn − z k n | → + ∞ for 1 ≤ k < k ≤ m ; u n ( · + z kn ) → w k = 0 for 1 ≤ k ≤ m ; E per ( w k ) = 0 for 1 ≤ k ≤ m. Then (1) if sup z ∈ R N Z B ( z, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u n − u − m X k =1 w k ( · − z kn ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dx → n → + ∞ (4.6) then u n − u − m X k =1 w k ( · − z kn ) → EMIRELATIVISTIC CHOQUARD EQUATIONS 19 (2) if there is { z m +1 n } n ⊂ Z N such thatlim inf n → + ∞ Z B ( z m +1 n , √ N ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u n − u − m X k =1 w k ( · − z kn ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dx > w m +1 ∈ H / ( R N ) such that (up to subsequences)(i) | z m +1 n | → + ∞ , | z m +1 n − z kn | → + ∞ for 1 ≤ k ≤ m ,(ii) u n ( · − z m +1 n ) * w m +1 = 0,(iii) E per ( w m +1 ) = 0.Define ξ n := u n − u − m X k =1 w k ( · − z kn ) . Suppose that (4.6) holds. Then Lions’ Concentration-Compactness Principle impliesthat ξ n → L t ( R N ) for any 2 < t < NN − . Repeating the argument of [5, Lemma4.3, Step 4] and using the identity E ( u )( ξ n ) = 0, we obtain that k ξ n k µ = −h u , ξ n i µ − m X k =1 h w k ( · − z kn ) , ξ n i µ + 12 D ( u n )( ξ n ) − Z R N K ( x ) | u n | q − u n ξ n dx + o (1)= − D ( u )( ξ n ) + Z R N K ( x ) | u | q − u ξ n dx − m X k =1 h w k ( · − z kn ) , ξ n i µ + 12 D ( u n )( ξ n ) − Z R N K ( x ) | u n | q − u n ξ n dx + o (1) . Taking into account that each w k is a critical point of E per , we obtain that k ξ n k µ = 12 D ( u n )( ξ n ) − D ( u )( ξ n ) − m X k =1 D ( w k ( · − z kn ))( ξ n ) − Z R N K ( x ) | u n | q − u n − | u | q − u − m X k =1 | w k ( x − z kn ) | q − w k ( x − z kn ) ! ξ n dx − m X k =1 Z R N V l ( x ) w k ( x − z kn ) ξ n dx + µ m X k =1 Z R N w k ( x − z kn ) ξ n | x | dx + o (1) . From [2, Lemma 2.5], we see that Z R N w k ( x − z kn ) ξ n | x | dx → . Similarly as in [5, Lemma 4.3, Step 4] we get that Z R N K ( x ) | u n | q − u n − | u | q − u − m X k =1 | w k ( x − z kn ) | q − w k ( x − z kn ) ! ξ n dx → Z R N V l ( x ) w k ( x − z kn ) ξ n dx → . Hence k ξ n k = 12 D ( u n )( ξ n ) − D ( u )( ξ n ) − m X k =1 D ( w k ( · − z kn ))( ξ n ) + o (1) . To show that |D ( u n )( ξ n ) | → F |D ( u n )( ξ n ) | ≤ Z R N × R N | F ( x, u n ( x )) || f ( y, u n ( y )) || ξ n ( y ) || x − y | N − α dx dy ≤ C Z R N × R N (cid:0) ε | u n | + C ε | u n | p (cid:1) (cid:0) | u n | αN + | u n | p − (cid:1) | ξ n || x − y | N − α dx dy ≤ C (cid:12)(cid:12) ε | u n | + C ε | u n | p (cid:12)(cid:12) r (cid:12)(cid:12)(cid:0) | u n | αN + | u n | p − (cid:1) | ξ n | (cid:12)(cid:12) r ≤ C (cid:0) | u n | r + | u n | ppr (cid:1) (cid:16) | u n | α/Nrα/N + | u n | p − p − r (cid:17) | ξ n | r → , (4.8)where r = NN + α < N/ ( N − D ( u )( ξ n ) → D ( w k ( · − z kn ))( ξ n ) →
0. Hence ξ n → { z m +1 n } n ⊂ Z N we have (4.7). (i) and (ii) are standard. Put v n := u n ( · + z m +1 n ). As in Step 3 we see that E per ( v n )( ϕ ) − E per ( w m +1 )( ϕ ) → E per ( v n )( ϕ ) → ϕ ∈ C ∞ ( R N ). Step 5:
Conclusion.
Iterating Step 4 we construct functions w k = 0 and sequences { z kn } n . Since w m are critical points of E per , there is ρ > k w k k ≥ ρ .Properties of the weak convergence yield0 ≤ lim n → + ∞ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) u n − u − ‘ X k =1 w k ( · − z kn ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = lim n → + ∞ k u n k − k u k − m X k =1 k w k k ! ≤ lim sup n → + ∞ k u n k − k u k − mρ . Hence ρ m ≤ lim sup n → + ∞ k u n k − k u k and the procedure will finish after finite number — say ‘ — of steps. Step 6: (v) holds. (v) follows from properties of the weak convergence, Hardy
EMIRELATIVISTIC CHOQUARD EQUATIONS 21 inequality and Lemma 4.2. Indeed, E ( u n ) = 12 h u n , u n i µ − D ( u n ) + 1 q Z R N K ( x ) | u n ( x ) | q dx = 12 h u , u i µ + 12 h u n − u , u n − u i µ + h u , u n − u i µ − D ( u n )+ 1 q Z R N K ( x ) | u n ( x ) | q dx = E ( u ) + E per ( u n − u ) + h u , u n − u i µ + 12 D ( u n − u ) − D ( u n ) + 12 D ( u ) − q Z R N K ( x ) [ | u n − u | q + | u | q − | u n | q ] dx + 12 Z R N V l ( x )( u n − u ) dx − µ Z R N ( u n − u ) | x | dx. By the weak convergence we obtain h u , u n − u i µ → . Lemma 4.1 imply that D ( u n − u ) − D ( u n ) + D ( u ) → . By a classical Brezis-Lieb lemma argument (see [6, Proposition 4.7.30]) we havealso that Z R N K ( x ) [ | u n − u | q + | u | q − | u n | q ] dx → . Let E ⊂ R N a measurable set, by (V1) and Hölder inequality we have Z E | V l ( x ) || u n − u | dx ≤ | V l χ E | N | u n − u | NN − and, since ( u n − u ) n is bounded in H / ( R N ), by Vitali convergence theorem Z R N V l ( x )( u n − u ) dx → . We note that Z R N ( u n − u ) | x | dx = Z R N ( u n − u ) u n | x | dx − Z R N ( u n − u ) u | x | dx and, as before, R R N ( u n − u ) u | x | dx →
0. Moreover Z R N ( u n − u ) u n | x | dx = Z R N (cid:16) u n − u − P ‘k =1 w k ( · − z kn ) (cid:17) u n | x | + ‘ X k =1 w k ( · − z kn ) u n | x | dx. Then (cid:12)(cid:12)(cid:12)(cid:12)Z R N w k ( · − z kn ) u n | x | dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:18)Z R N | w k ( · − z kn ) | | x | dx (cid:19) / (cid:18)Z R N | u n | | x | dx (cid:19) / → and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z R N (cid:16) u n − u − P ‘k =1 w k ( · − z kn ) (cid:17) u n | x | dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Z R N (cid:16) u n − u − P ‘k =1 w k ( · − z kn ) (cid:17) | x | dx (cid:18)Z R N | u n | | x | dx (cid:19) / ≤ C (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) u n − u − ‘ X k =1 w k ( · − z kn ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:18)Z R N | u n | | x | dx (cid:19) / → . Thus E ( u n ) = E ( u ) + E per ( u n − u ) + o (1)and it is sufficient to show that E per ( u n − u ) → P ‘k =1 E per ( w k ).Now, we compute E per ( u n − u ) = 12 k u n − u k µ − D ( u n − u ) + 1 q Z R N K ( x ) | u n − u | q dx − Z R N V l ( x )( u n − u ) dx + µ Z R N ( u n − u ) | x | dx = 12 (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) u n − u − ‘ X k =1 w k ( · − z kn ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) µ − D ( u n − u )+ 1 q Z R N K ( x ) | u n − u | q dx + 12 ‘ X k =1 k w k ( · − z kn ) k µ + o (1)= ‘ X k =1 E per ( w k ) + 12 ‘ X k =1 D ( w k ( · − z kn )) − q Z R N K ( x ) | w k ( · − z kn ) | q dx − D ( u n − u ) + 1 q Z R N K ( x ) | u n − u | q dx + o (1) . Iterating Lemma 4.1 and by Lemma 4.2 we obtain D ( u n − u ) − ‘ X k =1 D ( w k ( · − z kn )) → n → + ∞ and similarly we can prove Z R N K ( x ) | u n − u | q dx − Z R N K ( x ) | w k ( · − z kn ) | q dx → n → + ∞ ; hence E per ( u n − u ) → ‘ X k =1 E per ( w k )and the proof is complete. (cid:3) EMIRELATIVISTIC CHOQUARD EQUATIONS 23 E x i s t e n c e a n d n o n e x i s t e n c e o f g ro u n d s tat e s
The remaining part of the proof of the existence of solutions is similar to the proofof [5, Theorem 1.1] and [2, Theorem 1.1], and we include it here for the reader’sconvenience.
Proof of Theorem 1.5.
Put c per := inf N per E per , where E per is given by (4.5) and N per is the corresponding Nehari manifold. From Theorem 4.5(iii), (v) there followsthat c = lim n → + ∞ E ( u n ) = E ( u ) + ‘ X k =1 E per ( w k ) ≥ E ( u ) + ‘c per . Note that (1.6) implies that V ( x ) − µ | x | = V p ( x )+ V l ( x ) − µ | x | < V p ( x ) for a.e. x ∈ R N ,which gives the inequality c per > c >
0. Assume by contradiction that u = 0. Then c = E ( u ) + ‘ X k =1 E per ( w k ) = ‘ X k =1 E per ( w k ) ≥ ‘c per . If ‘ ≥ c ≥ ‘c per > ‘c , which is a contradiction. Hence ‘ = 0 and0 < c = E ( u ) = 0 , a contradiction. Thus u is a ground state solution. (cid:3) Proof of Theorem 1.8.
We assume by contradiction that u is a ground state for E .In particular c = inf N E = E ( u ) > . The inequality (1.7) implies that V ( x ) − µ | x | = V p ( x ) + V l ( x ) − µ | x | > V p ( x ) for a.e. x ∈ R N and therefore c > c per . On the other hand, fix u ∈ N per , where E per is given by(4.5) with the corresponding Nehari manifold N per . We may choose t z > t z u ( · − z ) ∈ N for any z ∈ Z N . Then E per ( u ) = E per ( u ( · − z )) ≥ E per ( t z u ( · − z ))= E ( t z u ( · − z )) − Z R N V l ( x ) | t z u ( · − z ) | dx + µ Z R N | t z u ( · − z ) | | x | dx ≥ c − Z R N V l ( x ) | t z u ( · − z ) | dx + µ Z R N | t z u ( · − z ) | | x | dx. Note that coercivity of E per on N per and the inequality E per ( t n ( u ( ·− z )) = E per ( t z u ) ≤ c per implies that sup z ∈ Z N t z < + ∞ . Hence Z R N V l ( x ) | t z u ( · − z ) | dx = t z Z R N V l ( x + z ) u dx → | z | → + ∞ . From [2, Lemma 2.5] there follows that Z R N | t z u ( · − z ) | | x | dx = t z Z R N | u ( · − z ) | | x | dx → | z | → + ∞ . Hence E per ( u ) ≥ c + o (1)and taking infimum over u ∈ N per we see that c per = inf N per E per ≥ c, which is a contradiction. (cid:3) C o m pac t n e s s o f g ro u n d s tat e s
Let { µ n } n ∈ (0 , µ ∗ ) be a sequence such that µ n → + as n → + ∞ and let E n bethe Euler functional for µ = µ n . We denote by E and N the Euler functional andthe Nehari manifold for µ = 0 and define c n := E n ( u n ) = inf N n E n , c := E ( u ) = inf N E , where u n ∈ N n is the ground state solution for E n , in particular E n ( u n ) = 0. Lemma 6.1.
There exists a positive radius r > such that inf n ≥ inf k u k µn = r E n ( u ) > . Proof.
Fix n ≥
1. Repeating the same computations as in Lemma 3.3 (J1) weobtain that E n ( u ) = 12 k u k µ n − D ( u ) + 1 q Z R N K ( x ) | u ( x ) | q dx ≥ k u k µ n = 14 r > k u k µ n = r and for every n ≥
1, and properly chosen r > (cid:3) Lemma 6.2.
The sequence { u n } n is bounded in H / ( R N ) .Proof. Suppose by contradiction that Q ( u n ) → + ∞ . Then, from Lemma 2.3, wesee that Q µ n ( u n ) ≥ min (cid:26) C (cid:18) N, (cid:19) − µ n C N, / , / , ess inf R N V − m (cid:27) (cid:0) [ u n ] + | u n | (cid:1) ≥ CQ ( u n ) , where C > µ n . Hence k u n k µ n → + ∞ . Thus c = lim n → + ∞ E n ( u n ) = lim n → + ∞ (cid:18) E n ( u n ) − q E ( u n )( u n ) (cid:19) = lim n → + ∞ (cid:20)(cid:18) − q (cid:19) k u n k µ n + 1 q Z R N × R N F ( x, u n ( x )) f ( y, u n ( y )) u n ( y ) | x − y | N − α dx dy − Z R N × R N F ( x, u n ( x )) F ( y, u n ( y )) | x − y | N − α dx dy (cid:21) = lim n → + ∞ (cid:20)(cid:18) − q (cid:19) k u n k µ n + Z R N × R N F ( x, u n ( x )) ϕ ( y, u n ( y )) | x − y | N − α dx dy (cid:21) where ϕ ( y, u n ( y )) = q f ( y, u n ( y )) u n ( y ) − F ( y, u n ( y )) ≥ c ≥ lim n → + ∞ (cid:18) − q (cid:19) k u n k µ n = + ∞ and we obtain a contradiction. (cid:3) EMIRELATIVISTIC CHOQUARD EQUATIONS 25
Lemma 6.3.
There holds c = lim n → + ∞ c n . Proof.
Consider t n > t n u n ∈ N ; note that c n = E n ( u n ) ≥ E n ( t n u n ) = E ( t n u n ) − µ n t n Z R N u n | x | dx ≥ c − µ n t n Z R N u n | x | dx. (6.1)Now, take s n > s n u ∈ N n , then c = E ( u ) ≥ E ( s n u ) = E n ( s n u ) + µ n s n Z R N u | x | dx ≥ c n + µ n s n Z R N u | x | dx. (6.2)By (6.1) and (6.2) we obtain c ≥ c n + µ n s n Z R N u | x | dx ≥ c n ≥ c − µ n t n Z R N u n | x | dx, that is c − µ n t n Z R N u n | x | dx ≤ c n ≤ c . Note that { u n } n is bounded in view of Lemma 6.2. Hence, from Lemma 2.2, weget that R R N u n | x | dx stays bounded. Hence, to complete the proof, it is sufficient toshow that { t n } n is bounded.Assume, by contradiction, that t n → + ∞ then, by the fact that t n u n ∈ N , we have E ( t n u n )( t n u n ) = t n Q ( u n ) − Z R N × R N F ( x, ( t n u n )( x )) f ( y, ( t n u n )( y ))( t n u n )( y ) | x − y | N − α dx dy + t qn Z R N K ( x ) | u n | q dx = 0 . Hence, Q ( u n ) t q − n = 12 D ( t n u n )( t n u n ) t qn − Z R N K ( x ) | u n | q dx. Note that, in view of Sobolev embeddings and Lemma 6.2, R R N K ( x ) | u n | q dx staysbounded. Moreover, Lemma 6.2 and q > Q ( u n ) t q − n → . Hence D ( t n u n )( t n u n ) t qn is bounded. On the other hand, (1.4), (F3) and Fatou’s lemmaimply that D ( t n u n )( t n u n ) t qn = 2 1 t qn Z R N × R N F ( x, t n u n ( x )) f ( y, t n u n ( y )) t n u n ( y ) | x − y | N − α dx dy ≥ · q t qn Z R N × R N F ( x, t n u n ( x )) F ( y, t n u n ( y )) | x − y | N − α dx dy = q Z R N × R N F ( x,t n u n ( x )) t q/ n F ( y,t n u n ( y )) t q/ n | x − y | N − α dx dy → + ∞ , which is a contradiction. (cid:3) Proof of Theorem 1.9.
Suppose thatlim n → + ∞ sup z ∈ R N Z B ( z, | u n | dx = 0 . From Lion’s Concentration-Compactness principle we obtain u n → L t ( R N ) for all t ∈ (cid:18) , NN − (cid:19) . Recall that0 = E n ( u n )( u n ) = k u n k µ n − D ( u n )( u n ) + Z R N K ( x ) | u n | q dx, therefore k u n k µ n = 12 D ( u n )( u n ) − Z R N K ( x ) | u n | q dx. Applying the same computation as in (4.8) we easily get that D ( u n )( u n ) → n → + ∞ and by ( K ) we have (cid:12)(cid:12)(cid:12)(cid:12)Z R N K ( x ) | u n | q dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ | K | ∞ | u n | qq → n → + ∞ . Hence, k u n k µ n → n → + ∞ , and since0 ≤ (cid:0) [ u n ] + | u n | (cid:1) ≤ Q µ n ( u n )min n C (cid:0) N, (cid:1) − µ n C N, / , / , ess inf R N V − m o → (cid:8) C (cid:0) N, (cid:1) , ess inf R N V − m (cid:9) = 0we have that u n → H / ( R N ).In view of Lemma 6.1 we have E n ( u n ) ≥ E n (cid:18) r u n | u n | (cid:19) > β > β , and by Lemma 6.2lim sup n → + ∞ E n ( u n ) = lim sup n → + ∞ (cid:18) − D ( u n ) (cid:19) ≤ { z n } n ⊂ Z N such thatlim inf n → + ∞ Z B ( z n , √ N ) | u n | dx > . In view of Lemma 6.2, there is u ∈ H / ( R N ) \ { } such that u n ( · + z n ) * u in H / ( R N ) ,u n ( · + z n ) → u in L ( R N ) ,u n ( x + z n ) → u ( x ) for a.e. x ∈ R N . EMIRELATIVISTIC CHOQUARD EQUATIONS 27
Let w n = u n ( · + z n ) and fix any ϕ ∈ C ∞ ( R N ). Observe that E ( w n )( ϕ ) = E n ( u n )( ϕ ( · − z n )) + µ n Z R N u n ϕ ( · − z n ) | x | dx = µ n Z R N u n ϕ ( · − z n ) | x | dx. By [2, Lemma 2.5] and Hölder’s inequality we have that Z R N | u n ( x ) || ϕ ( x − z n ) || x | dx → n → + ∞ , hence E ( w n )( ϕ ) →
0. From Corollary 4.4 there follows that E ( w n )( ϕ ) → E ( u )( ϕ )thus u is a nontrivial critical point of E . In particular, u ∈ N . By (1.4), Lemma6.3 and Fatou’s Lemma we have c = lim inf n → + ∞ E n ( u n ) = lim inf n → + ∞ (cid:18) E n ( u n ) − q E n ( u n )( u n ) (cid:19) = lim inf n → + ∞ (cid:20)(cid:18) − q (cid:19) Q ( u n ) + (cid:18) q − (cid:19) µ n Z R N u n | x | dx + 1 q Z R N × R N F ( x, u n ( x )) (cid:0) f ( y, u n ( y )) u n ( y ) − q F ( y, u n ( y )) (cid:1) | x − y | N − α dx dy = lim inf n → + ∞ (cid:20)(cid:18) − q (cid:19) Q ( w n ) + (cid:18) q − (cid:19) µ n Z R N u n | x | dx + 1 q Z R N × R N F ( x, w n ( x )) (cid:0) f ( y, w n ( y )) w n ( y ) − q F ( y, w n ( y )) (cid:1) | x − y | N − α dx dy ≥ (cid:18) − q (cid:19) Q ( u ) + 1 q Z R N × R N F ( x, u ( x )) (cid:0) f ( y, u ( y )) u ( y ) − q F ( y, u ( y )) (cid:1) | x − y | N − α dx dy = E ( u ) − q E ( u )( u ) = E ( u ) ≥ c , where we used the weak lower semicontinuity of the norm Q ( · ) and the fact that µ n R R N u n | x | dx → E ( u ) = c and u ∈ H / ( R N )is a ground state solution for E . (cid:3) Ac k n ow l e d g e m e n t s
Federico Bernini and Simone Secchi are members of INdAM-GNAMPA. BartoszBieganowski was partially supported by the National Science Centre, Poland (GrantNo. 2017/25/N/ST1/00531).
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