Nonlinear Analysis. Infinitely many positive solutions for a Schrödinger Poisson system

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1 Nonlinear Analysis 74 (0) ontents lists available at ScienceDirect Nonlinear Analysis journal homepage: wwwelseviercom/locate/na Infinitely many positive solutions for a Schrödinger Poisson system Pietro d Avenia a,, Alessio Pomponio a, Giusi Vaira b a Dipartimento di Matematica, Politecnico di Bari, Via E Orabona 4, I-705 Bari, Italy b SISSA Via Beirut, -4, I-3404 Trieste, Italy a r t i c l e i n f o a b s t r a c t Article history: Received 8 May 0 Accepted 7 May 0 ommunicated by S Ahmad Keywords: Non-autonomous Schrödinger Poisson system Perturbation method We are interested in the existence of infinitely many positive solutions of the Schrödinger Poisson system u + u + V( x )φu = u p u, x, φ = V( x )u, x, where V( x ) is a positive bounded function, < p < 5 and V(r), r = x, has the following decay property: V(r) = a r + O m r m+θ with a > 0, m > 3, θ > 0 The solutions obtained are non-radial 0 Elsevier Ltd All rights reserved Introduction and main results In this paper, we consider the following nonlinear Schrödinger Poisson system u + u + V(x)φu = u p u, x, φ = V(x)u, x, (SP ) where p (, 5) and V : R is a positive bounded function This ind of problem has been introduced in [] and arises in an interesting physical context In fact, according to a classical model, the interaction of a charged particle with an electromagnetic field can be described by coupling the nonlinear Schrödinger and Maxwell equations and so it is also nown as the Schrödinger Maxwell system In our case, V(x) can be interpreted as a changing pointwise charge distribution There is a great deal of literature concerning this type of problem in different situations; see for example [ 5] The equations in (SP ) are the Euler Lagrange equations of the -functional G : H ( ) D, ( ) R, where G(u, φ) = u + u dx + V(x)φu dx 4 φ dx p + u p+ dx, and the critical points of G are the solutions of (SP ) Let us observe that the functional G is unbounded, both from above and from below, also modulo compact perturbations As we shall see in Section, for all u H ( ), the Poisson equation in P d Avenia and A Pomponio are supported by MIUR - PRIN Metodi variazionali e topologici nello studio di fenomeni non lineari G Vaira is supported by MIUR - PRIN Variational Methods and Nonlinear Differential Equations orresponding author Tel: ; fax: addresses: pdavenia@polibait (P d Avenia), apomponio@polibait (A Pomponio), vaira@sissait (G Vaira) X/$ see front matter 0 Elsevier Ltd All rights reserved doi:006/jna005057

2 5706 P d Avenia et al / Nonlinear Analysis 74 (0) (SP ) admits a unique solution φ u D, ( ) Hence we can reduce ourselves to the study of the one variable functional I : H ( ) R, defined by I(u) = G(u, φ u ) = u + u dx + V(x)φ u u dx u p+ dx 4 p + The critical points of I are the solutions of the problem u + u + V(x)φ u u = u p u (SP ) If u H ( ) is a solution of (SP ), then (u, φ u ) H ( ) D, ( ) is a solution of (SP ) and so we will loo for solutions of (SP ) In this paper we assume that V is a positive bounded radial function, that is V(x) = V( x ) = V(r) Moreover we assume that V satisfies the following condition: (V) there are constants a > 0, m > 3, θ > 0 such that V(r) = a r + O, m r m+θ as r + In what follows, without any loss of generality, we assume a = The main result of this paper can be stated as follows: Theorem If V satisfies (V), then the problem (SP ) has infinitely many non-radial positive solutions To prove Theorem we will construct solutions with large number of bumps near infinity In fact, since V(r) 0 as r +, the solutions of (SP ) can be approximated by using the solution U of the following limit problem u + u = u p, in, u > 0, in R 3, () u(x) 0, as x + For any positive integer, let us define (j )π (j )π P j = r cos, r sin, 0, j =,,, with r [r 0 log, r log ] for some r > r 0 > 0 and z r (x) = U Pj (x), j= where U Pj ( ) := U( P j ) If x = (x, x, x 3 ), we set u is even in x, x 3 ; H s = u H ( ) u(r cos θ, r sin θ, x 3 ) = u r cos θ + πj, r sin θ + πj j =,, Theorem is a direct consequence of the following result Theorem If V satisfies (V), then there exists an integer 0 > 0 such that for all 0, (SP ) has a positive solution u of the form u = z r + w, where r [r 0 log, r log ], w H s and, as +, w 0 The proof of Theorem relies on a Lyapunov Schmidt reduction This technique is almost standard in the perturbation methods, prevalently in the presence of a small parameter (see [6]) and it has been applied to Schrödinger Poisson type system by several authors (see, for example, [7,9,3]) Here we use as parameter, that is we use the number of bumps of the solutions in the construction of spie solutions for (SP ) This idea has been introduced by Wei and Yan [7] They consider the nonlinear Schrödinger equation u + V( x )u = u p, x 3

3 P d Avenia et al / Nonlinear Analysis 74 (0) with V(r) V 0 > 0 as r + and they prove the existence of infinitely many positive non-radial solutions for such equation by using the technique outlined above This method has also been applied for the study of different problems (see for example [8,9]) In our case, however, many technical difficulties arise due to the presence of the non-local term φ u and a more careful analysis of the interaction between the bumps is required A note on our result was published in [0] Notation and preliminaries Hereafter we use the following notations: H ( ) is the usual Sobolev space endowed with the standard scalar product and norm (u, v) = [ u v + uv]dx, u = u + u dx; D, ( ) is the completion of 0 (R3 ) with respect to the norm u = u dx; D, L q ( ), q + denotes the usual Lebesgue space with the standard norm u q ; for any ρ > 0 and for any z, B ρ (z) denotes the ball of radius ρ centered at z and B ρ = B ρ (0);,, i, are various positive constants which may also vary from line to line; A = O h means that there exists a constant > 0 such that A h Let us summarize some properties of φ u Lemma For every u H ( ), there exists a unique φ u non-negative and the following representation formula holds φ u (x) = x y V(y)u (y) dy Moreover: D, ( ) solution of φ = V(x)u Such a solution φ u is (i) there exist, > 0 independent of u H ( ) such that and φ u D, u, 5 V(x)φ u u u 4 5 ; (ii) if u H s, then φ u D s, where φ is even in x, x 3 ; D s = φ D, ( ) φ(r cos θ, r sin θ, x 3 ) = φ r cos θ + πj, r sin θ + πj j =,, (iii) for all v, v H ( ) there holds φ v φ v D, ( v + v ) v v, x 3 ; Proof The existence and uniqueness of φ u D, ( ) is a direct application of the Lax Milgram theorem Moreover the inequalities in (i) are well nown (see, for instance, [,6,]) and here we give the proofs only of (ii) and (iii) (ii) If g O(3) and u : R, let us set (T g u)(x) = u(gx) () Fixed g O(3), if T g u = u, we have that (T g φ u ) = T g (φ u ) = T g (V( x )u (x)) = V( gx )u (gx) = V( x )u (x) = φ u and then T g φ u = φ u

4 5708 P d Avenia et al / Nonlinear Analysis 74 (0) Since H s and D s are the sets of the fixed points with respect to the action () for all g = (g i,j ) O(3) with cos πj sin πj 0 i = j, i =, 3 g ij = or g = δ ij otherwise sin πj cos πj 0, 0 0 we conclude that if u H s, then φ u D s (iii) We have φ v φ v = (φ D, v φ v ) dx = V(x)(v v)(φ v φ v ) dx v v 6 5 φ v φ v D, v + v v v φ v φ v D, 5 5 ( v + v ) v v φ v φ v D, Analogously, the following lemma holds Lemma For any u, v H ( ), let φ be the solution in D, ( ) of φ = V(x)uv Then φ D, V(x)uv 6 u v 5 and for any z, w H ( ) V(x) φzw dx φ D, V(x)zw 6 u v z w () 5 Let α (0, ), by (V), for any x B αr and for i =,,, we have that V(x + P i ) = x + P i + O m x + P i m+θ (3) Moreover, for any x B αr and β > 0 x + P i = x β P i β + O P i Let us evaluate now the distance between the various P i, i =,, Indeed, for i j, by maing a simple computation, we find (i P i P j = r j)π sin (5) Finally, let us recall the following elementary inequalities which hold for all a, b, b, b R and p > : a + b p a p + b p and, if a, a + b p a + b p p a p (b b ) b p + b p b b, if p, b p + b p + b + b b b, if p > where the constant depends only on p As mentioned in the introduction, we denote by U the unique positive radial solution in H ( ) of the problem () This solution satisfies the following decay property (see []): lim r + U(r)rer = > 0, U (r) lim r + U(r) =, r = x, for some constant The function U is a critical point of the functional I 0 : H ( ) R defined as I 0 (u) = u u p+ dx p + Furthermore the solution U is nondegenerate (up to translations) (4) (6) (7)

5 P d Avenia et al / Nonlinear Analysis 74 (0) More specifically, there holds Lemma 3 Define the operator Q : H ( ) R as Q [v] := I 0 (U)[v, v] = v + v pu p v dx We denote by U j = U x j Then there hold: Q [U] = ( p) U < 0; Q [U j ] = 0, j =,, 3; Q [v] v for all v U, v U j, j =,, 3 For the proof we refer, for instance, to [6, Lemma 86] Finally, let us recall the following results (see [, orollary 3] and [3, Lemma 37]) Lemma 4 Let β and β be two positive numbers Then we have U β P i U β P j dx = O e (min{β,β }δ) P i P j, i j, where δ > 0 is any small number Lemma 5 For all i, j =,,, i j, we have U p P i U pj dx = e P i P j ( + o()) P i P j 3 Proof of Theorem The proof of Theorem relies on a Lyapunov Schmidt reduction Let and define Z i,j = U P i x j, i =,, and j =,, 3, W := w H s : U p P i Z i,j w dx = 0, i =,, ; j =,, 3 and P W the orthogonal projection onto W Our approach is to find first a solution w W of the auxiliary equation P W I (z r + w) = 0 and then to solve the remaining finite dimensional equation, namely the bifurcation equation (I P W )I (z r + w) = 0 In what follows we always assume that V satisfies (V) and m m r S := π β log, π + β log, where m is the constant in (V) and β > 0 is a small constant 3 The auxiliary equation In what follows, we find a solution of the auxiliary equation, namely we prove the following proposition Proposition 3 There exists an integer 0 > 0, such that for each 0, there is a map w : S H s, w = w(r), satisfying w W and P W I (z r + w) = 0 Moreover, there is a small σ > 0, such that w m +σ We begin with some estimates

6 570 P d Avenia et al / Nonlinear Analysis 74 (0) Lemma 3 There exists an integer 0 > 0, such that for each 0, there is a small σ > 0 such that I (z r ) m +σ Proof Let v H s, with v = Taing into account that U Pi are solutions of (), we have I (z r )[v] = V(x)φ zr z r v dx z p R 3 R 3 r U p P i v dx i= (I) (II) Let us evaluate separately the two terms By using Hölder inequality we obtain: / (I) = V(x)φ zr U Pi v dx i= R (V(x)φ zr U Pi ) dx v 3 i= /3 φ zr D, (V(x)U Pi ) 3 dx i= Now, let be α (0, ) By (3) and (4) and since U decays exponentially outside a ball we find (V(x)U Pi (x)) 3 dx = (V(x + P i )U(x)) 3 dx R 3 = (V(x + P i )U(x)) 3 dx + (V(x + P i )U(x)) 3 dx and r 3m, \ φ zr = φ D, zr dx = V(x)φ zr z r dx = V(x)φ zr U i= P i dx + V(x)φ zr U Pi U Pj dx 5/6 φ zr D, (V(x)U P i ) 6/5 dx + φ 5/6 zr (V(x)U D, Pi U Pj ) 6/5 dx As done in (3) we find (V(x)U P i ) 6/5 dx i= r 6m/5 Moreover by Lemma 4 and (5), since r S, we find 5/6 (V(x)U Pi U Pj ) 6/5 dx Hence φ zr D, r m Thus, since r S and m > 3, = e (δ) P ip j i= e (δ) P ip e (δ) π r (δ)(mβπ) (3) (I) r m m +σ

7 P d Avenia et al / Nonlinear Analysis 74 (0) Finally we consider (II) These estimates have been done in [7]; we setch the proof for the sae of completeness Let us define for all j =,, x Ω j := x = (x, x 3 ) R R : x, P j cos π P j For any x Ω : U p P i U p P U Pi Thus, by using [7, Lemma A] (II) = z p r U p P i v dx Ω i= U Pj v dx U p P Ω U Ω p p+ j= (p)(p+) p P UP Ω p p+ e ηrπ (p+) (p)(p+) p j= p p+ e ηrπ (p+) j= j= p+ p U Pj j= U P p η p U Pj p e (δ) P P j p p+ e π p+ (η+p(δ))( m π β) log p p+ + π (η+p(δ))(, m p+ π β) with η, δ > 0 small Since we obtain p p + + pm p + > m, (II) m +σ Putting together (I) and (II) we find 3 U Pj dx dx p p+ p p+ U Pj dx j= p p+ Now we are concerned with the invertibility of the operator L := P W I (z r ) : W W Ω v p+ dx p p+ p+ Lemma 33 For sufficiently large L is invertible and L In order to prove Lemma 33 let us decompose W = A B where A := i= P W U Pi ; B := A W Lemma 33 follows immediately after showing the next result Lemma 34 For sufficiently large there exist two positive constants, such that (a) I (z r )[u, u] u, for all u A; (b) I (z r )[u, u] u, for all u B

8 57 P d Avenia et al / Nonlinear Analysis 74 (0) Proof The proof is very similar to [3, Lemma 34] We setch it for the sae of completeness Let u A Then u = λ i P W U Pi, λ i R, i =,, i= For i =,,, P W U Pi W Hence we can write P W U Pi = U Pi ψ i, i =,,, where ψ i are given by ψ i = l,j Z l,j (U Pi, Z l,j ) Z l,j, l i, and the functions Z l,j satisfy Z l,j + Z l,j = pu p P l Z l,j Since for + we have P i P l +, we get (U Pi, Z l,j ) = o() as + This implies ψ i = o() as + for i =,, Applying the bilinear form given by I (z r ), using the fact that I (z r ) maps bounded sets onto bounded sets and that ψ i = o() we obtain that I (z r )[u, u] = I (z r ) λ i U Pi, λ i U Pi + o() i= i= Furthermore, by maing simple computations, reasoning as in the proof of Lemma 3 and by Lemma 3, we find I (z r )[u, u] = λ i I 0 (U P i )[U Pi, U Pi ] + o() ( p) λ i U P i + o() < i= i= Then I (z r ) is negative definite on A We now prove that I (z r ) is positive definite on B hoose an arbitrary u B and we assume, for simplicity, that u = We denote by ˆφ the solution of ˆφ = V(x)z r u Then I (z r )[u, u] = R3 u + u + V(x)φ zr u + V(x) ˆφz r u pz p r u dx Since u is bounded and reasoning as in Lemma 3 we find V(x)φ zr u dx = V(x)φ u z r dx = V(x)φ u U i= P i dx + V(x)φ u U Pi U Pj dx 5/6 V(x)U 6/5 6/5 P i dx + V(x)UPi U Pj dx i= = o(), for sufficiently large In the same way one can prove that for + V(x) ˆφz r u dx = o() As done in Lemma 3, it can be proved that z p r u dx = U p P i u dx + o() Hence I (z r )[u, u] = i= u + u p i= U p P i u dx + o() At this point, arguing, for example, as in [3], we have that I (z r )[u, u] > 0, and (b) follows 5/6

9 P d Avenia et al / Nonlinear Analysis 74 (0) We are now ready to prove Proposition 3 Proof of Proposition 3 Let us consider J(w) = I(z r + w), w W and expand it as follows: J(w) = I(z r ) + I (z r )[w] + I (z r )[w, w] + R zr (w) = J(0) + l(w) + Lw, w + R z r (w), where l(w) = I (z r )[w], Lw, w = I (z r )[w, w], and R zr (w) = V(x)φ w w dx + V(x)φ w z r w dx 4 [ ] z r + w p+ z p+ p(p + ) r z p r w (p + )z p r p + w dx Since l(w) is a bounded linear functional in W, by Riesz Theorem there exists an l W such that l(w) = l, w Now we want to find a critical point of J, that is a w W such that 0 = J (w) = l + Lw + R z r (w) (3) Since by Lemma 33 L is invertible, we can rewrite (3) in the following way w = A(w) := L l L R z r (w) Thus the problem of finding a critical point of J(w) is equivalent to find a fixed point of A To this end, let B := w W : w, m +σ where σ > 0 is small We have to prove that A(B) B and that A is a contraction in B Let w B By using Lemmas 3 and 33 A(w) L l + R z r (w) + m +σ R z r (w) Let us evaluate R z r (w) We denote by φ the solution in D, ( ) of φ = V(x)vw, then R z r (w) = sup R z r (w)[v] v = sup V(x)φ w wv dx + V(x) φz r w dx + V(x)φ w z r v dx v = + z r + w p pz p r w z p r v dx If p then, by () and (7), [ ] R z r (w) sup v = w + w p v dx Thus, since w B, A(w) ( w + w p ) w p m +σ + p(m +σ ) m +σ

10 574 P d Avenia et al / Nonlinear Analysis 74 (0) If, now, p > then, by using again () and (7), R z r (w) sup w + w + w p v dx w v = and then A(w) m +σ Hence, in both cases, A maps B into B Finally, let us prove that A is a contraction Let be w, w B Then A(w ) A(w ) L ( l w w + R z r (w ) R z r (w ) ) w m +σ w + R z r (w ) R z r (w ) Moreover we have R z r (w ) R z r (w ) = sup R z r (w )[v] R z r (w )[v] v = sup V(x)φ w w w v dx + V(x) φ z r w w dx v = R 3 + V(x) φ w φ w z r v dx + V(x) φ φ z r w dx R 3 + V(x) φ w φ w w v dx + z r + w p z r + w p pz p r (w w ) v dx, where φ i is the solution in D, ( ) of φ i = V(x)z r w i, i =, If now p, by (iii) of Lemma and (7) [ R z r (w ) R z r (w ) sup w v = m +σ w + w p + w p ] w w v dx R [ ] 3 + w m +σ (p)(m +σ ) w w (p)(m +σ ) w If p > by using again (iii) of Lemma and (7) R z r (w ) R z r (w ) sup w v = m +σ w + w p + w p + w + w w w v dx (m +σ ) w w In both cases A is a contraction since for large for s = and s = p s(m +σ ) 3 The reduced functional This section is devoted to solve the finite dimensional equation, namely the bifurcation equation To this aim, let us define the reduced functional F : S R such that for all r S F(r) = I(z r + w),

11 P d Avenia et al / Nonlinear Analysis 74 (0) where w = w(r, ) is the unique solution of the auxiliary equation By Proposition 3 let us recall that w m +σ By Lemma 3 and since I maps bounded sets onto bounded sets then F(r) = I(z r ) + I (z r )[w] + I (ξ)[w, w] = I(z r ) + O m+σ where σ > 0 is small Then by Proposition A the reduced functional is given by F(r) = 0 + B r + B log B m r m+ 3 U p P U Pi dx + O, m+σ where 0, B, B, B 3 are positive constants The problem max {F(r) : r S } i= has a solution since F is continuous on a compact set We have to show that this maximum is an interior point of S Let us denote with F the function F(r) := B r + B log B m r m+ 3 U p P U Pi dx i= By Lemmas 5 and A6 e πr log U p P U Pi dx e So we define i= F (r) := B r + B log B m r m+ 5 e πr, F (r) := B r + B log m r m+ For sufficiently large, F (r) F(r) F (r) in S e πr B 4 log πr and, moreover, we have m m F π + β log F π + β log > 0, m m F π β log F π β log < 0, (33) and m F π + β log F m π + β log Hence F possesses a critical point (a maximum point) m r = π + o() log, m log m > 0 in the interior of S Finally it is easy to chec that (33) is achieved by some r, which is in the interior of S, and so we infer that r is a critical point of F(r) As a consequence, we can conclude that z r + w(r ) is a solution of (SP ) This prove the existence of infinitely many non-trivial non-radial solutions of (SP ) In order to get positive solutions, it suffices to repeat the whole procedure for the functional I + (u) = u + 4 V(x)φ u u dx p + u + p+ dx, where u + = max{0, u} It can be checed that Proposition 3, Lemmas 3 and 33 and Proposition A can be applied to I + Therefore we get infinitely many non-radial and non-negative solutions for the problem u + u + V(x)φ u u = (u + ) p Keeping in mind that φ u > 0 when u 0 then the maximum principle allows to find infinitely many positive solutions of (SP ) of the form z r + w(r ), and so the proof of Theorem is concluded

12 576 P d Avenia et al / Nonlinear Analysis 74 (0) Appendix In this section we prove the following result Proposition A For a small σ > 0 there holds I(z r ) = 0 + B r + B log B m r m+ 3 U p P U Pi dx + O where 0, B, B, B 3 are positive constants i= m+σ, First we give some preliminary We recall that I(z r ) = zr + z r dx + V(x)φ zr z 4 r dx z r p+ dx p + By maing simple computations we find V(x)φ zr z r dx = V(x)φ UPi z i= r dx + V(x)φ zr U Pi U Pj dx = V(x)φ UPi U i= P i dx + V(x)φ UPi U P j dx + V(x)φ UPi U Pj U Pl dx + V(x)φ zr U Pi U Pj dx i= j l = V(x)φ UP U P dx + V(x)φ UPi U P j dx + V(x)φ UPi U Pj U Pl dx + V(x)φ zr U Pi U Pj dx i= j l (A) (A) Lemma A There holds: V(x)φ UP U P dx = B r m + O where B = U (x)u (y) dxdy xy m+θ log m+θ Proof Let α (0, ), we have V(x)φ UP U P dx = V(x + P )φ UP (x + P )U (x)dx R 3 = V(x + P )φ UP (x + P )U (x)dx + O(e (τ)r ) V(x + P )V(y + P ) = U (x)u (y)dxdy + O(e (τ)r ) x y V(x + P )V(y + P ) = U (x)u (y)dxdy + O(e (τ)r ) x y, By (3) and (4), since P = r, we have: V(x + P )V(y + P ) U (x)u (y)dxdy = x y r m = B r m + O U (x)u (y) dxdy + O x y m+θ log m+θ r m+θ

13 P d Avenia et al / Nonlinear Analysis 74 (0) Lemma A3 For a suitable σ > 0, we have i= j l V(x)φ UPi U Pj U Pl dx = O m+σ (A3) Proof By using Hölder inequality we obtain i= j l V(x)φ UPi U Pj U Pl dx 6 φ UPi D, UPj U 5 Pl dx i= j l 5 6 Then, by Lemma 4, V(x)φ UPi U Pj U Pl dx φ UP D, e (δ) P jp l i= j l j l (A4) Let us evaluate φ UP D, φ UP = φ D, UP dx = V(x)φ UP U P dx 5/6 φ UP D, V(x)U 6/5 P dx 5/6 φ UP D, V(x + P )U 6/5 dx R 3 φ UP D, V(x + P )U 5/6 6/5 dx + O(e (τ)r ), where 0 < α < and τ > 0 sufficiently small By (6), we have φ UP D, V(x + P )U 6/5 dx (I) 5/6 + O(e 5 6 (τ)r ) Using again (6) we find [ ] 6/5 (I) = x + P + O m x + P m+θ U /5 dx U /5 dx + O x + P 6m 5 x + P U /5 dx 6 5 (m+θ) = U /5 dx + O P 6 5 m P 6 5 (m+θ) = U /5 dx + O r 6 5 m r 6 5 (m+θ) Hence φ UP D, r + O m r Then from (A4) we obtain i= j l 5 + e 6 (τ)r m+θ V(x)φ UPi U Pj U Pl dx r + O m r m+θ r + O m r m+θ e (δ) P jp l j l

14 578 P d Avenia et al / Nonlinear Analysis 74 (0) Since and e (δ) P jp l = j l j= j= e (δ) P P j e (δ) rπ, e (δ) P P j recalling that r S, we find V(x)φ UPi U Pj U Pl dx = O i= j l m log m (δ)(mπβ) = O, m+σ with σ := δ(πβ m) + m( η) πβ, where η > 0 small is such that log η for large Therefore, since m > 3, σ > 0 if δ, β and η are sufficiently small Lemma A4 For a suitable σ > 0, we have V(x)φ zr U Pi U Pj dx = O m+σ Proof We compute V(x)φ zr U Pi U Pj dx = V(x)φ UPl U Pi U Pj dx l= (A ) + V(x)φ l,t U Pi U Pj dx, l t (A ) where φ l,t D, ( ) is the unique solution of φ = V(x)U Pl U Pt By (A3) it follows immediately that (A ) = O m+σ Now, using Lemma 4, (A5) and (A6), since r S, we find (A ) φ l,t D, 5/6 6/5 UPi U Pj dx l t 5/6 6/5 UPl U Pt dx 6/5 UPi U Pj dx l t e (δ) P lp t e (δ) P ip j where l t (δ) 4rπ e = O, m+σ σ := δ(4βπ 4m) + m 4βπ Hence, since m >, σ > 0 if δ > 0 and β > 0 are sufficiently small 5/6 (A5) (A6)

15 P d Avenia et al / Nonlinear Analysis 74 (0) Lemma A5 For a suitable σ > 0, we have V(x)φ UPi U P j dx = r m P P j + r m+σ O P P j j= j= Proof First of all, let us observe that V(x)φ UPi U P j dx = j= V(x)φ UP U P j dx (A7) If 0 < α <, we have V(x)φ UP U P j dx = V(x + P j )φ UP (x + P j )U (x)dx R 3 = V(x + P j )φ UP (x + P j )U (x)dx + O(e (τ)r ) V(x + P j )V(y + P ) = U (x)u (y)dxdy + O(e (τ)r ) x y + P j P V(x + P j )V(y + P ) = U (x)u (y)dxdy + O(e (τ)r ) x y + P j P We claim that x U (x)u (y) x y + P j P dxdy = O Indeed, as in [7, Lemma 3], since x y P j + P U (x) dx, x y + P j P we have P P j x U (x)u (y) x y + P j P dxdy U x U (x) (y) x y + P j P dx dy U (y) y P j + P dy P j P = O Analogously, we can prove that y U (x)u (y) x y + P j P dxdy = O P P j x y U (x)u (y) x y + P j P dxdy = O P P j (A8) P P j, (A9) (A0) Therefore, since P = P j = r, by (3) and (4), together with (A8) (A0), we have: V(x)φ UP U P j dx = U (x)u (y) r m x y + P j P dxdy + r m+σ O P P j Hence, by [7], we infer that V(x)φ UP U P j dx = r m P P j + r m+σ O and by (A7), we get the conclusion P P j,

16 570 P d Avenia et al / Nonlinear Analysis 74 (0) Lemma A6 There holds: P P i = log + o() πr i= Proof First of all, let us observe i= P P i = r i= sin (i)π = r sin iπ i= + + () 4 Then it is sufficient to prove that lim log sin iπ = π i= (A) Since, for every i =,, and s [i, i + ], then sin iπ sin sπ sin (i+)π = i+ i (i + )π sin, sin (i+)π ds Hence, adding on i, we have that sin π sin iπ i= and so lim log i= sin iπ i+ i sin sπ sin sπ ds i+ ds sin π sin sπ ds = 0 i sin iπ ds = sin iπ Therefore (A) follows, being lim log sin sπ ds = π lim log log tan π tan π = π We are now ready to prove Proposition A Proof of Proposition A By (A) and using Lemmas A A6 we find V(x)φ zr z 4 r dx = B r + B + m r m P i= P i r m+θ O + O P i= P i m+σ [ B = r + B ] log + o() + O m r m r [ m+σ B = r + B ] log + O m r m+ m+σ Then, by (A), we have I(z r ) = U p j= i= P j U Pi dx + = U p+ dx + U p P U Pi dx + i= [ B r + B ] log + O m r m+ m+σ z r p+ dx p + [ B r + B log + O m r m+ m+σ ] p + z r p+ dx

17 P d Avenia et al / Nonlinear Analysis 74 (0) As done in [7, Proof of Proposition A3], for all r S, one can prove z r p+ dx = U p+ dx + (p + ) U p P U Pi dx + O m+σ At the end we find I(z r ) = p + = 0 + B r m + B log r m+ B 3 i= U p i= P U Pi dx + O m+σ U p P U Pi dx + O m+σ U p+ dx + B r m + B log r m+ B 3 i= Note added in proof After our paper was finished, we learned that a comparable result was obtained independently by Li et al in [4] References [] V Benci, D Fortunato, An eigenvalue problem for the Schrödinger Maxwell equations, Topol Methods Nonlinear Anal (998) [] A Azzollini, oncentration and compactness in nonlinear Schrödinger Poisson system, J Differential Equations 49 (00) [3] A Azzollini, P d Avenia, A Pomponio, On the Schrödinger Maxwell equations under the effect of a general nonlinear term, Ann Inst H Poincaré Anal Non Linéaire 7 (00) [4] A Azzollini, A Pomponio, Ground state solutions for the nonlinear Schrödinger Maxwell equations, J Math Anal Appl 345 (008) [5] G erami, G Vaira, Positive solutions for some non autonomous Schrödinger Poisson systems, J Differential Equations 48 (00) [6] T D Aprile, D Mugnai, Solitary waves for nonlinear Klein Gordon Maxwell and Schrödinger Maxwell equations, Proc Roy Soc Edinburgh Sect A 34 (004) [7] T D Aprile, J Wei, Standing waves in the Maxwell Schrödinger equation and an optimal configuration problem, alc Var Partial Differential Equations 5 (006) [8] P d Avenia, Non-radially symmetric solutions of nonlinear Schrödinger equation coupled with Maxwell equations, Adv Nonlinear Stud (00) 77 9 [9] I Ianni, G Vaira, On concentration of positive bound states for the Schrödinger Poisson problem with potentials, Adv Nonlinear Stud 8 (008) [0] H Kiuchi, On the existence of a solution for elliptic system related to the Maxwell Schrödinger equations, Nonlinear Anal TMA 67 (007) [] H Kiuchi, Existence and stability of standing waves for Schrödinger Poisson Slater equation, Adv Nonlinear Stud 7 (007) [] D Ruiz, The Schrödinger Poisson equation under the effect of a nonlinear local term, J Funct Anal 37 (006) [3] D Ruiz, G Vaira, luster solutions for the Schrödinger Poisson Slater problem around a local minimum of the potential, Rev Mat Iberoam 7 (0) 53 7 [4] Z Wang, HS Zhou, Positive solution for a nonlinear stationary Schrödinger Poisson system in, Discrete ontin Dyn Syst 8 (007) [5] L Zhao, F Zhao, On the existence of solutions for the Schrödinger Poisson equations, J Math Anal Appl 346 (008) [6] A Ambrosetti, A Malchiodi, Perturbation Methods and Semilinear Elliptic Problems on R n, Birhäuser Verlag, 005 [7] J Wei, S Yan, Infinitely many positive solutions for the nonlinear Schrödinger equations in R N, alc Var Partial Differential Equations 37 (00) [8] L Wang, J Wei, S Yan, A Neumann problem with critical exponent in non-convex domains and Lin Ni s conjecture, Trans Amer Math Soc 36 (00) [9] J Wei, S Yan, Infinitely many solutions for the prescribed scalar curvature problem on S N, J Funct Anal 58 (00) [0] P d Avenia, A Pomponio, G Vaira, Infinitely many positive solutions for a Schrödinger Poisson system, Appl Math Lett 4 (0) [] MK Kwong, Uniqueness of positive solutions of u + u = u p in R N, Arch Ration Mech Anal 05 (989) [] X Kang, J Wei, On interacting bumps of semi-classical states of nonlinear Schrödinger equations, Adv Differential Equations 5 (000) [3] A Ambrosetti, E olorado, D Ruiz, Multi-bump solitons to linearly coupled systems of nonlinear Schrödinger equations, alc Var Partial Differential Equations 30 (007) 85 [4] G Li, S Peng, S Yan, Infinitely many positive solutions for the nonlinear Schrödinger Poisson system, ommun ontemp Math (00)