Existence of positive solutions to Schrödinger Poisson type systems with critical exponent

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1 Communications in Contemporary Mathematics Vol. 16, No ) pages) c World Scientific Publishing Company DOI: /S Existence of positive solutions to Schrödinger Poisson type systems with critical exponent Fuyi Li,, Yuhua Li, and Junping Shi,, School of Mathematical Sciences Shanxi University Taiyuan 00006, Shanxi, P. R. China Department of Mathematics College of William and Mary Williamsburg, VA , USA fyli@sxu.edu.cn yhli@sxu.edu.cn jxshix@wm.edu Received 24 March 201 Revised 27 April 2014 Accepted 6 May 2014 Published 14 July 2014 The existence of positive solutions to Schrödinger Poisson type systems in with critically growing nonlocal term is proved by using variational method which does not require usual compactness conditions. A key ingredient of the proof is a new Brézis Lieb type convergence result. Keywords: Schrödinger Poisson type system; critical exponent; variational method; Brézis Lieb type convergence lemma. Mathematics Subject Classification 2010: 5J50, 5J47 1. Introduction The Schrödinger Poisson system is a standard model in quantum mechanics describing electrons moving on a positive charged background [14, 22]. In this paper, we consider positive solutions to the following nonlinear Schrödinger Poisson type system { u + bu + qφ u u = fu) in, 1.1) φ = u 5 in, where b 0andq Rare parameters. The investigation of 1.1) is motivated by recent studies of Schrödinger Poisson system { u + bu + qφgu) =fu) in, 1.2) φ =2Gu) in,

2 F.-Y.Li,Y.-H.Li&J.-P.Shi where gt) C t + t s )forsomes [1, 4) see, for example, [5, 19]). In [19], the system 1.2) was studied by using monotonicity technique for b>0andq 0. For the case of q = ±1, the system 1.2) on a bounded domain Ω was considered in [5]. Most of them focus on the case of b>0,q >0andgt) =t, thatis { u + bu + qφu = fu) in, φ = u 2 in 1.) see, for example, [1, 6, 7, 11, 1, 16, 18, 2, 24, 27, 28, 2]). For b = q =1 and ft) = t s 1 t, the existence of a positive radial solution of 1.) fors 2, 5) was proved in [12, 2], while the nonexistence was also shown for s 2. Multiple solutions to 1.) werealsopossibleforthecaseofq>0andft) = t s 1 t see [, 2]). These results can be extended to the case of more general f. In[2, 6], the following conditions on f were assumed: f 1 ) f CR, R) and < lim inf ft) ft) t 0 + t lim sup t 0 + t = m <0; ft) f 2 ) lim sup t t 0; 5 f ) there exists α>0 such that F α) = α 0 ft)dt > 0 and it was shown that system 1.) with b =0andq>0has a positive radial solution for q 0,q 0 )forsomeq 0 > 0. Related results for more general f were also obtained in [1, 28, 2], and solutions of Schrödinger Poisson systems with nonconstant potential where b is a function of x) wereconsideredin[17, 20, 21, 0, 1]. However, these aforementioned papers all considered the Schrödinger Poisson type system with subcritical nonlocal term which assumes that lim t Gt)/t 5 =0. To the best of our knowledge, the Schrödinger Poisson system with critical exponent Gt) = t 5 was only studied by [4]. In [4], a Schrödinger Poisson type system with a critically growing nonlinearity on a bounded domain was considered, and the equation studied there was in the following form: u = λu + qφ u u in B R, φ = q u 5 in B R, 1.4) u = φ =0 on B R, where B R is a ball in R n with radius R. Notethats = 5 is the critical exponent for the Sobolev embedding, and the critically growing nonlinearity in 1.4) appears in a nonlocal way as the second equation can be solved by a Green s function. In this paper, H 1 )andd 1,2 ) are the usual Sobolev spaces, and the norms and inner products are defined by here b>0 is a fixed parameter) ) 1/2 u H1 ) = [ u 2 + bu 2 ], u, v) = [ u v + buv], u,v H 1 ), ) 1/2 u D 1,2 ) = u 2, u, v) = u v, u, v D 1,2 )

3 Existence of positive solutions to Schrödinger Poisson type systems For convenience, we define H b = Hr 1 ) for the case b>0, and H b = Dr 1,2 ) when b = 0, where Hr 1 ) and Dr 1,2 ) denote the space of radial functions of H 1 )andd 1,2 ) respectively. Indeed we consider the system 1.1) inthe subspace Hr 1 )ofh 1 )whenb>0, or in the subspace Dr 1,2 )ofd 1,2 ) when b = 0. Throughout the paper, we use as the norm of H b,and, ) asthe inner product in H b. In this paper we consider the system 1.1) which is the critical exponent case for, and we assume the following conditions for the nonlinearity f which are weaker than the ones in previous work: H 1 ) f CR +, R + ) and lim ft) t 0 + bt+t =0,hereR 5 + =[0, + ); ft) H 2 ) lim t t =0; 5 H ) there is a function z H b such that F z) > b z 2,whereFz) = z 0 ft)dt; H 4 ) there exist r 4, 6),A>0,B >0 such that F t) At r Bt 2 for t 0. Sometimes the system 1.1) is labeled in two cases: b>0 positive mass), and b = 0 zero mass). If b = 0, then the condition H 1 ) is equivalent to lim ft) t 0 + t =0; 5 and if b>0, then the condition H 1 ) is equivalent to lim ft) t 0 + t =0.Inour result, we consider both the positive and zero mass cases. We also remark that the F t) condition H ) is weaker than lim t t =. Our main results in this paper are as follows. Theorem 1.1. Suppose that b 0 is fixed. Then there exists q 0 > 0 such that for any q [0,q 0 ), the system 1.1) possesses at least one positive radially symmetric solution u, φ) in H b D 1,2 ) if the conditions H 1 ), H 2 ) and H ) hold. For q = 1, the system 1.1) possesses at least one positive radially symmetric solution u, φ) in H b D 1,2 ) if the conditions H 1 ), H 2 ) and H 4 ) hold. Theorem 1.1 appears to be the first existence result for the system 1.1) which is the critical exponent case. For the subcritical case, { u + bu + qφ u s 1 u = fu) in, 1.5) φ = u s+1 in, where b>0ands [1, 4), we also have the following similar results. Theorem 1.2. Suppose that b>0 is fixed and s [1, 4). Thereexistsq 0 > 0 such that for any q [0,q 0 ), the system 1.5) possesses at least one positive radially symmetric solution u, φ) in H b D 1,2 ) if the conditions H 1 ), H 2 ) and H ) hold. For any q, 0), the system 1.5) possesses at least one positive radially symmetric solution u, φ) in H b D 1,2 ) if the conditions H 1 ) and H 2 ) hold. We remark that the results for the critical and subcritical cases appear to be similar, but the subcritical case has been studied in many papers for the case of q>0 though not the case of q<0. The critical case has not been studied in either q>0orq<0 case. From the technical point of view, there are several difficulties to

4 F.-Y.Li,Y.-H.Li&J.-P.Shi prove our results for the critical exponent case. Firstly, our result does not assume the classical Ambrosetti Rabinowitz hypothesis f 4 ) there exists µ>2 such that 0 <µft) ft)t for all t>0 and this makes it difficult to prove the boundedness of Palais Smale PS) sequences. Secondly, the critically growing nonlinearity in the equation sets an obstacle when showing the convergence. Lastly, for the zero mass case, the space D 1,2 )can only be embedded into L 6 ), which makes it difficult to deal with the integral φ u 5 in the energy function. Our result can be regarded as a generalization of the classical result in [8] for the semilinear scalar equation u = fu) in. 1.6) Note that when b =0andq =0,1.1) is reduced to 1.6). Then according to Theorem 1.2, we have the following result. Corollary 1.. If the conditions H 1 )with b = 0), H 2 ) and H ) hold, then Eq. 1.6) possesses at least one positive solution. Since Corollary 1. is a classical result proved in [8] for the zero mass case, then Theorem 1.2 is an extension of the classical result to the nonlocal case. From the Gagliardo Nirenberg Sobolev inequality, D 1,2 )canbecontinuously embedded into L 6 ), hence we can equivalently define D 1,2 )={u L 6 ): u L 2 )}. In the following we frequently use the inequality: and if u H 1 ), S u 2 L 6 ) u 2 D 1,2 ) 1.7) S u 2 L 6 ) u 2 D 1,2 ) u 2 H 1 ). 1.8) Note that we can choose the best embedding constant S here see [26, 29]). We denote by p the usual L p ) norm for p 1. In this paper, we only consider positive solutions to 1.1), then we may assume that ft) =0fort<0. In Sec. 2, we recall some basic setup and some useful lemmas. In Sec., we consider the case of q>0 and prove the first half of Theorem 1.1. We consider the case of q<0and prove the second half of Theorem 1.1 in Sec. 4. At the end of Sec. 4 we show that how proofs for the critical case can be adapted to the subcritical case. 2. Preliminaries In this section, we first provide the basic variational setup for 1.1), and we follow it with several technical ingredients in preparation of the proof of our main results. In Sec. 2.2, we prove a Brézis Lieb type convergence lemma which is the key in the convergence proof. Some other related convergence results and a monotonicity technique method are recalled in Secs. 2. and 2.4 respectively. In Sec. 2.5, a new Pohozaev identity which contains terms depending on the norms of functions in H b is proved. Note that the Brézis Lieb type convergence lemma and the

5 Existence of positive solutions to Schrödinger Poisson type systems Pohozaev identity with norm terms are new techniques which may be useful in other applications Setup For a fixed u H b, the second equation of 1.1) is a Poisson equation which is uniquely solvable for φ. Then the system 1.1) can be reduced to the first equation with φ represented by the solution of the Poisson equation. This is the basic strategy of solving 1.1). To be more precise about the solution φ of the Poisson equation, we have the following lemma for the critical case which is well known for the subcritical case of 1.5) with1 s<4see[2, 12, 2]). Lemma 2.1. For every u L 6 ), there exists a unique φ u D 1,2 ) which is the solution of Moreover, φ = u 5 in. 2.1) i) φ u 2 D 1,2 ) = φ u u 5 ; ii) φ u x) > 0 for every x ; iii) for any θ>0,φ uθ = θ 2 φ u ) θ, where u θ ) =u /θ); iv) for any t>0, φ tu = t 5 φ u ; v) for any u L 6 ), φ u D 1,2 ) S 1/2 u 5 6 and φ u u 5 S 1 u 10 6, where S is defined in 1.7); vi) if u is radially symmetric, so is φ u ; vii) if u n uin L 6 ) and u n u a.e. in as n, then φ un D 1,2 ); φ u in viii) if u n u in L 6 ) as n, then φ un u n 5 φ u u 5. φ u in D 1,2 ) and φ un Proof. The existence and uniqueness of φ u follow from the Lax Milgram theorem. The conclusions i), ii), iv) and vi) are clear from simple calculation. iii) It follows from 2.1) that φx/θ) = ux/θ) 5 = u θ x) 5 in.so θ 2 φx/θ)) = u θ x) 5. This implies that φ uθ ) =θ 2 φ u /θ). v) For any u L 6 ), φ u 2 D 1,2 ) = φ u u 5 φ u 6 u 5 6 S 1/2 φ u D 1,2 ) u

6 F.-Y.Li,Y.-H.Li&J.-P.Shi and φ u u 5 φ u 6 u 5 6 S 1/2 φ u D 1,2 ) u 5 6 S 1 u vii) For any v D 1,2 ), v L 6 ) from the Sobolev embedding. Since u n uin L 6 )andu n u a.e. in as n,then u n 5 u 5 in L 6/5 ). So φ un,v)= u n 5 v u 5 v =φ u,v). Therefore, φ un φ u in D 1,2 ). viii) For any v D 1,2 ), since φ un φ u,v) = u n 5 u 5 )v R 5 u + θu n u) 4 u n u v 40 u n u 4 6 ) u n u 6 v 6 40S 1/2 u n u 4 6 ) u n u 6 v D 1,2 ), where θ [0, 1], φ un φ u in D 1,2 )andthenφ un φ u in L 6 ). Hence φ un u n 5 φ u u 5. The results in Lemma 2.1 imply that the system 1.1) is reduced to a nonlocal semilinear elliptic equation u + bu + qφ u u u = fu), in, 2.2) where φ u is defined in Lemma 2.1. We look for positive solutions of 2.2) inthespaceh b. Define a functional J q in the space H b by J q u) = 1 2 u R q φ u u 5 F u), u H b. 2.) Then from H 1 ), H 2 )andh )wehavethatj q is well defined on H b and is of C 1 class for any q R, and J qu),v =u, v)+q φ u u uv fu)v, u, v H b. It is standard to verify that a critical point u of the functional J q corresponds to a weak solution u, φ u )of1.1). Hence in the following, we consider critical points of J q using variational method

7 Existence of positive solutions to Schrödinger Poisson type systems 2.2. ABrézis Lieb type convergence lemma To prove the convergence of nonlocal term in the critically growing Schrödinger Poisson system 1.1), we need the following key lemma which is inspired by the Brézis Lieb convergence lemma see [10]). Lemma 2.2. Let r 1 and let Ω be an open subset of R N. Suppose that u n u in L r Ω), and u n u a.e. in Ω as n, then for p [1,r], as n, u n p u n u p u p 0 in L r/p Ω), 2.4) u n p 1 u n u n u p 1 u n u) u p 1 u 0 in L r/p Ω). 2.5) Proof. If p = r, the conclusions are proved in [10]. If p =1<r,since u n u n u u 2 u and u n u a.e. in Ω as n, by Lebesgue s Dominated Convergence Theorem, we have u n u n u u 0inL r Ω). In the following, we assume that p 1,r). Let v n = u n u. Thenv n 0 in L r Ω) and u n = v n + u. From the intermediate value theorem, there exists θ L Ω) and 0 θ 1 such that v n + u p v n p = p v n + θu p 1 u 2 p 1 p v n p 1 u + u p ). For any ε>0, by using Young s inequality we obtain that, there exists C ε > 0such that v n + u p v n p u p ε v n p + C ε u p. We consider the sequence defined by h n =max{ v n + u p v n p u p ε v n p, 0}, which satisfies h n 0 a.e. in Ω, 0 h n C ε u p L r/p Ω). Then by Lebesgue s Dominated Convergence Theorem, we have h r/p n 0. From the definition of h n, it follows that Ω v n + u p v n p u p h n + ε v n p. Thus, we obtain the following inequality lim sup v n + u p v n p u p r/p Cε, n Ω where C is a positive constant so that Ω v n r C. Thisimpliesthat v n + u p v n p u p 0 in L r/p Ω), which is equivalent to 2.4)

8 F.-Y.Li,Y.-H.Li&J.-P.Shi For 2.5), the case of p = 1 is trivial as now u n p 1 u n u n u p 1 u n u) u p 1 u =0. If p 1,r], then from the intermediate value theorem, there exists θ L Ω) and 0 θ 1 such that v n + u p 1 v n + u) v n p 1 v n = p v n + θu p 1 u 2 p 1 p v n p 1 u + u p ). Then we can follow a similar proof as in the last paragraph to prove 2.5). By using the basic convergence result in Lemma 2.2, wehavethefollowing convergence estimates which are also related to φ u. Lemma 2.. If u n uin L 6 ) and u n u a.e. in, then as n, u n 5 u n u 5 u 5 0 in L 6/5 ), 2.6) u n u n u n u u n u) u u 0 in L /2 ), 2.7) φ un φ un u φ u 0 in D 1,2 ), 2.8) φ un u n 5 φ un u u n u 5 φ u u ) Proof. The convergences in 2.6)and2.7) follow from Lemma 2.2.Letv n = u n u. Since for every w D 1,2 ), φ vn+u φ vn φ u,w = w v n + u 5 v n 5 u 5 ) w 6 v n + u 5 v n 5 u 5 6/5, then φ vn+u φ vn φ u 0, in D 1,2 ), which implies 2.8). Since u n u in L 6 )andu n u a.e. x, then u n u 0inL 6 )andu n u 0a.e.x. By Lemma 2.1vii), φ un u 0 in D 1,2 ). Therefore, as n, φ un u n 5 φ un u u n u 5 φ u u 5 R = [φ un φ un u φ u ] u n 5 + φ un u[ u n 5 u n u 5 u 5 ] R + φ un u u 5 + φ u u n 5 φ u u

9 Existence of positive solutions to Schrödinger Poisson type systems 2.. Convergence of nonlinear terms For the convergence of nonlinear terms, we first recall the following lemma from [8]. Lemma 2.4. Let N. Every radial function u in D 1,2 R N ) is almost everywhere equal to a function U, continuous for x 0, such that where C N only depends on N. Ux) C N x 2 N)/2 u D 1,2 R N ), x 1, The convergence of fu n )u n fu)u and F u n ) F u) needs the following compactness result due to [25], see also [8]. Theorem 2.5. Let P, Q : R R be two continuous functions satisfying P s) Qs) 0, s. Let {u n } be a sequence of measurable functions on R N such that sup Qu n ) < n R N and P u n ) v a.e. on R N. Then for any bounded Borel set B one has P u n ) v 0. B If one further assumes that P s) Qs) 0, s 0 and u n x) 0 as x, uniformly with respect to n, then {P u n )} converges to v in L 1 R N ). Lemma 2.6. Suppose that H 1 ) and H 2 ) hold. If u n uin H b, then F u n ) F u) and fu n )u n fu)u. Proof. Let P s) =F s) andqs) = 1 2 bs s6. By the conditions H 1 ), H 2 ), Lemma 2.4 and Theorem 2.5, the conclusion of Lemma 2.6 holds Monotonicity technique Here we recall a monotonicity method due to Struwe [26] and Jeanjean [15] which will be used in the proof. The version here is from [15]

10 F.-Y.Li,Y.-H.Li&J.-P.Shi Theorem 2.7. Let X, ) be a Banach space and I R + be an interval. Consider the family of C 1 functionals on X J λ = A λb, λ I, with B nonnegative and either Au) or Bu) as u and such that J λ 0) = 0. For any λ I we set Γ λ = {γ C[0, 1],X):γ0) = 0,J λ γ1)) < 0}. If for every λ I the set Γ λ is nonempty and c λ = inf max J λγt)) > 0, γ Γ λ t [0,1] then for almost every λ I there is a sequence {u n } X such that i) {u n } is bounded; ii) J λ u n ) c λ ; iii) J λ u n) 0 in the dual X 1 of X A generalized Pohozaev identity Pohozaev type integral identities have been used in previous work as an important tool see [8, 18]). Here we prove a generalized Pohozaev identity which involves functions of norms of solutions. Lemma 2.8. If u H b is a weak solution of { a1 u) u + a 2 u)u + a u)φ u p 1 u = λfu) in, φ = u p+1 in, 2.10) where p [1, 4], a 1,a 2,a CH b, R) and a 1 u) 0for all u 0, then the following Pohozaev type identity holds: 1 2 a 1u) u 2 + R 2 a 2u) u R 2p +1) a u) φ u p+1 =λ F u). 2.11) Proof. By the assumption, a 1 u),a 2 u),a u) are constants for a given u H b. Since u H b is a weak solution of 2.10), by the elliptic standard regularity results, we have u W 2,6/5 loc ). The first equation of 2.10) can be rewritten to a 1 u) u =λfu)/u a 2 u) a u)φ u p 1 )u. By the conditions H 1 )and H 2 ), fu) C u 5 + u ) for some positive constant C so fu)/u L /2 loc R ). Since φ, u L 6 ), we have that φ u p 1 L /2 loc R )aslongasp [1, 4]. Now the Brézis Kato theorem [9] implies that u L r loc R ) for any r [1, ), and consequently u W 2,r loc R ) for any r [1, ) fromthel p estimates. Again the

11 Existence of positive solutions to Schrödinger Poisson type systems elliptic regularity theory implies that u, φ C 2 ). It follows that, for R>0and B R = {x : x <R}, wehave ux u) = 1 u 2 1 x u 2 + R u 2, B R 2 B R R B R 2 B R ux u) = u 2 + R u 2, B R 2 B R 2 B R φ u p 1 ux u) = 1 u p+1 x φ B R p +1 B R φ u p+1 + p +1 B R fu)x u) = B R F u)+r B R F u), B R B R φx φ) = 1 2 B R φ 2 1 R R φ u p+1, p +1 B R B R x φ 2 + R 2 B R φ 2. Multiplying the first equation of 2.10) by x u, multiplying the second equation by x φ, integratingonb R, and using the estimates above we get that a 1 u) 1 u 2 1 x u 2 + R ) u 2 2 B R R B R 2 B R + a 2 u) u 2 + R ) u 2 2 B R 2 B R + a u) 1 u p+1 x φ φ u p+1 + R ) φ u p+1 p +1 B R p +1 B R p +1 B R = λ F u)+λr B R F u) B R and u p+1 x φ) = 1 φ 2 1 x φ 2 + R B R 2 B R R B R 2 B R φ 2. It follows that a 1 u) 1 u 2 1 x u 2 + R 2 B R R B R 2 + a 2 u) u 2 + R ) u 2 2 B R 2 B R ) u 2 B R

12 F.-Y.Li,Y.-H.Li&J.-P.Shi 1 + a u) φ x φ 2 2p +1) B R p +1)R B R φ u p+1 + R ) φ u p+1 p +1 B R p +1 B R = λ F u)+λr F u). B R B R Letting R, we obtain that a 1 u) 1 ) u 2 2 R 2 a 2u) u a u) φ 2 ) φ u p+1 2p +1) R p +1 By using that R φ 2 = R φ u p+1,weobtain2.11). R φ 2 2p +1) B R = λ F u).. TheCaseofPositiveq In this section, we consider the case of q>0for1.1), and we assume that the conditions H 1 ), H 2 )andh ) are satisfied. Using the setup in Sec. 2.1, we seek for critical points of the function J q defined in 2.). To overcome the difficulty of finding bounded PS sequences for the associated functional J q, following [16, 18], we use a cut-off function ψ C R +, [0, 1]) satisfying ψt) =1, t [0, 1], ψt) =0, t [2, ), ψ 2 and study the following modified functional Jq T : H b R defined by Jq T u) =1 2 u qh T u) φ u u 5 F u), u H b, where for every T>0, ) u 2 h T u) =ψ. With this penalization, for T > 0 sufficiently large and q sufficiently small, we are able to find a critical point u of Jq T such that u T and so u is also a critical point of J q. We will apply the monotonicity method described in Theorem 2.7. Inourcase, X = H b, Au) = 1 2 u qh T u) φ u u 5, Bu) = F u). T

13 Existence of positive solutions to Schrödinger Poisson type systems So the perturbed functional which we shall study is Jq,λu) T = 1 2 u qh T u) φ u u 5 λ F u) and Jq,λ T ) u),v =u, v)+qh T u) φ u u uv + q ) u 2 5T 2 ψ T 2 u, v) φ u u 5 λ fu)v..1) Lemmas.1 and.2 below imply that Jq,λ T satisfies the conditions of Theorem 2.7. First we show that the path set Γ λ as defined in Theorem 2.7 is not empty. Lemma.1. For every λ I [1/2, 1], q 0 and T>0, define the path set Γ λ = {γ C[0, 1],H b ):γ0) = 0,Jq,λ T γ1)) < 0}. Let z H b be the function defined in H ). Choose θ>0 satisfying ) θ R 2 F z) b z 2 =2 z ) φ z z 5. R 5 R 2 If q [0, 1/θ 4 ], then Γ λ. Proof. Let λ I. Setw ) =z /θ). Define γ :[0, 1] H b in the following way { 0, t =0, γt) = w /t), t 0, 1]. It is easy to see that γ is a continuous path connecting 0 and w. Moreover, we have that Jq,λγ1)) T = 1 2 w b w qh T w) φ w w 5 λ F w) 1 2 θ z θ b z qθ5 φ z z 5 1 R 2 θ F z) 1 2 θ z θ b z θ φ z z 5 1 R 2 θ F z) = 1 ) 4 R θ F z) b z 2 < 0. Secondly we show that the critical level c λ is uniformly bounded from below by a positive lower bound for all λ I. Lemma.2. Suppose that λ I, q [0, 1/θ 4 ] and T>0, and define c λ = inf max J q,λ T γt)). γ Γ λ t [0,1] Then there exists a constant c>0 such that c λ c for all λ I

14 F.-Y.Li,Y.-H.Li&J.-P.Shi Proof. From the conditions H 1 )andh 2 ), there exists C 0 > 0 such that ft)t 1 4 b t 2 + C 0 t 6 and F t) 1 4 b t 2 + C 0 t 6, t R..2) Then for any u H b and λ I, wehave Jq,λu) T 1 2 u qh T u) φ u u 5 1 R 4 b u 2 C 0 u u R b u 2 C 0 u 6. By the Sobolev s embedding theorem, we conclude that there exists ρ>0such that Jq,λ T u) > 0 for any λ I and u H b with 0 < u ρ. Inparticular,for u = ρ, we have Jq,λ T u) c>0. For any given λ I and γ Γ λ, by the definition of Γ λ, we have γ1) >ρ. According to the continuity of γt), we then deduce that there exists t γ 0, 1) such that γt γ ) = ρ. Therefore, for any λ I, c λ inf γ Γ λ J T q,λ γt γ)) c>0. Now we can show that the modified functional Jq,λ T satisfies the PS condition. Lemma.. Assume that q and T satisfy qs 6 T 8 1,.) where S is defined in 1.7). Then for any λ I, each bounded PS sequence of the functional Jq,λ T admits a convergent subsequence. Proof. Let λ I and let {u n } be a bounded PS sequence of Jq,λ T,thatis,{u n} and {Jq,λ T u n)} are bounded, and Jq,λ T ) u n ) 0inH b,whereh b is the dual space of H b.since{u n } is bounded, we may assume that there exists u H b such that It follows from Lemma 2.6 that u n u in H b, u n u a.e. on. fu n )u n u) 0. Since u n uin H b, in view of the Sobolev s embedding theorem, we may assume that u n u in L 6 ), φ un φ u in D 1,2 r R )

15 Existence of positive solutions to Schrödinger Poisson type systems Let v n = u n u. Since v n 0inL 6 )and u uv n 0inL 6/5 ), then φ un u uv n 0. By using Lemma 2., we obtain that φ un u n u n u n u) = φ un v n 5 + φ un u uv n + o1) R = φ un v n 5 + o1) R = φ vn v n 5 + φ u v n 5 + o1) R = φ vn v n 5 + o1). Thus, Jq,λ T ) u n ),u n u =u n,u n u)+qh T u n ) φ un u n u n u n u) + q un 2 ) 5T 2 ψ T 2 u n,u n u) φ un u n 5 λ fu n )u n u) = 1+ q un 2 ) ) 5T 2 ψ T 2 φ un u n 5 u n,u n u) R + qh T u n ) φ un u n u n u n u)+o1) R = 1+ q un 2 ) ) 5T 2 ψ T 2 φ un u n 5 v n 2 + qh T u n ) φ vn v n 5 + o1), and then 1+ q un 2 ) ) 5T 2 ψ T 2 φ un u n 5 v n 2 + qh T u n ) φ vn v n 5 0. From Lemma 2.1 and.), we have q un 2 ) 5T 2 ψ T 2 φ un u n 5 64 R 5 qs 6 T 8 1 2, it follows that v n 0. This implies that u n u in H b. The proof is completed. From the results in Lemmas.1. and Theorem 2.7, we can now show the existence of a critical point for the modified functional J T q,λ. Lemma.4. Suppose that q and T satisfy.). Then for almost every λ I, there exists u λ H b such that J T q,λ ) u λ )=0and J T q,λ uλ )=c λ. Proof. By using the results in Lemmas.1. and Theorem 2.7, for almost every λ I, there exists a bounded sequence {u λ n} H b such that J T q,λ uλ n) c λ and

16 F.-Y.Li,Y.-H.Li&J.-P.Shi Jq,λ T ) u λ n) 0. By Lemma., we can assume that there exists u λ H b such that u λ n uλ in H b, then the assertion follows. According to Lemma.4, there exist sequences {λ n } I with λ n 1 and {u n } H b such that J T q,λ n u n )=c λn, J T q,λ n ) u n )=0. The following lemma shows that this critical sequence {λ n,u n )} satisfies u n T for all n, which is the key for this paper. Lemma.5. Let u n be a critical point of Jq,λ T n at the energy level c λn, where {λ n } I. Then there exists T 0 > 0 sufficiently large so that for any T T 0, there exists q T = qt ) satisfying { } 5 qt ) min 128 S6 T 8,θ 4,.4) such that for any q [0,q T ), subject to a subsequence, u n T for all n N. Proof. We define and a 1 u) = 1+ q 5T 2 ψ a 2 u) =b 1+ q 5T 2 ψ ) u 2 ) T 2 φ u u 5, ) u 2 ) T 2 φ u u 5 ) u 2 a u) =qh T u) =qψ. By viii) of Lemma 2.1, a 1,a 2,a CH b, R) hence the assumptions in Lemma 2.8 are satisfied. Since λ n,u n ) satisfies that Jq,λ T n ) u n ) = 0, that is 2.10) withλ, u, φ) = λ n,u n,φ un )witha 1,a 2,a defined above, then it follows from 2.11) in Lemma 2.8 that the following Pohozaev type identity holds: 1+ q un 2 ) ) 1 5T 2 ψ T 2 φ un u n 5 u n 2 + ) R 2 R 2 b u n qh T u n ) φ un u n 5 =λ n F u n )..5) By using Jq,λ T n u n )=c λn,wehavethat 1 2 u n qh T u n ) φ un u n 5 λ n F u n )=c λn..6) T 2

17 Existence of positive solutions to Schrödinger Poisson type systems Combining.5) and.6), we obtain that 1 u n 2 1 q un 2 ) ) 2 R 10T 2 ψ T 2 φ un u n 5 u n 2 =c λn qh T u n ) φ un u n 5 + bq un 2 ) 10T 2 ψ T 2 φ un u n 5 u n 2..7) We now estimate the right-hand side of.7). By the min max definition of the mountain pass level, Lemma.1 and H ), we have that c λn max J q,λ T t [0,1] n γt)) { 1 max t [0,1] 2 tθ z bt θ z 2 2 t θ λ n + max t [0,1] 1 10 qψ 1 2 θ z θ } F z) tθ z bt θ z 2 ) 2 T 2 t 5 θ 5 φ z z 5 φ z z 5 C 1,.8) where γ isthepathdefinedinlemma.1. Wealsohavethat 1 5 qh T u n ) φ un u n 5 2 R 5 qs 6 T 10,.9) ) bq un 2 10T 2 ψ T 2 φ un u n 5 u n 2 96 R 5 bqs 6 T 8 u n qs 6 T ) Thus it follows from.7),.8),.9) and.10) that u n 2 6C R 5 qs 6 T qs 6 T 10 =6C qs 6 T ) On the other hand, since Jq,λ T n ) u n ),u n =0,wehavethat u n 2 + b u n 2 + qh T u n ) φ un u n 5 + q un 2 ) 5T 2 ψ T 2 u n 2 φ un u n 5 = λ n fu n )u n

18 F.-Y.Li,Y.-H.Li&J.-P.Shi It follows from.4) and.2) that 1 2 b u n 2 1+ q un 2 ) ) R 5T 2 ψ T 2 φ un u n 5 b u n 2 = λ n fu n )u n 1+ R q un 2 ) ) 5T 2 ψ T 2 φ un u n 5 u n 2 R qh T u n ) φ un u n R b u n 2 + C 0 u n 6, where C 0 is defined in.2). Therefore, ) 1 4 b u n 2 C 0 u n 6 6 C 0 S u n 2 C 0 S 6C ) 5 qs 6 T ) Thus from.11) and.12), we obtain that where u n 2 6C qs 6 T 10 +4C 0 S = C 2 + C qt 10 + C 4 q 2 T 20 + C 5 q T 0, 6C ) 5 qs 6 T 10 C 2 =6C C 0 S C1, C = S C 0 C S 9 5, ) 2 ) C 4 =12C 0 C 1 S 15, C 5 =4C 0 S Choose T0 2 C 2 +1, T T 0 and q T = qt ) such that C qt 10 + C 4 q 2 T 20 + C 5 q T 0 < 1. Then the conclusion holds. Now we can complete the proof of Theorem 1.1 for the case of q>0. Proof of Theorem 1.1 for q > 0. Let T 0 and q T = qt )bechosenasin Lemma.5, andletu n be a critical point for J T0 q,λ n at the level c λn.thenfrom Lemma.5 we may assume that u n T 0. Therefore J T0 q,λ n u n )= 1 2 u n q φ un u n 5 λ n F u n )

19 Existence of positive solutions to Schrödinger Poisson type systems Since λ n 1asn, we can show that {u n } is a PS sequence of J q. Indeed, the boundedness of {u n } implies that {J q u n )} is bounded. Also for v H b,wehave J q u n),v = J T0 q,λ n ) u n ),v +λ n 1) fu n )v. It follows that J qu n ) 0 and hence {u n } is a bounded PS sequence of J q.by Lemma., {u n } has a convergent subsequence. Without loss of generality, we may assume that u n u. Consequently,J qu) = 0. According to Lemma.2, wehave that J q u) = lim n J q u n ) = lim n J T0 q,λ n u n ) c>0, and u is a positive solution by the condition H 1 ). The proof is completed. 4. TheCaseofNegativeq In this section, we assume q<0, and we assume that the conditions H 1 ), H 2 ) and H 4 ) are satisfied. Indeed let p = q, we may consider the following equivalent system { u + bu pφ u u = fu) in, 4.1) φ = u 5 in. Here we define Au) = 1 2 u 2, Bu) = 1 10 R p φ u u 5 + F u) and J p,λ u) = 1 2 u pλ φ u u 5 λ F u). Then we have J p,λ ) u),v =u, v) pλ φ u u uv λ fu)v. 4.2) The following Lemmas 4.1 and 4.2 imply that J p,λ satisfies the conditions of Theorem 2.7. First we show that the path set Γ p,λ is not empty. Lemma 4.1. For every λ I [1/2, 1] and p>0, define the path set Then Γ p,λ. Γ p,λ = {γ C[0, 1],H b ):γ0) = 0,J p,λ γ1)) < 0}. Proof. Let λ I. For any u H b \{0} and t>0, J p,λ tu) = 1 2 t2 u t10 pλ φ u u 5 λ F tu) 1 2 t2 u t10 pλ φ u u 5. Hence J p,λ tu) as t

20 F.-Y.Li,Y.-H.Li&J.-P.Shi Secondly we show that the critical level c p,λ is uniformly bounded from below by a positive lower bound for all λ I. Lemma 4.2. For every λ I and p>0, define c p,λ = inf γ Γ p,λ max J p,λγt)). t [0,1] Then there exists a constant c p > 0 such that c p,λ c p for all λ I. Proof. For any u H b and λ I, the estimates.2) hold from the conditions H 1 )andh 2 ). Then we have J p,λ u) 1 2 u p φ u u 5 1 R 4 b u 2 C 0 u u ps 1 u b u 2 2 C 0 u u ps 6 u u 2 C 0 S u 6. Hence we can reach the desired conclusion in the same way as in the proof of Lemma.2. Next we give an estimate of the upper bound of the critical level c p,λ by using a test function. Lemma 4.. For any p>0 and λ [1/2, 1], let c p,λ be defined as in Lemma 4.2. Then we have c p,λ < pλ)/2 S/2. 4.) 6pλ) 1/2 Proof. It is well known that the best Sobolev embedding constant S is attained by the functions ξ ε x) = ε 1/4 ε + x 2 ) 1/2 for ε>0. We define u ε x) =ξ ε x)ψx), where ψ C0 B 2r0)) such that 0 ψx) 1, and ψx) =1onB r 0) for some r>0. By simple calculations, we can derive that as ε 0 +, R u ε 2 x 2 = R 1 + x 2 ) + Oε1/2 ):=K 1 + Oε 1/2 ), 4.4) u ε 6 1 = R 1 + x 2 ) + Oε/2 ):=K 2 + Oε /2 ) 4.5)

21 Existence of positive solutions to Schrödinger Poisson type systems and for every ε>0, K t ε 6 t 4, t, 6), u ε t = K ε 4 ln ε, t =, K t ε t 4, t [2, ), 4.6) where K 1,K 2, K t 2 t<6) are positive constants, and S = K 1 K 1/ 2.By4.4) and 4.5), we have u ε 2 R ) 1/ = S + Oε 1/2 ). 4.7) u ε 6 Multiplying the second equation of 4.1) by u and integrating, we obtain that u 6 6 = φ u u 1 R 2 φ u u 2 2 = 1 φ u u R 2 u ) Define 1 I p,λ u) = ) 10 pλ u b u pλ u 6 6 λa u r r + λb u 2 2 and 1 H p,λ u) = ) 10 pλ u b u pλ u λb u 2 2, where A and B are defined in the condition H 4 ). From 4.8), Lemma 4.2 and the condition H 4 ), it can be derived that Since c p,λ sup J p,λ tu ε ) sup I p,λ tu ε ). t [0, ) t [0, ) d dt [I p,λtu ε )] = ) pλ t u ε bt u ε pλt5 u ε 6 6 then there exists a unique t ε > 0 such that d dt [I p,λtu ε )] = 0. Thus rλat r 1 u ε r r +2λBt u ε 2 2, 4.9) sup I p,λ Itu ε )=I p,λ t ε u ε )=H p,λ t ε u ε ) λt r εa u ε r r H p,λ t εu ε ) λt r εa u ε r r, t [0, ) where t ε is the unique positive solution of the equation d dt [H p,λtu ε )] =

22 F.-Y.Li,Y.-H.Li&J.-P.Shi From d dt [H p,λtu ε )] = pλ ) t u ε bt u ε pλt5 u ε λBt u ε 2 2, 4.10) it is easy to see that t ε <t ε. From 4.4) 4.7) and4.10), we obtain that, for small ε>0, t 5 + pλ) u ε = 4 ε b u ε λB u ε 2 2 6pλ u ε pλ)k = Oε 1/2 ) 5+pλ K = Oε 6pλK 2 + Oε /2 ) 6pλ K 1/2 ). 2 And from 4.4) 4.7) and4.9), we calculate that ) pλ K 1 + Oε 1/2 )= ) pλ u ε b u ε λB u ε 2 2 = 6 5 pλt4 ε u ε rλatr 2 ε u ε r r 6 5 pλtr 2 ε t ε )6 r u ε rλatr 2 ε u ε r r ) 6 tε r 2 = 5 pλt ε) 6 r u ε rλa u ε r r 6 5 pλt ε) 6 r K 2 + Oε 6 r)/4 ) and hence we have that t ε >t ε K p,λ > 0, where K p,λ is a positive constant only depending on p, λ and K 1,K 2. Summarizing these estimates, we get ) t r 2 ε sup I p,λ tu ε )=H p,λ t ε u ε ) λat r ε u ε r r H p,λ t εu ε ) λt r εa u ε r r t [0, ) ) 10 pλ t ε) 2 u ε bt ε) 2 u ε Bλt ε) 2 u ε pλt ε) 6 u ε 6 6 λat r ε u ε r r pλ)/2 S/2 + Oε 1/2 ) λak r 15 6pλ) K 1/2 p,λ r ε 6 r)/4 < pλ)/2 S/2 15 6pλ). 1/2 The last inequality holds if we choose ε>0 small enough as r 4, 6) here. Thus we obtain the estimate of c p,λ in 4.)

23 Existence of positive solutions to Schrödinger Poisson type systems Now we can show that the modified functional J p,λ satisfies the PS condition. Lemma 4.4. For any λ I and p satisfying pλ = 1, each bounded PS) cp,λ sequence of the functional J p,λ admits a convergent subsequence. Proof. Let λ I and {u n } be a bounded PS) cp,λ sequence of J p,λ,thatis,{u n } is bounded and J p,λ u n ) c p,λ,j p,λ ) u n ) 0inH b,whereh b is the dual space of H b.since{u n } is bounded, we may assume that there exists u H b such that u n u, in H b, u n u, a.e. on. From Lemma 2.6, wehave fu n )u n fu)u, F u n ) F u). 4.11) Since u n uin H b, in view of Sobolev s embedding theorems, we have that J p,λ u) =0. By using Lemma 2.8 and 4.11), we obtain that 1 u R 2 b u 2 1 R 2 pλ φ u u 5 =λ F u), hence, J p,λ u) = 1 2 u R pλ φ u u 5 λ F u) 1 u R 15 pλ φ u u 5 0. Now let v n = u n u, by using Lemma 2. and 4.11), we have 0 J p,λ ) u n ),u n = J p,λ ) u n ),u n J p,λ ) u),u = u n 2 u 2 pλ φ un u n 5 + pλ φ u u 5 λ fu n )u n + λ fu)u = v n 2 pλ φ vn v n 5 + o1) 4.12) and c p,λ J p,λ u) =J p,λ u n ) J p,λ u)+o1) = 1 2 u n u pλ φ un u n R 10 pλ φ u u 5 R λ F u n )+λ F u)+o1) = 1 2 v n pλ φ vn v n 5 + o1). 4.1)

24 F.-Y.Li,Y.-H.Li&J.-P.Shi Suppose that v n 0inH b, we may assume that v n 2 l>0. So 4.12) andthe estimate pλ φ vn v n 5 pλs 1 v n 10 6 pλs 6 v n 10, implies that Hence, we have that l pλs 6 l 5 or l S/2 pλ) 1/4. S /2 c p,λ c p,λ J p,λ u) = 1 2 l 1 10 l + o1) = 2 5 l + o1) 2 + o1). 4.14) 5 pλ) 1/4 Since pλ = 1, by Lemma 4., wehave c p,λ < pλ)/2 S/2 = 2 S /2, 4.15) 15 6pλ) 1/2 5 pλ) 1/4 which is a contradiction with 4.14). Therefore v n 0inH b,orequivalently, u n u in H b as n. From the results in Lemmas and Theorem 2.7, we can now show the existence of a critical point for the functional J p,λ. Lemma 4.5. For almost every λ I and p =1/λ, there exists u λ p H b such that J p,λ ) u λ p)=0and J p,λ u λ p)=c p,λ. Proof. By Theorem 2.7, for almost every λ I, there exists a bounded sequence {u λ n } H b such that J p,λ u λ n ) c p,λ and J p,λ ) u λ n ) 0. From Lemma 4.4, for p =1/λ, we can assume that there exists u λ H b such that u λ n uλ in H b,then the assertion follows. From Lemma 4.5, there exists a sequence {λ n,p n,u n )} I R H b such that as n, λ n 1, p n λ n =1and J pn,λ n u n )=c pn,λ n, J pn,λ n ) u n )=0. We now show that the sequence {u n } is uniformly. Lemma 4.6. Let u n be a critical point of J pn,λ n at the level c pn,λ n as defined above, where λ n I and p n λ n =1. Then there exists M>0 such that u n M for all n N. Proof. Firstly, since J pn,λ n ) u n ) = 0, it follows from 2.11) in Lemma 2.8 that u n satisfies the following Pohozaev type identity: 1 u n R 2 b u n 2 1 R 2 p nλ n φ un u n 5 =λ n F u n ). 4.16)

25 Existence of positive solutions to Schrödinger Poisson type systems By using J pn,λ n u n )=c pn,λ n,wehavethat 1 u n R 2 b u n 2 1 R 10 p nλ n φ un u n 5 λ n F u n )=c pn,λ n. 4.17) Thus we obtain from 4.16) and4.17) that u n R 5 p nλ n φ un u n 5 =c pn,λ n. 4.18) We now estimate the right-hand side of 4.18). Since J pn,λ n u) = 1 2 u 2 1 φ u u 5 λ n F u) J 10 2,1/2 u), u H b. Hence for any γ Γ 2,1/2,wehavethatγ Γ pn,λ n,andthen So c pn,λ n Then 4.18) implies that max t [0,1] J 2,1/2γt)), γ Γ 2,1/2. c pn,λ n c 2,1/2 1 C 6. u n 2 2 C 6, p n λ n φ un u n 5 5C ) On the other hand, from the equation J pn,λ n ) u n ) = 0, we have the relation u n 2 + b u n 2 p n λ n φ un u n 5 λ n fu n )u n = ) It follows from 4.20) and.2) that b u n 2 = λ n fu n )u n u n 2 + p n λ n φ un u n R b u n 2 + C 0 u n 6 +5C b u n 2 + C 0 S C 6 +5C 6, 4.21) which implies 1 2 b u n 2 C 0 S C6 +5C ) Combining 4.19) and4.22), we obtain that u n 2 11C 6 +2C 0 S C 6 M 2. Now we are at the position to prove the second part of Theorem 1.1 where q is negative

26 F.-Y.Li,Y.-H.Li&J.-P.Shi ProofofTheorem1.1 for q < 0. Let λ n [1/2, 1], p n λ n =1andλ n 1as n.letu n be a critical point of J pn,λ n at the critical level c pn,λ n.sinceλ n 1, then we can show that {u n } is a PS sequence of J 1,1.Indeed,byLemma4.6, we may assume that {u n } is bounded, which also implies that {J 1,1 u n )} is bounded. Also for any v H b, J 1,1u n ),v = J p n,λ n u n ),v +λ n 1) fu n )v. It follows that J 1,1 u n) 0, and thus {u n } is a bounded PS) c1,1 sequence of J 1,1. Therefore Lemma 4.4 holds and {u n } has a convergent subsequence. Without loss of generality, we assume that u n u and consequently J 1,1 u) = 0. From the proof of Lemma 4.2, wehavethatj pn,λ n u n ) c 2,1/2, and hence J 1,1 u) = lim J 1,1u n ) = lim J p n,λ n u n ) c 2,1/2 > 0. n n Therefore u is a positive solution by the condition H 1 ), and the proof is completed. To conclude the paper we briefly remark on the proof of Theorem 1.2. Forthe subcritical equation 1.5) withs [1, 4), there exists q 0 > 0 such that 1.5) hasa positive solution for any q [0,q 0 ). This has been proved in [19]. For the case of q<0, by using the same method in this section for the critical case, we can prove that 1.5) has a positive solution for any q<0astheresultsinlemmas4.1, 4.2, 4.4, 4.5, 4.6 hold for any q<0. We need only to notice that H b can be compactly embedded into L s+1 r ). So the conclusion of Lemma 4.4 hold for all q<0and λ I. Acknowledgments The authors thank anonymous referees for a careful reading and some helpful comments, which greatly improve the manuscript. The research was partially supported by National Natural Science Foundation of China Grant Nos , 11011), and Science Council of Shanxi Province , , ) and Shanxi 100 Talent program. References [1] C.O.Alves,M.A.S.SoutoandS.H.M.Soares,Schrödinger Poisson equations without Ambrosetti Rabinowitz condition, J. Math. Anal. Appl. 772) 2011) [2] A. Ambrosetti, Concentration and compactness in nonlinear Schrödinger Poisson system with a general nonlinearity, J. Differential Equations 2497) 2010) [] A. Ambrosetti and D. Ruiz, Multiple bound states for the Schrödinger Poisson problem, Commun. Contemp. Math. 10) 2008) [4] A. Azzollini and P. d Avenia, On a system involving a critically growing nonlinearity, J. Math. Anal. Appl. 871) 2012)

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