On Existence and Multiplicity for Schrödinger–poisson Systems Involving Weighted Sublinear Nonlinearities

We deal with existence and multiplicity for the following class of nonhomo-geneous Schrödinger–Poisson systems


Introduction
In this paper we consider the following class of Schrödinger-Poisson systems (also called Schrödinger-Maxwell systems) in both nonhomogeneous case g(x) ≡ 0, namely in R 3 , and in the homogeneous case g(x) ≡ 0, that is −∆φ = K(x)u 2 in R 3 .(P 0 )

S. Barile
This class of systems has a strong physical meaning since it arises in several applications from mathematical physics, in particular in quantum mechanics models where it describes the mutual interactions of charged particles in the electrostatic case (see e.g.[5,6] and references therein for more detailed physical aspects).For this reason, many authors have devoted their attention to systems of this type and they have widely studied them by using variational methods under various conditions on the potentials V(x) and K(x) and the nonlinearity f (x, u) especially when it is superlinear or asymptotically linear at infinity in u.On the contrary, up to now, there is no extensive literature dealing with the case of nonlinearities f (x, u) sublinear at infinity especially involving suitable weights and this motivates our work.Let us start with the homogeneous case g(x) ≡ 0.
In 2012, Sun [12] proved the existence of infinitely many small negative energy solutions to (P 0 ) in the case K(x) ≡ 1 by a variant fountain theorem established in [16] under the following assumptions (V ) V ∈ C(R 3 , R) satisfies inf x∈R 3 V(x) ≥ a > 0 with a a real constant; (V ) for any M > 0, meas{x ∈ R 3 : V(x) ≤ M} < +∞ where meas denotes the Lebesgue measure on R 3 ; (F ) F(x, u) = W 1 (x)|u| w 1 where F(x, u) = u 0 f (x, t) dt, W 1 : R 3 → R is a positive continuous function such that In particular, conditions (V )-(V ) imply a coercive condition on V which was first introduced by Bartsch and Wang [4] in order to overcome the loss of compactness due to the unboundedness of the domain R 3 .Clearly, thanks to (F ) only the one-weight nonlinearity f (x, u) = w 1 W 1 (x)|u| w 1 −1 is allowed.
In 2013, Liu, Guo and Zhang [8] generalized the results in Sun [12] since they showed for (P 0 ) with K(x) ≡ 1 the existence of a nontrivial solution by minimization arguments [10] and the multiplicity of solutions with negative energy which goes to zero by a symmetric mountain pass lemma based on genus properties in critical point theory (see Salvatore [11]) by removing assumption (V ) and relaxing assumption (F ) with the following | f (x, u)| ≤ w 1 W 1 (x)|u| w 1 −1 + w 2 W 2 (x)|u| w 2 −1 for a.e.x ∈ R 3 and for all u ∈ R with W 1 ∈ L 2 2−w 1 (R 3 ) and W 1 > 0, W 2 ∈ L 3 (R 3 ) and W 2 ≥ 0 where w 1 ∈ (1, 2) and w 2 ∈ [4/3, 2).This assumption makes indefinite nonlinearities f (x, u) can be also considered and the presence of the weights W 1 and W 2 ensures a good property of compactness for these f (x, u).
In 2013, Lv [9] also generalized the result in Sun [12] by showing existence of a nontrivial solution by minimization arguments [10] and multiplicity of solutions with vanishing and negative energy levels by the dual fountain theorem [14] to (P 0 ) in the case K(x) ≡ 1 without the coercive assumption (V ) and under only (V ) on V.This has been possible since the odd nonlinearity f (x, u) is supposed to satisfy suitable sublinear growth hypotheses which imply the existence of three weights and allow to recover compact embeddings of the functional space in the weighted space 3 and for all u ∈ R so that the case of indefinite nonlinearities f (x, u) is also covered.This result improves and completes the paper by Liu, Guo and Zhang [8].
Few years later, in 2015 Ye and Tang [15] improved the results in Sun [12] since they showed the existence of infinitely many small solutions with small negative energy to (P 0 ) by means of a new version of symmetric mountain pass lemma developed by Kajikiya [7] with the non-negative potential Besides a suitable local assumption, on the continuous and odd nonlinearity f (x, u) is assumed in particular the following sublinear growth condition Regarding the nonhomogeneous case g(x) ≡ 0, Wang, Ma and Wang [13] in 2016 established only existence to (P g ) with non-negative g ∈ L 2 (R 3 ), for a class of potentials and the same sublinear growth condition assumed in [15] with two weights W 1 and W 2 .In this case the authors work without using for compactness this last condition.
The aim of this paper is to study (P g ) (resp.(P 0 )) under more generic conditions in order to generalize or to give complementary results to the ones listed above.More precisely, we investigate existence (resp.multiplicity) of solutions to (P g ) (resp.(P 0 )) under the following assumptions: (V) V : R 3 → R is a Lebesgue measurable function with ess inf R 3 V(x) ≥ a > 0 where a is a real constant; x ∈ R 3 and for all s ∈ R; ( f 3 ) there exist Ω ⊂ R 3 with meas(Ω) > 0, w r+1 ∈ (1, 2), η > 0 and δ > 0 such that F(x, s) ≥ η|s| w r+1 for a.e.x ∈ Ω and for all s ∈ R, |s| ≤ δ where F(x, s) = s 0 f (x, t) dt; x ∈ R 3 and for all s ∈ R; (G) g ∈ L 2 (R 3 ).

S. Barile
Thus, we obtain the following results.For the definition of the functional spaces E V and D 1,2 (R 3 ) and of the energy functional J 0 which appear in next theorems, see Section 2. First, let us state the existence result for the nonhomogeneous case and for the homogeneous case.
Theorem 1.1 (Existence).Suppose that (V), (K), ( f 1 ) and ( f 2 ) hold.Then, we get the following: (i) (nonhomogeneous case g(x) ≡ 0) if in addition (G) holds, problem (P g ) admits at least a nontrivial weak solution (u, ) is also assumed, problem (P 0 ) possesses both a trivial weak solution and at least a non-trivial weak solution (u, Now, let us provide the multiplicity result obtained in the case g(x) ≡ 0.
Theorem 1.2 (Multiplicity).Assume that (V), (K), ( f 1 ), ( f 2 ), ( f 3 ), ( f 4 ) hold.Then, problem Remark 1.3.Thanks to Remark 2.8 and the properties of J 0 and φ u stated in Section 2, we remark that Theorem 1.2 gives in particular the existence of a sequence {(u k , φ u k )} of critical points of J 0 such that J 0 (u k , φ u k ) ≤ 0, u k = 0 and then φ u k = 0, lim k u k = 0 from which we get lim k φ u k = 0; consequently, lim k J 0 (u k , φ u k ) = 0 − .Let us observe that, as concerns the existence result in the homogeneous case g(x) ≡ 0, we complete the papers by Sun [12] and by Ye and Tang [15] where no existence result has been stated.Moreover, we improve the existence of solutions to (P 0 ) for not necessarily constant potentials K(x) by relaxing (V ) with (V) in Lv [9] and in Liu, Guo and Zhang [8].
Moreover, we generalize the existence of multiple solutions obtained in Sun [12], Liu, Guo and Zhang [8] and Lv [9] to (P 0 ) for K(x) ≡ 1 to a more general class of potentials satisfying (K) thus providing the existence of infinitely many small solutions with small negative energy.
In the nonhomogeneous case g(x) ≡ 0, we improve the existence result established by Wang, Ma and Wang [13] since we relax condition (V ) by (V), skip (V ) and recover compactness by the different requirement ( f 2 ) involving r weights.Furthermore, we do not impose any sign condition on g.Remark 1.4.Let us observe that, from ( f 2 ) by integration it follows that x ∈ R 3 and for all s ∈ R. (1.1) The paper is organized as follows: in Section 2 we introduce the variational formulation of the problem and we recall a generalized version of Weierstrass theorem, Mazur theorem and a convexity criterion.Moreover, we recall a variant of the symmetric mountain pass theorem for "subquadratic" problems stated in [7].In Section 3 we prove Theorem 1.1 and in Section 4 we show Theorem 1.2.

Variational tools
In order to introduce the variational structure of the problem, let E = H 1 (R 3 ) be the usual Sobolev space endowed with the standard scalar product and the corresponding norm We denote by L s (R 3 ), 1 < s < +∞, the Lebesgue space endowed with the norm Moreover, let us introduce By assumption (V), E V is a Hilbert space endowed with the scalar product and the related norm with dual space (E V , • E V ).From now on, let 1 < s < ∞ and . Clearly, ) and by (V) we have that E V → E.Moreover, the following continuous embeddings hold From now on, c and C will denote real positive constants changing line from line.
At this point, we prove the following result which allows us to state the compact embedding of E V in a weighted Lebesgue space with a specific weight W(x); the result will be applied to the Lebesgue measurable weight W i and to the constant w = w i for any i ∈ {1, . . ., r} in assumption ( f 2 ).
Under assumption (V), we get the following compact embedding Therefore, by Hölder's inequality and Sobolev embeddings we get for every ε > 0 there exists n ε ∈ N such that for every n > n ε one has Let us point out that in Sun [12] and Wang, Ma and Wang [13] potential V satisfies stronger assumptions (V )-(V ).These conditions allow to prove that E V → → L s (R 3 ) for all 2 ≤ s < 6. Differently, here above in Proposition 2.1 we show that E V → → L w W (R 3 ) with w ∈ (1, 2).In the following (see Proposition 2.3 and Proposition 4.1) we will exploit only this weaker result in order to overcome the lack of compactness due to the unboundedness of the domain R 3 .
Under our assumptions, it is not difficult to see that system (P g ) (resp.(P 0 )) has a variational structure, that is, it is possible to find its solutions by looking for critical points of the functional ).But the functional J g (resp.J 0 ) is strongly indefinite, namely it is unbounded from below and above on infinite dimensional subspaces.In order to remove its indefiniteness and to reduce to study a not strongly indefinite functional, we can use the following reduction method introduced in [5] (see also [6]).This method relies on the fact that, for every u ∈ E V , the Lax-Milgram theorem implies the existence of a unique It is well known that φ u can be written with the following integral formula So, substituting φ = φ u in J g (resp.J 0 ) it is possible to consider the functional Now, by multiplying −∆φ u = K(x)u 2 by φ u and integrating by parts we get then the reduced functional I g (resp.I 0 ) takes the form for every u ∈ E At the same time, problem (P g ) (resp.problem (P 0 )) can be reduced to an equivalent single Schrödinger equation with a nonlocal term.Indeed, substituting φ = φ u in (P g ) (resp.(P 0 )) we get the following equation As we will prove in Proposition 2.3, and every critical point of I g (resp.I 0 ) corresponds to a solution u ∈ E V to (S g ) (resp.(S 0 )) and provides a solution (u, φ) ∈ E V × D 1,2 (R 3 ) to (P g ) (resp.(P 0 )).
Remark 2.2.Since by (K), it is K(x) ≥ 0 for a.e.x ∈ R 3 , we get φ u ≥ 0 for any u ∈ E V .Now, as just noticed, by hypothesis (V) it is E V → H 1 (R 3 ); this fact together with the well known continuity of φ u : 2), Hölder's inequality and Sobolev embeddings we obtain Therefore, in the first case we get while in the second At this point we can state the following variational principle and recover the compactness of the problem.
Proposition 2.3.Assume that (V), (K), ( f 1 ), ( f 2 ) and (G) hold.Then, the weak solutions of (P g ) (resp.(P 0 )) are the critical points of the energy functional I g : E V → R (resp.I 0 : E V → R) defined by and its derivative dI g : ) is a solution of problem (P g ) (resp. Proof.Let us start by showing that the functional I g (resp.I 0 ) is well defined and its Fréchet derivative given in (2.5) is a continuous operator from E V to E V .For the sake of completeness, we give here all the details of the proof.We define and study separately the following maps is a linear continuous map on E V .
Concerning the map ϕ K , we need to show that then by Sobolev embeddings ϕ K (u) ∈ R for any u ∈ E V .Now we prove that ), by Hölder's inequality and (2.3) we get the following Similarly, if K ∈ L ∞ (R 3 ) by Hölder's inequality and (2.4) we obtain By Sobolev embeddings in both cases we have done.It is not difficult to find that the Gâteaux derivative of ϕ K at u is as in (2.6) and it is linear and continuous from E V to R. It remains to prove that dϕ K is continuous from E V to E V , i.e.
First, observe that by adding and subtracting K(x)φ u n u ζ in the integral we have ), by Hölder's inequality and Sobolev embeddings it follows As u n → u in E V , by the continuity of φ u from E V in D 1,2 (R 3 ) ensured in Remark 2.2 we get φ u n → φ u as n → +∞ and consequently the boundedness of φ u n in D 1,2 (R 3 ); therefore, the right terms in these four inequalities above go to zero and by (2.8) the convergence in (2.7) follows.
Now, we have to prove that also Let us point out that, by (1.1) in Remark 1.4 and Hölder's inequality, we have where µ i = 2 w i = 2 2−w i and similarly by ( f 2 ) we obtain Hence, by Sobolev embeddings it follows that ϕ Moreover, standard tools imply that the Gâteaux derivative of ϕ F at u is as in (2.9) and it is linear and continuous from E V to R. At this point, we have to prove that dϕ F is continuous from (2.10) Indeed, by Hölder's inequality and Sobolev embeddings, Now, by ( f 2 ) we get for a.e.x ∈ R 3 1) .By Fatou's lemma, it follows that Now, we observe that, since u n → u in E V it is u n (x) → u(x) a.e.x ∈ R 3 , therefore x ∈ R 3 and for all i = 1, . . ., r and also by ( On the other hand, by Hölder's inequality and Sobolev embeddings we get for all i = 1, . . ., r and, since u n → u in L 2 (R 3 ) by continuous embeddings, also the left-hand side term goes to zero as n → +∞ for every i = 1, . . ., r.Consequently, (2.11) and (2.10) is proved.
By exploiting the arguments carried out in [5,6], we get that the pair (u, φ) ) is a solution of problem (P g ) (resp.(P 0 )) if and only if u ∈ E V is a critical point of I g (resp.I 0 ) and φ = φ u .
Finally, we prove that dϕ F is compact from (2.12) Fixed i = 1, . . ., r and taken α i = 2 w i ∈ (0, 2), by Hölder's inequality we get w i ,W i , hence, by (2.12) it follows that Then, arguing as in the proof of the continuity of dϕ F , as soon as and the proof is completed.Now, in order to prove in next Section 3 the existence result by minimization arguments, we will exploit the following generalized version of the Weierstrass theorem.
Theorem 2.4.Let (X, • ) be a reflexive Banach space and M ⊆ X be a weakly closed subset of X. Suppose that the functional I : M → R is coercive and (sequentially) weak lower semi-continuous on M.
Then, I is bounded from below on M and there exists u 0 ∈ M such that I(u 0 ) = min u∈M I(u).

S. Barile
Proof.Since in order to get the thesis, it is sufficient to prove that for every u, v ∈ E V .By Hölder's inequality we get Now, if we multiply first by φ u then by φ v the following two equations by integration by parts we get and By substituting equalities (3.3)-(3.6) in the last line of (3.2) and by applying again Hölder's inequality we get Recall that the following inequality holds (xy) 1/2 (x + y) ≤ x 2 + y 2 for every x, y ≥ 0.
By applying it to the last line of (3.7) and by exploiting equalities (3.3) and (3.6) we obtain thus we get (3.1) and this completes the proof.
Thanks to Lemma 3.1, we can prove the next proposition.
Proposition 3.2.Suppose that (V) and (K) are satisfied.Then, the functional Proof.By Proposition 2.3 we already know for any u, v ∈ E V .Then, by Lemma 3.1 we can apply Proposition 2.6 to the functional I = ϕ K and to the Banach space X = E V and we obtain the thesis.
Consequently, we get the following result.
Proposition 3.3.Under assumptions (V) and (K), the C 1 functional ϕ K is weak lower semicontinuous on E V .
Proof.Since ϕ K ∈ C 1 (E V , R) is convex by Proposition 3.2, the thesis is a direct consequence of Theorem 2.5.
Proof of Theorem 1.1.
(i) First, let us consider the nonhomogeneous case g(x) ≡ 0. Observe that, since −∆φ u = K(x)u 2 , by multiplying by φ u and integrating by parts we get R 3 K(x)φ u u 2 dx = φ u 2 D ≥ 0. Therefore, from (1.1), Hölder's inequality and Sobolev embeddings, we obtain Then, since w i ∈ (1, 2) for any i ∈ {1, . . ., r}, it follows that I g is coercive and bounded from below on the reflexive Banach space E V .Moreover, the functional I g is weak lower semicontinuous on E V .In order to show it, it is useful to write it again as I g = ϕ V + 1 4 ϕ K − ϕ F − ϕ g by using the notations introduced in Proposition 2.3.Clearly, ϕ V is weak lower semicontinuous by the norm properties while ϕ K is weak lower semicontinuous on E V by Proposition 3.3.In addition, ϕ F is weak continuous as it is of class C 1 on E V and its derivative dϕ F is compact by Proposition 2.3.Moreover, ϕ g is linear continuous then it is weak continuous on E V .
Consequently, by the generalized Weierstrass theorem stated in Theorem 2.4 there exists u ∈ E V such that I g (u) = min u∈E V I g (u).Hence, u is a critical point of I g and, by applying Proposition 2.3 we get u is a solution of problem (S g ) and then (u, φ u ) is a solution to (P g ).Now, since g(x) ≡ 0, equation (S g ) does not admit the trivial solution.Therefore, u is a not trivial solution to (S g ) and we obtain that (u, φ u ) is a not trivial solution to (P g ).
(ii) Now we consider the homogeneous case g(x) ≡ 0. By ( f 2 ), equation (S 0 ) admits always the trivial solution u = 0 with I 0 (0) = 0. Clearly, the generalized Weierstrass theorem also applies to I 0 by adapting previous arguments with g(x) ≡ 0. In any case, since we assume also ( f 3 ) holds, the solution u to (S 0 ) is not trivial; indeed, recall that φ tu = t 2 φ u for every t > 0 and any u ∈ E V and let us fix u 1 ∈ E V ∩ C c (R 3 ) with u 1 = 0 and supp(u 1 ) ⊆ Ω; by ( f 3 ) and 1 < w r+1 < 2 we get for ε > 0 small enough.Therefore, system (P 0 ) has a non-trivial weak solution (u, φ u ).
Remark 3.4.Here above we have exploited the weak lower semicontinuity of ϕ K in particular and then of the functional I g (resp.I 0 ).Really, the existence result can be also found by applying [10,Theorem 2.7] (see also [14,Corollary 2.5]) since is bounded from below on E V and, as proved in the next Proposition 4.1, satisfies (PS) condition by neglecting the presence of dϕ K thanks to Lemma 3.1.

Proof of Theorem 1.2
Let us take in mind from now on we treat only the homogeneous case g(x) ≡ 0. As proved in Proposition 2.3, compact embeddings stated in Proposition 2.1 allow us to recover the compactness of dϕ F .
On the contrary, Proposition 2.1 which is weaker with respect compactness results E V → → L s (R 3 ) for all 2 ≤ s < 6 obtained by assumptions (V )-(V ) does not enable us to show that dϕ K is compact.Fortunately, this problem is overcome thanks to Lemma 3.1 and we can state the following proposition.Proposition 4.1.Suppose (V), (K), ( f 1 ) and ( f 2 ) hold.Then, the functional I 0 satisfies (PS) condition.
Proof.Let {u k } ⊂ E V be a (PS) sequence of I 0 , namely {I 0 (u k )} is bounded and dI 0 (u k ) → 0 in E V as k → +∞.As observed in the proof of Theorem 1.1, by (3.8) we get that I 0 is coercive on E V and this implies that {u k } is bounded in E V .Thus, up to subsequence, there exists u ∈ E V such that u k u as k → +∞.By (2.5), Lemma 3.1 and Hölder's inequality we get The first term in the last line goes to zero since dI 0 (u k ) → 0 in E V ; the second one also tends to zero because dI 0 (u) is linear and continuous from E V to E V and u k u as k → +∞.The same occurs for the third term since, by Proposition 2.3, we have that the function u Then, we can conclude that u k → u in E V and (PS) condition is proved.Now, we prove the following result which allows us to show (A 2 ) in Theorem 2.7.
Proposition 4.2.Assume that (V), (K), ( f 1 ), ( f 2 ) and ( f 3 ) hold.Then, for every k ∈ N there exists A k ∈ Γ k such that sup Proof.Fixed k ∈ N, let us consider k disjoint open sets Ω 1 , . . ., Ω k such that k j=1 Ω j ⊂ Ω with Ω as in assumption ( f 3 ).For every ε > 0 and for every j = 1, . . ., k there exist a closed set H j and an open set G j such that H j ⊂ Ω j ⊂ G j , meas(G j \ Ω j ) < ε and meas(Ω j \ H j ) < ε.Without loss of generality, we can assume k j=1 G j = ∅.Moreover, for every G j there exists ϕ j ∈ C ∞ 0 (G j , R) such that ϕ j | H j = 1 and 0 ≤ ϕ j ≤ 1.Now, let us consider ν j = ϕ j ϕ j V and denote by ν j again its null extension on R N \ G j .Clearly, ν 1 , . . ., ν k are linearly independent functions in E V .
Denoted by E V,k the k-dimensional vector space generated by ν 1 , . . ., ν k , for every u ∈ E V,k we get u = ∑ k j=1 λ j ν j with λ j ∈ R, j = 1, . . ., k.Therefore, for all u ∈ E V,k it results

(a) |u|
Since ν j V = 1 for every j = 1, . . ., k by the definition of ν j we get (c) There exists c k > 0 such that c k u V ≤ |u| w r+1 .
(c) follows since E V,k has finite dimension and then all norms are equivalent in E V,k .
which joint to (a) and (c) gives Proof of Theorem 1.2.Since ( f 4 ) holds, the functional I 0 is even.As proved in the proof of Theorem 1.1, thanks to (3.8) we get I 0 is bounded from below on E V ; moreover, from ( f 2 ) it is I 0 (0) = 0.By Proposition 4.1, I 0 satisfies (PS) condition.Hence, I 0 satisfies assumption (A 1 ) in Theorem 2.7.
Furthermore, (A 2 ) holds by Proposition 4.2.By Theorem 2.7 (see also Remark 2.8), there exists a sequence {u k } in E V of critical points of I 0 such that u k = 0, lim k u k = 0 in E V and lim k I 0 (u k ) = 0. Therefore, by Proposition 2.3, {u k } is a sequence of non-trivial solutions to (S 0 ) such that u k → 0 in E V and I 0 (u k ) → 0 as k → +∞; hence, by the continuity of φ u and J 0 we obtain that {(u k , φ u k )} is a sequence of solutions to system (P 0 ) with u k → 0 in E V , φ u k → 0 in D 1,2 (R 3 ) and J 0 (u k , φ u k ) = I 0 (u k ) → 0 as k → +∞.

. 2 )
At this point, by(b)  taken any u ∈ E V,k with u 2 V = ∑ k j=1 |λ j | 2 = r 2 k ,we can choose r k small enough such that, by exploiting the equivalence of the norms | • | ∞ and • and (4.5),I 0 (u) < 0 holds for everyu ∈ E k ∩ S r k , S r k = {u ∈ E V : u ρ = r k }.Consequently, for every k ∈ N it results sup u∈E k ∩S r k I 0 (u) < 0,then, by well known properties of the genus, the thesis follows with A k = E k ∩ S r k .Now, we are ready to prove the multiplicity result stated in Theorem 1.2.