Existence and multiplicity of solutions for Schrödinger equations with sublinear nonlinearities

Abstract: In the paper, we investigate a class of Schrödinger equations with sign-changing potentials V(x) and sublinear nonlinearities. We remove the coercive condition on V(x) usually required in the existing literature and also weaken the conditions on nonlinearities. By proving a Hardy-type inequality, extending the results in [1], and using it together with variational methods, we get at least one or infinitely many small energy solutions for the problem.


Introduction and main results
In this paper, we are devoted to studying the existence and multiplicity of solutions for the following Schrödinger equation with a sublinear nonlinearity, where N ≥ 3, V is sign-changing, K is positive and f ∈ C(R). We remove the coercive condition usually imposed on V(x) and obtain the existence of at least one or infinitely many small energy solutions to (1.1) for sublinear nonlinearities K(x) f (u). As mentioned in [1,10,12], this type of equations is essentially related to seeking for the standing waves ψ(t, x) = e −iωt u(x) for the time-dependent Schrödinger equation, where the potential V is given by V(x) = U(x) − ω. Hence V may be indefinite in sign for large ω(see [1,23]).
Much attention has been paid on the following equation, involving a continuous term V(x). We refer, for instance, to [2-8, 11, 14, 15, 17-20, 22, 23] and the references therein. It is known to all that the main difficulty in dealing with problem (1.3) arises from the lack of the compactness of Sobolev embeddings, which prevents from checking directly that the energy functional associated with (1.3) satisfies the PS-condition.
To obtain the compactness in R N , some feasible methods are provided in the existing papers. For example, Bartsch, Pankov and Wang [6] have studied a class of Schrödinger equations, where V(x) is continuous function verifying the following conditions, (v 1 ) ess inf V(x) > 0; (v 2 ) for any M > 0, there exists x 0 such that lim |y|→∞ meas({x ∈ R N : |x − y| ≤ x 0 , V(x) ≤ M}) = 0, where meas devotes the Lebesgue measure on R N . Under conditions (v 1 ) and (v 2 ), the compactness of Sobolev embedding can be recovered. With the assumptions (v 1 ) and (v 2 ), equation (1.3) has been investigated by the variational methods by [6] and some other authors.
In [22], the authors studied a class of sublinear Schrödinger equations, where f (x, u) = ξ(x)|u| µ−2 u with 1 < µ < 2 and ξ(x) : R N → R being a positive continuous function. Under conditions (v 1 ) and (v 2 ), they established a theorem on the existence of infinitely many small energy solutions.
The results of [22] were improved in the recent paper [7], where they improved the results of [22] by removing assumption (v 2 ) and relaxing the assumptions on f (x, t). By using the genus properties in critical point theory, they established some existence criteria to guarantee that the problem has at least one or infinitely many nontrivial solutions.
In [8], for problem (1.3), Cheng and Wu studied a sublinear problem and used conditions on V(x) below: Under some additional conditions of f , two theorems are obtained in [8]. One theorem states that equation (1.3) possesses at least one nontrivial solution. By using a variant fountain theorem, they obtained the existence of infinitely many small energy solutions in another theorem.
Bao and Han [4] also considered a nonlinear sublinear Schrödinger equation, Under some conditions on V(x) and by using bounded domain approximation technique, infinitely many small energy solutions are obtained. In those above papers, (v 2 ), (V2) or the coercive condition on V plays an important role in obtaining the compact embedding. In this paper, we remove the coercive condition of V(x) and also weaken the conditions on f.
We remark that there have been many interesting results for the similar sublinear problems (1.1) but on bounded domains Ω ⊂ R N . We refer to [13] for some results for p−Laplacian equation problems and the references therein.
measΩ > 0 and there exists a large constant R 0 such that V(x) > 0 for a.e. |x| ≥ R 0 . (V 2 ) There exists a constant η 0 > 1 such that Conditions similar to (V 2 ) can be found in [9] and [16]. By condition (V 1 ) and the Hölder and Sobolev inequalities, where S is the best constant for the Sobolev embedding of D 1,2 By (V 2 ) and a simple calculation, More details on condition (V 2 ) can be found in [9]. We emphasize that the conditions on V(x) in this paper are essentially different from those in [8] and [22]. In fact, we are dealing with the vanishing potentials V(x). As far as we know, for problem (1.1) with sublinearity, few works in this case seem to have appeared in the literature. Since V(x) is sign-changing and vanishing, it seems not to be obvious from the literature to obtain the compactness suitable to deal with the problem. By proving a Hardy-type inequality, which extends the results in [1], we can obtain the needed compactness. Our theorems also extend the results in [8,22] and our hypotheses on nonlinearities are more general.
The paper is organized as follows. In Section 2, we introduce the variational setting and state some preliminary results which will be needed later. In Section 3, the proofs of our main results are given.

Variational setting
In this paper, we define We know E is a separable Hilbert space with the inner product It is well known that the embedding E ⊂ L s (R N )(2 ≤ s ≤ 2 * ) is continuous. Now we give a Hardy-type inequality which extends the one in [1] and is suitable for dealing with our sublinear problems. Before stating the result, we recall condition (A), As stated in [1], if K ∈ L 1 (R N \B ρ (0)) for some ρ > 0, we know that K satisfies condition (A).
is well defined and belongs to C 1 (E, R). Moreover, Proof. By (V 1 ) and Lemma 2.13 in [21], we know R N V − (x)|u| 2 dx is well defined for u ∈ E. By virtue of ( f 1 ), (2.21) Hence, by Lemma 2.1, we get which means that J is well defined for u ∈ E. By a direct computation, it is not difficult to prove that (2.20) holds. Furthermore, by a standard argument, we obtain that the critical points of J in E are solutions of problem (1.1).
Finally, we will show that J (u) is weakly continuous, that is, if u n u in E, then Arguing directly, by u n u in E, choose a subsequence {u n k } of {u n } such that u n k (x) → u(x) a.e. in R N and Q 1 (x) ∈ L 2 K (R N ), where It is clear that 1 (x)+K(x)|u| 2(τ i −1) . By (V 1 ) and (KV), we obtain that K ∈ L 1 (R N ). By 1 τ i −1 > 1 and 1 2−τ i > 1, one has

(2.25)
This together with Lebesgue's Dominated Convergence Theorem implies that Therefore, for any v ∈ E, Hence, J (u) is weakly continuous in E. The proof is complete. Let {e j } be an orthonormal basis of the Hilbert space E and define X j = Re j , For the statement of Dual Fountain Theorem, we need the following condition. More details can be found in [21].
(A 1 ) A compact group G acts isometrically on the Hilbert space E = j∈N X j , the spaces X j are invariant and there exists a finite dimensional space V such that, for every j ∈ N, X j V and the action of G on V is admissible. Lemma 2.5. (Theorem 3.18 in [21] Dual Fountain Theorem, Bartsch-Willem, 1995) Assume that condition (A 1 ) holds and let J ∈ C 1 (E, R) be an invariant functional. If, there exist two sequences 0 < r k < ρ k → 0 as k → ∞ and the following conditions (D 1 )−(D 4 ) hold, then J has a sequence of negative critical values converging to 0, where (D 1 ) a k := inf (D 4 ) for every c ∈ [d k , 0), J satisfies the (PS ) * c condition, that is, every sequence u n j ∈ E satisfying u n j ∈ Y n j , J(u n j ) → c, J| Y n j (u n j ) → 0, n j → ∞ contains a subsequence converging to a critical point of J. Proof. It is obvious to obtain that
By Lemma 3.1 and Ekeland's variational method, there exists a minimizing sequence {u n } such that J(u n ) → inf E J and J (u n ) → 0, as n → ∞.
By Lemma 3.1, it is clear that the minimizing sequence {u n } is bounded in E. Proof. By Lemma 3.1, the minimizing sequence {u n } is bounded. Passing to a subsequence, one has By a direct computation, we derive that We can see that This together with (3.2), for large n, we obtain that Similarly, for that ε and large n, The proof is complete. Proof of Theorem 1.2. Assume that conditions (K 1 ), (V 1 ), (V 2 ), (KV), ( f 1 ) and ( f 2 ) hold. Then the limit u 0 of the minimizing sequence {u n } is nontrivial.
This proof is complete.
Proof of Theorem 1.3. We just need to prove the (PS ) * c condition. Consider a sequence {u n j } such that n j → ∞, u n j ∈ Y n j , J(u n j ) → c, J | Y n j (u n j ) → 0.
Since {u n j } is bounded in E, by Lemma 2.1, we get that the sequence {u n j } has strong convergent subsequence in E. Passing to a sequence, we suppose that u n j → u k in E. Thus, by Lemma 2.5, for each k, {u k } is a critical point of J and J(u k ) → 0, as k → ∞. Hence, (1.1) possesses infinitely many small energy solutions. The proof of Theorem 1.3 is complete.