EXISTENCE OF POSITIVE SOLUTIONS OF SCHR¨ODINGER EQUATIONS WITH VANISHING POTENTIALS

. We prove the existence of at least one positive solution for a Schr¨odinger equation in R N of type − ∆ u + V ( x ) u = f ( x,u ) in R N with a vanishing potential at inﬁnity and subcritical nonlinearity f . Our hy- potheses allow us to consider examples of nonlinearities which do not verify the Ambrosetti-Rabinowitz condition, neither monotonicity conditions for the function f ( x,s ) s . Our argument requires new estimates in order to prove the boundedness of a Cerami sequence. phrases. Cerami sequences, variational methods, Schr¨odinger equation, nonau-tonomous


(Communicated by Bernhard Ruf)
Abstract. We prove the existence of at least one positive solution for a Schrödinger equation in R N of type 1. Introduction. In the present paper, we prove the existence of at least one positive solution of the equation for N ≥ 3 and assuming that f : R N × R → R is a Carathéodory nonnegative function and V : R N → R is a nonnegative potential. Here, we assume that f is superlinear at the origin and at infinity and has subcritical growth. Also, we consider cases where the potential V : R N → R can vanish at infinity. Recently several authors studied equation (1) using variational techniques such as fractional Sobolev spaces, reduction methods, generalized mountain pass theorem, dual variational formulation, generalized fountain theorem and generalized linking theorems, see for instance [3,6,12,13,14,16], where these authors consider several kinds of behaviour for the potential V : R N → R and for the nonlinearity f .
Equation (1) appears in various applications, such as chemotaxis, population genetics, chemical reactor theory and also the solutions of this class of problems are related to the existence of standing wave solutions ψ(x, t) = exp −iEt ε v(x) for nonlinear Schrödinger equation where ε > 0, E ∈ R and v is a real function. Equation (2) is one of the main objects of the quantum physics, since it appears in problems which involve nonlinear optics, plasma physics and condensed matter physics. We notice that ψ satisfies (2) if, and only if, the function v(x) satisfies equation (1) with f (u) = g(|u|)u |u| .
An interesting class of problems associated with (1) is the zero mass case. This case happens when the potential V : R N → R vanishes at infinity, which means, In [1], the authors define that (V, K) ∈ K if the following conditions hold: is a sequence of Borel sets such that |A n | ≤ R, for all n and some R > 0, then lim r→+∞ An∩B c r (0) K(x)dx = 0, uniformly in n ∈ N.
(K 1 ) (III): One of the below conditions occurs: The main advantage on considering such hypothesis relies on the fact that the space D 1 V (R N ) endowed with the norm is compactly embedded into the weighted Lebesgue space L p K (R N ), for all p ∈ (2, 2 * ) (see [1,Proposition 2.1]). We notice that this set of conditions generalizes the (V K) condition stated in [2].
More recently, in [9], the author introduces many different conditions in order to prove that the embedding described above is compact for p ∈ [2, 2 * ). See [9,Theorem 4.1].
In order to obtain a nontrivial solution of equation (1) in the zero mass case, previous results always consider such equation in the following form where this function K : R N → R is used to obtain some compactness results, see [1,2,3,4,9] and the references therein for instance.
The novelty here is that we will introduce hypotheses on the nonlinearities that allows us to consider examples which do not verify the condition of Ambrosetti-Rabinowitz or certain monotonicity conditions. Also, our approach enables us to include examples which can not be treated as equation (3). Note that we use variational methods involving Cerami sequences where the most difficult part in our argument is to prove that such sequences are bounded.
For instance, we can consider where a : R N × R + → R is a nondecreasing function, such that 0 ≤ a(x, s) ≤ 1 and a(x, s) → 1 as s → ∞ uniformly in x and q > N . An example of such function is a(x, s) = 2 π arctan ((1 + |x|)s). This function f does not verify the classical Ambrosetti-Rabinowitz condition. Another example that verifies our hypotheses is may not be monotone in s according to the shape of b(x, s). We notice that both examples of nonlinearities can not be written as in (3). It should be noted that the results we will use to handle compactness are mainly included in [9, Theorem 4.1].
Since we are interested in obtaing positive solutions to problem (1), we assume that f : R N × R → R verifies f (x, s) > 0 for s > 0 and f (x, s) = 0 for s ≤ 0.
Firstly we introduce our hypotheses on the functional V : (V 1 ) V (x) > 0 for all x ∈ R N is measurable in R N and V (x) → 0 almost everywhere (a.e.) as |x| → +∞; (V 2 ) There is a positive measurable function K : and ω(x) := K(x)V −1 (x) > 0, satisfies a.e. as |x| → +∞. We notice that if hypotheses (V 1 ) and (V 2 ) hold, then we may use some compactness results of [9].
Concerning the function f we assume the following conditions: uniformly for x ∈ R N ; and for any s 0 < +∞, there exists a function ψ ∈ L 1 (R N ) (not necessarily positive) such thatF Remark 1. Condition (f 4 ) will be essential in the proof of the boundedness of the Cerami sequence (see Lemma 3.3 below). To the best of our knowledge, a similar condition to (f 4 ) was firstly introduced in [7] for a Schrödinger equation in the case of a periodic potential and superlinear nonlinearities. Since we are assuming a different type of superlinearity given by hypothesis (f 3 ), it is natural to consider a modified condition such as (f 4 ). We also notice that since K ∈ L ∞ (R N ), it follows that hypothesis (f 4 ) implies that hypothesis (N 4) in [7] holds. To the best of our knowledge, our work is the first to consider an hypothesis such as (f 4 ) in the context of Schrödinger equations with vanishing potentials.

Remark 2.
Our choice for conditions similar to the ones presented in [9, Theorem 4.1] is based on the fact that the compactness result presented in [9] holds for p = 2 which is essential in the proof of Lemma 3.3 below. For more results concerning these type of embeddings we refer the reader to [2,4].
Remark 3. Hypotheses (V 2 ) and (f 5 ) provide the following relation between the nonlinearity f and the potential V : Our main result is Then, problem (1) possesses at least one nonnegative solution u. If in addition we suppose that the potential V is bounded then, problem (1) possesses at least one positive solution u.
Note that in [9, Theorem 1.2], in order to obtain the existence of a positive solution, the author imposes the following assumption: There exists η ≥ 1 such thatF (s) ≤ ηF (t) for all 0 ≤ s ≤ t, which is a type of monotonicity inF and implies thatF ≥ 0 for all s ≥ 0. We notice that our assumption inF allows us to produce examples whereF (x, s) < 0 for some s ∈ R even in the case when f (x, s) = K(x)g(s).
The main idea for the proof of Theorem 1.1 is to use variational methods. In order to prove that the functional associated with equation (1) has at least one positive critical point, we first obtain a Cerami sequence of this functional showing that it possesses a mountain pass structure, then we prove that the sequence is bounded, which is the most difficult part of the present paper since, for example, we are not assuming conditions such as Ambrosetti-Rabinowitz, neither monotonicity of f (x, s) s . After that, we prove some convergence results and finally we prove that the Cerami sequence converges to a nontrivial critical point of the functional.
The rest of this paper is organized in the following way: In section 2 we gather some preliminary results and present our variational setting involving Cerami sequences. In section 3, we show that the Cerami sequence is bounded. Finally, in section 4 we prove Theorem 1.1 and present examples of our main result.
2. Preliminary results. In this section we present the main tools in order to prove Theorem 1.1. In [9], the author introduces some conditions in order to prove that the embedding is compact for all q ∈ [2, 2 * ), where for some q > 1 is the weighted Lebesgue space, where as usual See [9,Theorem 4.1]. We notice that under the hypotheses of Theorem 1.1 we can apply Theorem 4.1 of [9] to show that the embedding (4) is compact for q ∈ [2, 2 * ).
Since we are working with the subcritical case we shall work with a smaller space than D 1 V (R N ). This space is We denote the norm in H 1 V (R N ) by The energy functional associated to (1) is given by defined on E : where ε > 0 is sufficiently small and c ε > 0 is sufficiently large, for p ∈ (2, 2 * ).
Remark 5. It follows from Remark 4 that f (x, s) is bounded in compact sets of R N × [0, +∞).
For the convenience of the reader we state here a direct consequence of [9, Theorem 4.1] Proposition 1. Let 2 ≤ t < 2 * and V, ω : R N → R be two measurable functions for each subset Ω of R N having finite measure |Ω| < ∞, and such that |{x ∈ R N ; ω(x) ≥ c}| < ∞. Then, the embedding D 1 is continuous and therefore we have the following result, which is a direct consequence of (6) and Proposition 1.
is compact is compact for q ∈ [2, 2 * ) Notice that by Remark 4 and the embeddings described above and by standard Dominated Convergence Theorem arguments we have that the functional J is well defined on E and J ∈ C 1 (E, R) with Fréchet derivative given by for any u, v ∈ E. Therefore, positive solutions of (1) correspond to positive critical points of J on E.
3. Boundedness of the Cerami sequence. In this section, we obtain a Cerami sequence for the functional (5) and show that this sequence is bounded. Finally, we prove some convergence results which will be essential in the proof of Theorem 1.1. Let (X, · ) be a real Banach space with dual space (X * , · * ), I ∈ C 1 (X, R) and c ∈ R. We call a sequence {x n } ⊂ X a Cerami sequence at level c and denote (C) c for short, if I(x n ) → c and (1+ x n ) I (x n ) * → 0 as n → ∞ and we say that I satisfies the Cerami condition if every (C) c sequence has a strongly convergent subsequence in X.
The following result is a version of the mountain pass theorem for (C)c sequences. See [15]. This result states that the mountain pass geometry is sufficient to obtain a (C) c sequence.  In the next result we prove that the functional J possesses the mountain pass geometry.
Then the functional J satisfies (I 1 ) and (I 2 ).
Proof. Firstly, notice that J(0) = 0. By Remark 4 and Proposition 1 it follows that Thus, for u = ρ sufficiently small. Now, we show that there is a e ∈ E such that e > ρ and J(e) < 0. Hypoth- Let φ(x) ≡ 1 in B(x 0 , δ), φ(x) ≥ 0, for all x ∈ R N and φ ∈ C ∞ 0 (R N ). We claim that there is R 0 > 0 such that, for any R > R 0 , we have J(Rφ) < 0. If that is the case, we take e = Rφ with R > 0 large enough.
For any R ≥ s M , we have Thus, Therefore, taking M sufficiently large, we obtain K(x)dx < 0 which means that for R > R 0 (s M ), J(Rφ) < 0 and this concludes the proof.
In the following lemma, we prove that the (C) c -sequence obtained in the previous result is bounded in E.
Proof. Suppose that {u n } ⊂ E is a (C) c -sequence of J, i.e., J(u n ) → c and (1 + u n ) J (u n ) E * → 0 as n → ∞. Then, as n → ∞. By (9), for n sufficiently large, We may assume, by contradiction, that for some subsequence, still denoted by {u n }, u n → ∞. We set Then {w n } is bounded in E with w n = 1 and up to a subsequence we can assume that w n w in E and by Corollary 1 a.e. in R N . Notice that Thus, Therefore, Let 0 ≤ a < b ≤ +∞ and we define It follows from (10) that for n sufficiently large we have Let ε > 0 be arbitrary and C 3 > 0 be such that w 2 . By (f 1 ), there exists a = a ε > 0 such that |f (x, s)| ≤ ε 3C 3 K(x)s for each |s| ≤ a. Then, for any n ∈ N, we have An(0,a) An(0,a) for all n ∈ N. Now we deal with the set A n (b, +∞). Firstly, we notice that by hypotheses (f 3 ), (f 4 ) and (12), where the first inequality follows from hypothesis (f 4 ). Therefore, as b → +∞ uniformly in n. Hence, we obtain for any s ∈ (1, 2 * ) as b → ∞ uniformly in n. Thus, by hypothesis (f 4 ), (12) and (14) it follows that An(b,+∞) Finally, we deal with the set A n (a, b), where a and b were chosen in the previous steps. By Remark 3, there is R > 0 sufficiently large such that Since f (x, s) is bounded in compact sets of R N × [0, +∞), we have that there exists a constant C = C a,b,R > 0 such that for n sufficiently large since we are assuming that u n → +∞ as n → +∞. Gathering all these informations, we obtain and this is a contradiction with (11). Therefore, {u n } is bounded in E and the lemma is proved.
{u n } ⊂ E be a bounded sequence. If u n is such that u n u in E, then as n → ∞.
Proof. As noticed in Remark 4, for all ε > 0 with some sufficiently large c ε > 0. Since we are assuming that u n u in E it follows from Corollary 1 that → 0 (16) as n → ∞. Also, by Corollary 1 we have that u n (x) → u(x) a.e. in R N and u n → u in L t K (R N ) for all t ∈ [2, 2 * ), which means that as n → +∞. Thus, by the reciprocal of the dominated convergence theorem (see [5,Theorem 4.9]) we have that up to a subsequence, there exists a function h ∈ L t (R N ) where h 1 ∈ L 2 (R N ) and h 2 ∈ L p (R N ) were obtained by the reciprocal of the dominated convergence theorem. Thus, by the dominated convergence theorem we obtain as n → ∞.
Gathering (16) and (17) we obtain as n → ∞ and the lemma is proved.
It follows from Lemma 3.3 that {u n } is bounded and thus up to a subsequence, we can assume that there is u ∈ E such that u n u in E.
Also, since J (u n )u = o n (1), we obtain Hence, by (18) and (19), lim n→∞ u n 2 = u 2 which shows that u n → u in E. Therefore, u is a nontrivial solution of problem (1) with J(u) = c. Aplying u − (x) = max{−u(x), 0} as a test function in the weak formulation of problem (1) we conclude that u is a nonnegative solution of (1). Besides that, if we assume that the potential V in (1) is bounded, we obtain as a direct consequence of [8,Theorem 8.18] that u > 0 which completes the proof. Example 1. Consider the following equation where 0 < α < 1, p ∈ (2, 2 * ) and the function a : R N × R → R is such that 0 < a 1 < a(x, s) < a 2 with a 1 2 − a 2 p > 0. One can take a function K : R N → R