Entire solutions of Schr\"{o}dinger elliptic systems with discontinuous nonlinearity and sign-changing potential

We establish the existence of an entire solution for a class of stationary Schr\"{o}dinger systems with subcritical discontinuous nonlinearities and lower bounded potentials that blow-up at infinity. The proof is based on the critical point theory in the sense of Clarke and we apply Chang's version of the Mountain Pass Lemma for locally Lipschitz functionals. Our result generalizes in a nonsmooth framework a result of Rabinowitz \cite{rabi} related to entire solutions of the Schr\"{o}dinger equation.


Introduction and the main result
The Schrödinger equation plays the role of Newton's laws and conservation of energy in classical mechanics, that is, it predicts the future behaviour of a dynamic system. The linear form of Schrödinger's equation is where ψ is the Schrödinger wave function, m is the mass, denotes Planck's constant, E is the energy, and V stands for the potential energy. The structure of the nonlinear Schrödinger equation is much more complicated. This equation describes various phenomena arising in: self-channelling of a highpower ultra-short laser in matter, in the theory of Heisenberg ferromagnets and magnons, in dissipative quantum mechanics, in condensed matter theory, in plasma physics (e.g., the Kurihara superfluid film equation). We refer to [7,14] for a modern overview, including applications. Consider the model problem where p < 2N/(N − 2) if N ≥ 3 and p < +∞ if N = 2. In the study of this equation Oh [12] supposed that the potential V is bounded and possesses a non-degenerate critical point at x = 0. More precisely, it is assumed that V belongs to the class (V a ) (for some a) introduced in Kato [10]. Taking γ > 0 and > 0 sufficiently small and using a Lyapunov-Schmidt type reduction, Oh [12] proved the existence of a standing wave solution of Problem (1), that is, a solution of the form ψ(x, t) = e −iEt/ u(x) . (2) Note that substituting the ansatz (2) into (1) leads to The change of variable y = −1 x (and replacing y by x) yields In a celebrated paper, Rabinowitz [13] continued the study of standing wave solutions of nonlinear Schrödinger equations. After making a standing wave ansatz, Rabinowitz reduces the problem to that of studying the semilinear elliptic equation under suitable conditions on b and assuming that f is smooth, superlinear and subcritical. Inspired by Rabinowitz' paper, we consider the following class of coupled elliptic systems in We point out that coupled nonlinear Schrödinger systems describe some physical phenomena such as the propagation in birefringent optical fibers or Kerr-like photorefractive media in optics. Another motivation to the study of coupled Schrödinger systems arises from the Hartree-Fock theory for the double condensate, that is a binary mixture of Bose-Einstein condensates in two different hyperfine states, cf. [6]. System (3) is also important for industrial applications in fiber communications systems [8] and all-optical switching devices [9].
Throughout this paper we assume that a, b ∈ L ∞ loc (R N ) and there exist a , b > 0 such that a(x) ≥ a , b(x) ≥ b a.e. in R N , and esslim |x|→∞ a(x) = esslim |x|→∞ b(x) = +∞. Our aim in this paper is to study the existence of solutions to the above problem in the case when f, g are not continuous functions. Our goal is to show how variational methods can be used to find existence results for stationary nonsmooth Schrödinger systems.
Throughout this paper we assume that f (x, ·, ·), g(x, ·, ·) ∈ L ∞ loc (R 2 ). Denote: Under these conditions we reformulate Problem (3) as follows: Let H 1 = H(R N , R 2 ) denote the Sobolev space of all U = (u 1 , u 2 ) ∈ (L 2 (R N )) 2 with weak derivatives ∂u 1 ∂x j , ∂u 2 ∂x j (j = 1, . . . , N ) also in L 2 (R N ), endowed with the usual norm Given the functions a, b : R N → R as above, define the subspace Then the space E endowed with the norm We assume throughout the paper that f, g : R N × R 2 → R are nontrivial measurable functions satisfying the following hypotheses: where p < 2 * ; f and g are chosen so that the mapping F : there exists µ > 2 such that for any Our main result is the following.

Auxiliary results
We first recall some basic notions from the Clarke gradient theory for locally Lipschitz functionals (see [4,5] for more details). Let E be a real Banach space and assume that I : E → R is a locally Lipschitz functional. Then the Clarke generalized gradient is defined by where I 0 (u, v) stands for the directional derivative of I at u in the direction v, that is, Let Ω be an arbitrary domain in R N . Set Then E Ω becomes a Hilbert space.
Proof. . We first observe that is a Carathéodory functional which is locally Lipschitz with respect to the second variable. Indeed, by where V is a neighbourhood of (t 1 , t 2 ), (s 1 , s 2 ). Set It is obvious that if U = (u 1 , u 2 ), V = (v 1 , v 2 ) belong to E Ω , then (χ 1 , χ 2 ) ∈ E Ω . So, by Hölder's inequality and the continuous embedding E Ω ⊂ L p (Ω; R 2 ), which concludes the proof.
The following result is a generalization of Lemma 6 in [11].

Lemma 2.
Let Ω be an arbitrary domain in R N and let f : . Then f and f are Borel functions. Proof. Since the requirement is local we may suppose that f is bounded by M and it is nonnegative. Denote by By the second inequality in (9) we obtain Let By the first inequality in (9) and the definition of the essential supremum we obtain that |A| > 0 and Since (10) and (11)  (ii) f (k) ∈ M and f (k) k → f imply f ∈ M.
Since N contains obviously the continuous functions and (ii) is also true for N then, by the Lebesgue dominated convergence theorem, we obtain that M = N . For f we note that f = −(−f )) and the proof of Lemma 2 is complete.
Proof. By the definition of the Clarke gradient we have in Ω. Thus, by Lemma 2, Analogously we obtain Arguing by contradiction, suppose that (12) is false. Then there exist ε > 0, a set A ⊂ Ω with |A| > 0 and w 1 as above such that Taking v = 1 A in (14) we obtain which contradicts (16). Proceeding in the same way we obtain the corresponding result for g in (13).
Proof. By the definition of the Clarke gradient we deduce that W,Ṽ ≤ Ψ 0 (U,Ṽ ) for all V in E Ω Ψ 0 (U,Ṽ ) = lim sup H→U, H∈E λ→0 By Lemmas 3 and 4 we obtain that for any W ∈ ∂Ψ(U ) (with U ∈ E), W Ω satisfies (12) and (13). We also observe that for Ω 1 , Then W 0 is well defined and 3 Proof of Theorem 1 Define the energy functional I : E → R The existence of solutions to problem (4) will be justified by a nonnsmooth variant of the Mountain-Pass Theorem (see [3]) applied to the functional I, even if the P S condition is not fulfilled. More precisely, we check the following geometric hypotheses: there exist β, ρ > 0 such that I ≥ β on {U ∈ E; U E = ρ}.
Verification of (19). It is obvious that I(0) = 0. For the second assertion we need the following lemma.
Proof. We first observe that (8) implies which places us in the conditions of Lemma 5 in [11].
Set λ I (U ) = min Thus, by the nonsmooth version of the Mountain Pass Lemma [3], there exists a sequence {U M } ⊂ E such that So, there exists a sequence {W m } ⊂ ∂Ψ(U m ); W m = (w 1 m , w 2 m ) such that Note that, by (8), Therefore, by (17), for every U ∈ E and W ∈ ∂Ψ(U ). Hence, if ·, · denotes the duality pairing between E * and E, we have This relation in conjunction with (22) implies that the Palais-Smale sequence {U m } is bounded in E. Thus, it converges weakly (up to a subsequence) in E and strongly in L 2 loc (R N ) to some U . Taking into account that W m ∈ ∂Ψ(U m ) and U m ⇀ U in E, we deduce from (23) that there exists W ∈ E * such that W m ⇀ W in E * (up to a subsequence). Since the mapping U −→ F (x, U ) ia compact from E to L 1 , it follows that W ∈ ∂Ψ(U ). Therefore These last two relations show that U is a solution pf the problem (4). It remains to prove that U ≡ 0. If {W m } is as in (23), then by (8), (17), (22) and for large m Now, taking into account the definition of f , f , g, g we deduce that f , f , g, g verify (19), too. So by (24) we obtain So, {U m } does not converge strongly to 0 in L p+1 (R N ; R 2 ). From now on, a standard argument implies that U ≡ 0, which concludes our proof.