STANDING WAVE SOLUTIONS OF SCHRÖDINGER SYSTEMS WITH DISCONTINUOUS NONLINEARITY IN ANISOTROPIC MEDIA

We establish the existence of an entire solution for a class of stationary Schrödinger 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 (1992) related to entire solutions of the Schrödinger 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 behavior 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 high-power 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 nondegenerate critical point at x = 0.More precisely, it is assumed that V belongs to the class (V a ) (for some a) introduced by Kato in [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 (1.2), that is, a solution of the form ψ(x,t) = e −iEt/ u(x). (1.3) Note that substituting the ansatz (1.3) into (1.2) leads to The change of variable y = −1 x (and replacing y by x) yields where 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 R N (N ≥ 3): (1.7) 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.[5]).System (1.7) 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.
Teodora-Liliana Dinu 3 Throughout this paper, we assume that f (1.8) Under these conditions, we reformulate problem (1.7) as follows: (1.9) 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 becomes a Hilbert space.Since a(x) ≥ a > 0, b(x) ≥ b > 0, we have the continuous embeddings We assume throughout the paper that f ,g : R N × R 2 → R are nontrivial measurable functions satisfying the following hypotheses: where p < 2 * ; lim f and g are chosen so that the mapping F : (1.18) Our main result is the following.

Auxiliary results
We first recall some basic notions from the Clarke gradient theory for locally Lipschitz functionals (see [3,4] 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, Teodora-Liliana Dinu 5 Let Ω be an arbitrary domain in R N .Set which is endowed with the norm Then E Ω becomes a Hilbert space.
Lemma 2.1.The functional Proof.We first observe that is a Carathéodory functional which is locally Lipschitz with respect to the second variable.Indeed, by (1.13), (2.6) Similarly, where V is a neighborhood of (t 1 ,t 2 ), (s 1 ,s 2 ).Set (2.9) 6 Schrödinger systems with discontinuous nonlinearity Hölder's inequality and the continuous embedding which concludes the proof.
The following result is a generalization of [11,Lemma 6].
Lemma 2.2.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 , we deduce that for every ε > 0, there exists n ∈ N * such that for By the second inequality in (2.12), we obtain By the first inequality in (2.12) and the definition of the essential supremum, we obtain that |A| > 0 and it suffices to prove that f m,n is Borel.Let ᏹ is the smallest set of functions having the following properties (cf.[1, page 178]): Since ᏺ contains obviously the continuous functions and (ii) is also true for ᏺ, then by the Lebesgue dominated convergence theorem, we obtain that ᏹ = ᏺ.For f , we note that f = −(− f ) and the proof of Lemma 2.2 is complete.
Let us now assume that Ω ⊂ R N is a bounded domain.By the continuous embedding L p+1 (Ω;R 2 ) L 2 (Ω;R 2 ), we may define the locally Lipchitz functional

Under the above assumptions and for any
) Proof.By the definition of the Clarke gradient, we have (2.23) Analogously, we obtain Arguing by contradiction, suppose that (2.20) is false.Then there exist ε > 0, a set A ⊂ Ω with |A| > 0, and w 1 as above such that (2.25) (2.28) Proof.By the definition of the Clarke gradient, we deduce that W, V ≤ Ψ 0 (U, V ) for all (2.29) Teodora-Liliana Dinu 9 By Lemmas 2.3 and 2.4, we obtain that for any W ∈ ∂Ψ(U) (with U ∈ E), W Ω satisfies (2.20) and (2.21).We also observe that for Then W 0 is well defined and a.e.x ∈ R N . (2.31)

Proof of Theorem 1.2
Define the energy functional I : E → R by The existence of solutions to problem (1.9) will be justified by a nonsmooth variant of the mountain pass theorem (see [2]) applied to the functional I, even if the PS condition is not fulfilled.More precisely, we check the following geometric hypotheses: Verification of (3.2).It is obvious that I(0) = 0.For the second assertion, we need the following lemma.
Lemma 3.1.There exist two positive constants C 1 and C 2 such that Proof.We first observe that (1.16) implies that which places us in the conditions of [11,Lemma 5].
Thus by Lemma 3.1, we obtain for t > 0 large enough.
Verification of (3.3).We observe that (1.14), (1.15), and (1.16) imply that for any ε > 0, there exists a constant A ε > 0 such that By (3.7) and Sobolev's embedding theorem, we have for any U ∈ E, where ε is arbitrary and C 4 = C 4 (ε).Thus for U E = ρ, with ρ, ε, and β sufficiently small positive constants.Denote Teodora-Liliana Dinu 11 Thus, by the nonsmooth version of the mountain pass lemma [2], 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 (1.16), Therefore, by (2.31), 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 (3.12) 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 (3.13) that there exists W ∈ E * such that W m W in E * (up to a subsequence).Since the mapping U Now, taking into account the definition of f , f , g, g, we deduce that f , f , g, g verify (3.2) too.So by (3.18), we obtain So, {U m } does not converge strongly to 0 in L p+1 (R N ;R 2 ).From now on, with the same arguments as in the proof of [6, Theorem 1], we deduce that U ≡ 0, which concludes our proof.

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable: