Multiplicity and concentration of solutions for elliptic systems with vanishing potentials

Abstract

In this article we use variational methods to study a strongly coupled elliptic system depending on a positive parameter λ. We suppose that the potentials are nonnegative and the intersection of the sets where they vanish has positive measure. A technical condition, imposed on the product of the potentials, allows us to consider a setting where we do not assume any positive lower bound for the potentials. Considering the associated functional, defined on an appropriated subspace of D 1,2 (R N ) × D 1,2 (R N ), we are able to establish results on the existence and multiplicity of solutions for the system when the parameter λ is sufficiently large.We also study the asymptotic behavior of these solutions when λ → ∞.

Infinitely many small energy solutions for fractional coupled Schrodinger system with critical growth

“…Obviously, the form of (1.1) is more general than that of (1.2)-(1.3). Hence our results can be viewed as an extension to the results in [4,10,11,14,16,22,23,28].…”

“…By applying the generalized mountain pass lemma, the author established the existence of one nontrivial solution of (1.2). From then on, the coupled quasi-linear Schrödinger system had attracted more and more attention, see [4,10,14,22,23,28] and the references therein. But for the coupled quasi-linear Schrödinger system involving fractional Laplacian, there are very few work.…”

Ground state solutions for a coupled Kirchhoff‐type system

“…We also note that when a i =1, b i =0, problem reduces to the nonlinear Schrödinger system. Different approaches have been taken to deal with this nonlinear Schrödinger system under various hypotheses on the potentials and the nonlinearity; see , and the references therein.…”

“…The following Brézis‐Lieb type lemma is proved in Lemma 4.2 of . Lemma Let be a sequence such that weakly in .…”

On a critical Schrödinger system in R4 with steep potential wells

“…Most of the papers are devoted to the subcritical case. In recent years, the steep potential wells were also introduced to some other elliptic equations and systems, see for example [24,25,27,41,49] and the references therein. In particular, in [49], the Gross-Pitaevskii equations in R 3 (subcritical case) with steep potential wells were considered and some existence results of the solitary wave solutions were established.…”