Ground state solutions of non‐linear singular Schrödinger equations with lack of compactness

Abstract

SUMMARYWe study a class of time-independent non-linear Schr odinger-type equations on the whole space with a repulsive singular potential in the divergence operator and we establish the existence of non-trivial standing wave solutions for this problem in an appropriate weighted Sobolev space. Such equations have been derived as models of several physical phenomena. Our proofs rely essentially on critical point theory tools combined with the Ca arelli-Kohn-Nirenberg inequality.

Existence and multiplicity of weak solutions for a class of degenerate nonlinear elliptic equations

“…In this paper our aim is to study the problem 2) where N ≥ 3 and 2 ≤ p < N. We point out the fact that in the case when a(x,∇u) = |x| α ∇u, α ∈ (0,2) and p = 2 problem (2.2) was studied by Mihȃilescu and Rȃdulescu in [10]. In that paper the authors present the connections between such equations and some Schrödinger equations with Hardy potential and show that (2.2) has a nontrivial weak solution.…”

“…In that paper the authors present the connections between such equations and some Schrödinger equations with Hardy potential and show that (2.2) has a nontrivial weak solution. A discussion of some physical applications for equations of type (2.2) and a list of papers devoted with the study of such problems is also included in [10].…”

On the Existence of Weak Solutions for a Degenerate and Singular Elliptic System in ℝ N

“…For more information and connection on problems of this type, the readers may consult in [13,16] and the references therein. Our paper is motivated by the recent results in [4,6,10,12] and [17]. In order to state our main result, we first introduce some hypotheses below.…”

“…It should be observed that a model example for (H) is h i (x) = |x| α , (i = 1, 2) (see [10,12]). The case α = 0 covers the "isotropic" case corresponding to the Laplacian operator.…”

EXISTENCE RESULT FOR NONUNIFORMLY DEGENERATE SEMILINEAR ELLIPTIC SYSTEMS INN

“…Note that by hypothesis (H), the problem which was considered here contains the situations in [6] and [7]. We also do not require the coercivity for the functions a(x) and b(x) as in [12]. Theorem 1.2 will be proved by using variational techniques based on a variant of the Mountain pass theorem [8].…”