Abstract
We consider a class of parametric Schrödinger equations driven by the fractional $p$-Laplacian operator and involving continuous positive potentials and nonlinearities with subcritical or critical growth. Using variational methods and Ljusternik-Schnirelmann theory, we study the existence, multiplicity and concentration of positive solutions for small values of the parameter.
Highlights
In the first part of this paper we focus our attention on the existence, multiplicity and concentration of positive solutions for the following fractional p-Laplacian type problem εsp(−∆)spu + V (x)|u|p−2u = f (u)
We introduce the assumptions on the potential V and the nonlinearity f
The paper is organized as follows: in Section 2 we collect some facts about the involved fractional Sobolev spaces and we provide some useful technical lemmas
Summary
In the first part of this paper we focus our attention on the existence, multiplicity and concentration of positive solutions for the following fractional p-Laplacian type problem. U(x) > 0 for all x ∈ RN , where ε > 0 is a parameter, s ∈ (0, 1), 1 < p < ∞, N > sp, W s,p(RN ) is defined as the set of the functions u : RN → R belonging to Lp(RN ) such that [u]pW s,p(RN ) =. |u(x) − u(y)|p R2N |x − y|N+sp dxdy < ∞. (−∆)sp is the fractional p-Laplacian operator which may be defined, up to a normalization constant, by setting (−∆)spu(x). For any u ∈ Cc∞(RN ); see [18, 34] for more details and applications. Fractional Schrodinger equation, fractional p-Laplacian operator, Nehari manifold, Ljusternik-Schnirelmann theory, critical growth
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