%0 Journal Article
%T Pseudo-Index Theory for a Schrödinger Equation with Competing Potentials
%A Rui Sun
%J Open Access Library Journal
%V 10
%N 11
%P 1-22
%@ 2333-9721
%D 2023
%I Open Access Library
%R 10.4236/oalib.1110885
%X In this paper, we study a nonlinear Schr&ouml;dinger equation with competing potentials -&epsilon;<sup>2</sup>&Delta;&nu; V(x)&nu;=W<sub>1</sub>(x)|&nu;|<sup>p-2</sup>&nu; W<sub>2</sub>(x)|&nu;|<sup>q-2</sup>&nu;, &nu;&isin;H<sup>1</sup>(R<sup>N</sup>), where &epsilon;>0, p,q&isin;(2,2*), p>q, , V(x), W<sub>1</sub>(x) and W<sub>2</sub>(x) are continuous bounded positive functions. Under suitable assumptions on the potentials, we consider the existence, concentration, convergence and decay estimates of the ground state solution for this equation. Furthermore, the multiplicity of semi-classical solutions is established by using Benci pseudo-index theory, and the existence of sign-changing solutions is obtained via Nehari method.
%K Pseudo-Index
%K Multiplicity
%K Concentration
%K Sign-Changing Solution
%U http://www.oalib.com/paper/6808147