Multiple solutions for a class of nonhomogeneous fractional Schr\"odinger equations in $\mathbb{R}^{N}$

TL;DR

This paper proves the existence of multiple positive solutions for a class of nonhomogeneous fractional Schrödinger equations in ^N using variational methods and the s-harmonic extension technique, especially when the nonhomogeneous term is small.

Contribution

It introduces a novel application of variational methods combined with the s-harmonic extension to establish multiple solutions for nonhomogeneous fractional Schrd6dinger equations.

Findings

Existence of at least two positive solutions under small ^2 norm of h.

Solutions are obtained for equations with asymptotically linear or superlinear nonlinearities.

The approach extends previous methods to nonhomogeneous fractional problems.

Abstract

This paper is concerned with the following fractional Schr\"odinger equation \begin{equation*} \left\{ \begin{array}{ll} (-\Delta)^{s} u+u= k(x)f(u)+h(x) \mbox{ in } \mathbb{R}^{N}\\ u\in H^{s}(\R^{N}), \, u>0 \mbox{ in } \mathbb{R}^{N}, \end{array} \right. \end{equation*} where $s\in (0,1)$, $N> 2s$, $(-\Delta)^{s}$ is the fractional Laplacian, $k$ is a bounded positive function, $h\in L^{2}(\mathbb{R}^{N})$, $h\not \equiv 0$ is nonnegative and $f$ is either asymptotically linear or superlinear at infinity.\\ By using the $s$-harmonic extension technique and suitable variational methods, we prove the existence of at least two positive solutions for the problem under consideration, provided that $|h|_{2}$ is sufficiently small.