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

Abstract

This paper is concerned with the following fractional Schrödinger equation \begin{equation*} \left\{ \begin{array}{ll} (-Δ)^{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.

Cited by in corpus (3)

• On the fractional Schrödinger-Kirchhoff equations with electromagnetic fields and critical nonlinearity
• Sign-changing solutions for a class of zero mass nonlocal Schrödinger equations
• Multiple solutions for a fractional Schrodinger equation with potentials