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

摘要

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 positivefunction, $h\in L^{2}(\mathbb{R}^{N})$, $h\not \equiv 0$ 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 theexistence of at least two positive solutions for the problem underconsideration, provided that $|h|_{2}$ is sufficiently small.