Fractional Schrödinger equations involving potential vanishing at infinity and supercritical exponents

Abstract

This work studies the existence of positive solutions for fractional Schrodinger equations of the form $$\begin{aligned} (-\Delta )^s u + V(x) u = g(u)+\lambda |u|^{q-2}u\quad \text{ in }\quad {\mathbb {R}}^N, \end{aligned}$$ where $$s\in (0,1)$$ , $$N>2s$$ , $$V:{\mathbb {R}}^N\rightarrow {\mathbb {R}}$$ is a potential function which can vanish at infinity, $$g:{\mathbb {R}}\rightarrow {\mathbb {R}}$$ is superlinear and has subcritical growth, the exponent $$q\ge 2^*_s:=2N/(N-2s)$$ and $$\lambda $$ is a nonnegative parameter. Our approach is based on a truncation argument in combination with variational techniques and the Moser iteration method.