Existence of Solutions for a Fractional Laplacian Equation with Critical Nonlinearity

Abstract

We study the fractional Laplacian equation ${(-\mathrm{\Delta })}^{s}u+\lambda A(x)u=\mu u+{\left|u\right|}^{{\mathrm{2}}^{*}(s)-\mathrm{2}}u$, $x\in {ℝ}^N$, here $N>2s$, $s\in (0,1)$, $2^{*}(s)=2N/(N-2s)$ is the critical exponent, and $A(x)\ge 0$ is a real potential function. Employing the variational method we prove the existence of nontrivial solutions for $\mu$ small and $\lambda$ large.