On the Existence of Solutions for the Critical Fractional Laplacian Equation in ℝN

Abstract

We study existence of solutions for the fractional Laplacian equation ${\left(-\Delta \right)}^{s}u+V\left(x\right)u={\left|u\right|}^{2^{*}\left(s\right)-2}u+f\left(x, u\right)$ in ${ℝ}^N$, $u\in H^s(R^N)$, with critical exponent $2^{*}\left(s\right)=2N/(N-2s)$, $N>2s$, $s\in \left(0, 1\right)$, where $V\left(x\right)\ge 0$ has a potential well and $f:{ℝ}^N\times ℝ\to ℝ$ is a lower order perturbation of the critical power ${\left|u\right|}^{2^{*}\left(s\right)-2}u$. By employing the variational method, we prove the existence of nontrivial solutions for the equation.