On the existence of solutions for the critical fractional Laplacian equation in \(\mathbb{R}^N\) (Q1722178)

Summary: 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 \(\mathbb{R}^N\), \(u \in H^s(R^N)\), with critical exponent \(2^* \left(s\right) = 2 N /(N - 2 s)\), \(N > 2 s\), \(s \in \left(0, 1\right)\), where \(V \left(x\right) \geq 0\) has a potential well and \(f : \mathbb{R}^N \times \mathbb{R} \rightarrow \mathbb{R}\) 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.