On the Existence of Solutions for the Critical Fractional Laplacian Equation in ℝ N

Abstract EN

We study existence of solutions for the fractional Laplacian equation - Δ s u + V x u = u 2 * s - 2 u + f x , u in ℝ N , u ∈ H s ( R N ) , with critical exponent 2 * s = 2 N / ( N - 2 s ) , N > 2 s , s ∈ 0 , 1 , where V x ≥ 0 has a potential well and f : ℝ N × ℝ → ℝ is a lower order perturbation of the critical power u 2 * s - 2 u .

By employing the variational method, we prove the existence of nontrivial solutions for the equation.