Existence of solutions for a fractional Laplacian equation with critical nonlinearity. (English) Zbl 1470.35412

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


35R11 Fractional partial differential equations
35A15 Variational methods applied to PDEs
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