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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.

MSC:

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