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Weak solutions for a fractional \(p\)-Laplacian equation with sign-changing potential. (English) Zbl 1331.35380

Summary: In this paper, we use some critical point theorems to discuss the existence of weak solutions for the fractional \(p\)-Laplacian equation in \(\mathbb R^N\) \[ (-\Delta)^\alpha_pu+V(x)|u|^{p-2}u-\lambda|u|^{p-2}u=f(x,u)+g(x)|u|^{q-2}u,\quad x\in\mathbb R^N, \] where \(N\), \(p\geq 2\), \(\alpha\in (0,1)\) \(\lambda\) is a parameter, \((-\Delta)^\alpha_p\) is the fractional \(p\)-Laplacian and \(f:\mathbb R^N\times\mathbb R\to\mathbb R\) is a Carathéodory function.

MSC:

35R11 Fractional partial differential equations
35A15 Variational methods applied to PDEs
35J92 Quasilinear elliptic equations with \(p\)-Laplacian
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35D30 Weak solutions to PDEs
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References:

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