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Existence of semiclassical ground state solutions for a generalized Choquard equation. (English) Zbl 1309.35036

Authors’ abstract: In this paper, we study a generalized quasilinear Choquard equation \[ \epsilon ^p \Delta _p u + V(x) | u | ^{p-2} u = \epsilon ^{\mu -N}\left( \int _{{\mathbb R}^N} \frac{Q(y)F(u(y))}{| x - y | ^{\mu }}\right) Q(x) f(u) \text{ in } {\mathbb R}^N, \] where \(\Delta _p\) is the \(p\)-Laplacian operator, \(1<p<N\), \(V\) and \(Q\) are two continuous real functions on \({\mathbb R}^N\), \(0<\mu <N\), \(F(s)\) is the primate function of \(f(s)\) and \(\epsilon \) is a positive parameter. Under suitable assumptions on \(p,\mu \) and \(f\), we establish a new concentration behavior of solutions for the quasilinear Choquard equation by variational methods. The results are also new for the semilinear case \(p=2\).

MSC:

35J92 Quasilinear elliptic equations with \(p\)-Laplacian
35A15 Variational methods applied to PDEs
35J50 Variational methods for elliptic systems
35J60 Nonlinear elliptic equations
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