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Infinitely many bound state solutions of Choquard equations with potentials. (English) Zbl 1401.35055

Summary: In this paper, we consider the following Choquard equation \[ \begin{cases} - \Delta u+a(x)u=(I_\alpha *|u|^p)|u|^{p-2}u, &\quad x\in {\mathbb {R}}^N,\\ u(x)\rightarrow 0,&\quad |x|\rightarrow +\infty , \end{cases} \tag{CH} \] where \(N\geq 3, I_\alpha \) is a Riesz potential, \(\frac{N+\alpha}{N}<p<\frac{N+\alpha}{N-2}\) and \(a(x)\) is a given nonnegative potential function. Under some assumptions of asymptotic properties on \(a(x)\) at infinity and according to a concentration compactness argument, we obtain infinitely many solutions of (CH), whose energy can be arbitrarily large.

MSC:

35J20 Variational methods for second-order elliptic equations
35J65 Nonlinear boundary value problems for linear elliptic equations
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