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Multiple positive solutions for a nonlinear Choquard equation with nonhomogeneous. (English) Zbl 1404.35200

Summary: In this paper, we study the existence of multiple positive solutions for the following equation: \[ -\Delta u+u=(K_\alpha(x)\ast|u|^p)|u|^{p-2} u+\lambda f(x), \quad x\in\mathbb R^N, \] where \(N\geqslant 3\), \(\alpha\in(0,N)\), \(p\in(1+\alpha/N,(N+\alpha)/(N-2))\), \(K_\alpha(x)\) is the Riesz potential, and \(f(x)\in H^{-1}(\mathbb R^N)\), \(f(x)\geqslant 0\), \(f(x)\not\equiv 0\). We prove that there exists a constant \(\lambda^\ast>0\) such that the equation above possesses at least two positive solutions for all \(\lambda\in(0,\lambda^\ast)\). Furthermore, we can obtain the existence of the ground state solution.

MSC:

35J91 Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian
35B09 Positive solutions to PDEs
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