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Groundstates for Kirchhoff-type equations with Hartree-type nonlinearities. (English) Zbl 1412.35124

Summary: In this paper, we consider the following nonlinear problem of Kirchhoff-type with Hartree-type nonlinearities: \[ \begin{cases}-\left( a+b\int _{\mathbb {R}^N}|Du|^2\right) \Delta u+V(x)u=(I_{\alpha }*|u|^{p})|u|^{p-2}u,\quad x\in \mathbb {R}^N,\\ u\in H^1(\mathbb {R}^N), \quad u>0, \end{cases} \] where \(N\geq 3\), \(\max \{0,N-4\}<\alpha <N\), \(2<p<\frac{N+\alpha }{N-2}\), \(a>0,b\geq 0\) are constants, \(I_{\alpha }\) is the Riesz potential and \(V:\mathbb {R}^N\rightarrow \mathbb {R}\) is a potential function. Under certain assumptions on \(V\), we prove that the problem has a positive ground state solution by using global compactness lemma, monotonicity technique and some new tricks recently given in the literature.

MSC:

35J60 Nonlinear elliptic equations
35B09 Positive solutions to PDEs
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References:

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