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Existence and qualitative properties of solutions for Choquard equations with a local term. (English) Zbl 1412.35123

Summary: In this paper, a nontrivial solution \(u \in H^1(\mathbb{R}^N)\) to the autonomous Choquard equation with a local term \[- \Delta u + \lambda u = (I_\alpha \ast | u |^p) | u |^{p - 2} u + | u |^{q - 2} u \quad\text{in } \mathbb{R}^N\] is obtained, where \(N \geq 3\), \(\alpha \in(0, N)\), \(\lambda > 0\) is a constant, \(I_\alpha\) is the Riesz potential, \(\frac{N + \alpha}{N} < p < \frac{N + \alpha}{N - 2}\) and \(q \in(2, 2^\ast = \frac{2 N}{N - 2})\). Under some further assumptions on \(p\) and \(q\), the regularity and the Pohožaev identity of the solution are established, and then it is shown that the obtained solution is a groundstate of mountain pass type. Moreover, the positivity and symmetry of the groundstate are also considered. By using the results obtained for the autonomous equation, a positive groundstate solution for the nonautonomous equation \[- \Delta u + V(x) u = (I_\alpha \ast | u |^p) | u |^{p - 2} u + | u |^{q - 2} u \quad\text{in } \mathbb{R}^N\]is also found under some assumptions on \(V(x)\).

MSC:

35J60 Nonlinear elliptic equations
35B09 Positive solutions to PDEs
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