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Ground state solutions for a Choquard equation with lower critical exponent and local nonlinear perturbation. (English) Zbl 1436.35118

Summary: This paper deals with the following Choquard equation with a local nonlinear perturbation: \[ \begin{cases} - \Delta u + u = \left( I_\alpha \ast | u |^{\frac{ \alpha}{ N} + 1}\right) | u |^{\frac{ \alpha}{ N} - 1} u + f (u ), \quad x \in \mathbb{R}^N; \\ u \in H^1 (\mathbb{R}^N), \end{cases}\] where \(I_\alpha : \mathbb{R}^N \to \mathbb{R}\) is the Riesz potential, \(N \geq 3\), \(\alpha \in (0, N )\), the exponent \(\frac{ \alpha}{ N} + 1\) is critical with respect to the Hardy-Littlewood-Sobolev inequality, and the nonlinear perturbation \(f\) is only required to satisfy some weak assumptions near 0 and \(\infty\). Our results improve the previous related ones in the literature.

MSC:

35J20 Variational methods for second-order elliptic equations
35J62 Quasilinear elliptic equations
35Q55 NLS equations (nonlinear Schrödinger equations)
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