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Remarks about a fractional Choquard equation: ground state, regularity and polynomial decay. (English) Zbl 1373.35111

Summary: With appropriate hypotheses on the nonlinearity \(f\), we prove the existence of a ground state solution \(u\) for the problem \[ \biggl\{ (-\Delta_p)^s u+A|u|^{p-2}u=\left( \frac{1}{|x|^\mu}\ast F(u) \right) f(u)\quad \text{in}\; \mathbb{R}^N\text{,} \] where \(0<\mu<N\), \((-\Delta_p)^s\) stands for the \((s,p)\)-Laplacian operator, \(F\) is the primitive of \(f\) and \(A\) is a positive constant. When \(\mu<p\), we also show that \(u\in L^\infty (\mathbb{R}^N)\cap C^0(\mathbb{R}^N)\) and has polynomial decay.

MSC:

35J20 Variational methods for second-order elliptic equations
35Q55 NLS equations (nonlinear Schrödinger equations)
35B38 Critical points of functionals in context of PDEs (e.g., energy functionals)
35R11 Fractional partial differential equations
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