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

Belchior, P.; Bueno, H.; Miyagaki, O. H.; Pereira, G. A.

Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 164, 38-53 (2017).

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

Keywords:

variational methods; polynomial decay; fractional Laplacian; Choquard equations

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