Fractional Choquard Equation with Critical Nonlinearities

Abstract

In this article, we study the Brezis-Nirenberg type problem of nonlinear Choquard equation involving a fractional Laplacian \[ (-\De)^s u = \left( \int_{\Om}\frac{|u|^{2^*_{μ,s}}}{|x-y|^μ}\mathrm{d}y \right)|u|^{2^*_{μ,s}-2}u +\la u \; \text{in } \Om,\] where $\Om $ is a bounded domain in ${\mathbb{R}}^n$ with Lipschitz boundary, $\la $ is a real parameter, $s\in (0,1)$, $n>2s$ and $2_{\mu ,s}^{\ast }=(2n-\mu )/(n-2s)$ is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality. We obtain some existence, multiplicity, regularity and nonexistence results for solution of the above equation using variational methods.

32 pages. arXiv admin note: text overlap with arXiv:1604.00826 by other authors

References in corpus (2)

• A guide to the Choquard equation
• Nodal solutions for the Choquard equation

Cited by in corpus (8)

• Multiple and concentration of nontrivial nonnegative solutions for a fractional Choquard equation with critical exponent
• Choquard equation involving mixed local and nonlocal operators
• On the $p$-fractional Schrödinger-Kirchhoff equations with electromagnetic fields and the Hardy-Littlewood-Sobolev nonlinearity
• Multiple positive solutions for degenerate Kirchhoff equations with singular and Choquard nonlinearity
• On elliptic problems with Choquard term and singular nonlinearity
• Compact embedding from variable-order Sobolev space to $L^{q(x)}(\Omega )$ and its application to Choquard equation with variable order and variable critical exponent
• Degenerate fractional Kirchhoff-type system with magnetic fields and upper critical growth
• On degenerate fractional Schrödinger-Kirchhoff-Poisson equations with upper critical nonlinearity and electromagnetic fields