Groundstates of nonlinear Choquard equations: Hardy-Littlewood-Sobolev critical exponent

Abstract

We consider nonlinear Choquard equation $$ - Δu + V u = \bigl(I_α\ast |u|^{\fracα{N}+1}\bigr) |u|^{\fracα{N}-1} u\quad\text{in (\mathbb{R}^N)},$$ where $N\ge 3$, $V\in L^{\infty }({\mathbb{R}}^N)$ is an external potential and $I_{\alpha }(x)$ is the Riesz potential of order $\alpha \in (0,N)$. The power $\frac{\alpha }{N}+1$ in the nonlocal part of the equation is critical with respect to the Hardy-Littlewood-Sobolev inequality. As a consequence, in the associated minimization problem a loss of compactness may occur. We prove that if ${\mathrm{liminf}}_{|x|\to \infty }(1-V(x))|x{|}^2>\frac{N^2(N-2)}{4(N+1)}$ then the equation has a nontrivial solution. We also discuss some necessary conditions for the existence of a solution. Our considerations are based on a concentration compactness argument and a nonlocal version of Brezis-Lieb lemma.

11 pages

References in corpus (1)

• Nonexistence and optimal decay of supersolutions to Choquard equations in exterior domains

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