Choquard equations under confining external potentials
英文摘要：
We consider the nonlinear Choquard equation $$ -Δu+V u=(I_α\ast \vert u\vert ^p)\vert u\vert ^{p-2}u \qquad \text{ in } \mathbb{R}^N $$ where $N\geq 1$, $I_α$ is the Riesz potential integral operator of order $α\in (0, N)$ and $p > 1$. If the potential $ V \in C (\mathbb{R}^N; [0,+\infty)) $ satisfies the confining condition $$ \liminf\limits_{\vert x\vert \to +\infty}\frac{V(x)}{1+\vert x\vert ^{\frac{N+α}{p}-N}}=+\infty, $$ and $\frac{1}{p} > \frac{N - 2}{N + α}$, we show the existence of a groundstate, of an infinite sequence of solutions of unbounded energy and, when $p \ge 2$ the existence of least energy nodal solution. The constructions are based on suitable weighted compact embedding theorems. The growth assumption is sharp in view of a Pohožaev identity that we establish.
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