| |
| This paper are concerned with the following linear Choquard equation\begin{equation*} -\Delta u +V(x)u =K(x)(\int_{\mathbb{R}^{N}}\frac{|u|^{2^{*}_{\mu}}}{|x-y|^{\mu}}dy)|u|^{2^{*}_{\mu}-2}u + g(x,u) \quad x \in \mathbb{R}^{N},\end{equation*}where $N \geq 3$, $0<\mu<N$ and $ 2^{*}_{\mu}$ is the critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality. V(x) is a continuous function such that the spectrum $\sigma(-\Delta+V(x))$ of $-\Delta+V(x)$ in $L^{2}(R^{N})$ has a negative part, K(x) is a bounded positive function, g is of subcritical growth. The existence of nontrivial solutions has been obtained by variational methods.ssume the following hypotheses:$ $\\$(V_{1})$: $V(x)\in C(\mathbb{R}^{N})\bigcap L^{\infty}(\mathbb{R}^{N})$ and $\liminf_{| x |\rightarrow \infty} V(x)=v_{\infty} >0.$\\$(V_{2})$: $(W_{1}(x)-v_{\infty}) \in L^{N/2}(\mathbb{R}^{N})$, $0\notin \sigma(-\Delta+V)$ and $\sigma(-\Delta+V)\bigcap (-\infty,0)\neq \emptyset$,$\quad$ where $\sigma$ denotes the spectrum in $L^{2}(\mathbb{R}^{N})$ and $W_{1}(x)=max\{V(x),v_{\infty}\}.$\\$(K_{1})$: $K(x)\in C(\mathbb{R}^{N})$ attains its maximum at 0. $K_{M}:=K(0)=\max_{\mathbb{R}^{N}}K(x)$ and$\quad$ there exist positive constants $K_{min}$ and $\alpha$ such that $K(x)\geq K_{min}$ and $K(0)$$\quad$ $-K(x)=O(|x|^{\alpha})$.\\$(G_{1})$: $g\in C(\mathbb{R}^{N}\times\mathbb{R}, \mathbb{R})$ and $|g(x,s)| \leq \omega(x)|s|+h(x)|s|^{p-1}$, where $\omega(x)$$\in L^{N/2}(\mathbb{R}^{N})$$\quad$ $\bigcap L^{\infty}(\mathbb{R}^{N})$, $2<p<$ $2^{*}$ and $h(x)\in L^{\frac{2^{*}}{2^{*}-P}}(\mathbb{R}^{N})\bigcap L^{\infty}(\mathbb{R}^{N}).$\\$(G_{2})$: $\lim_{s\rightarrow 0}g(x,s)/s=0$ uniformly on $\mathbb{R}^{N}.$\\$(G_{3})$: $0\leq 2G(x,s)\leq sg(x,s)$ for a.e. $x \in \mathbb{R}^{N}, \forall s \in \mathbb{R},$ where $G(x,s):=\int_{0}^{s}g(x,t)dt.$$ $ |
|
| Keywords:Choquard equation,Hardy-Littlewood-Sobolev inequality,Indefinite problem; |
|
