## Multi-bump solutions for Choquard equation with deepening potential well.(English)Zbl 1347.35097

This paper is concerned with the following Choquard equation $-\Delta u+ (\lambda a(x)+1) u = \left(\frac{1}{| x|^\mu} \ast | u|^p\right)| u|^{p-2}u \text{ in } \mathbb R^3, {(C)_\lambda}$ where $$\mu \in (0,3)$$, $$p\in (2, 6-\mu)$$ and $$0\leq a(x)\in C(\mathbb R^3)$$ with $$\Omega=\int(a^{-1}(0))$$ being non-empty bounded open set with smooth boundary $$\partial \Omega$$. Moreover, there exists $$M_0>0$$ such that $$| \{x\in \mathbb R^3: a(x)\leq M_0\}| <+\infty$$ and $$\Omega =\bigcup_{j=1}^k \Omega_j$$ with $$\mathrm{dist}(\Omega_i,\Omega_j)>0$$ if $$i\neq j$$. Under these conditions, by variational method, the authors prove that, there is a constant $$\lambda_0>0$$ such that for any non-empty subset $$\Gamma \subset \{1, \cdots, k\}$$ and $$\lambda \geq \lambda_0$$, the problem $$(C)_\lambda$$ has a positive solution $$u_\lambda$$. Furthermore, for any sequence $$\lambda_n \rightarrow \infty$$, passing to a subsequence, $$\{u_{\lambda_n}\}$$ converges strongly in $$H^1(\mathbb{R}^3)$$ to a function $$u$$ with $$u=0$$ outside $$\Omega_\Gamma = \bigcup_{j\in \Gamma} \Omega_j$$ and $$u|_{\Omega_\Gamma}$$ is a least energy solution for the following problem $-\Delta u +u =\left(\int_{\Omega_\Gamma}\frac{| u|^p}{| x-y|^\mu}dy\right) | u|^{p-2}u \text{ in } \Omega_\Gamma, u \in H_0^1(\Omega_\Gamma).$

### MSC:

 35J20 Variational methods for second-order elliptic equations 35J65 Nonlinear boundary value problems for linear elliptic equations
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### References:

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