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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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