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| We consider the equation $-\varepsilon^2\Delta u + V(x)u - u^p = 0$ in $\mathbb R^N$ where for each $k \in \{1,\cdots,K\}$, $M_k$ is a $q_k$-dimensional smooth compact manifold in $\mathbb R^N$ with $q_k \in \{1,\cdots,N-1\} $, and $V(x)=|\textrm{dist}(x,M_k)|^{m_k}+O(|\textrm{dist}(x,M_k)|^{m_k+1})$ as $\textrm{dist}(x,M_k) \to 0 $ for some $m_k > 0$. For a sequence of $\varepsilon$ converging to zero, we will find a positive solution $u_{\varepsilon}$ of the equation
which concentrates on $M_1\cup \dots \cup M_K$. |
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