Multiple Solutions for Singularly Perturbed Semilinear Elliptic Problems in Bounded and Unbounded Domains

We are concerned with the $\coprod 1\mathrm{U}\mathrm{l}\mathrm{t}\mathrm{i}\mathrm{P}^{\mathrm{l}\mathrm{i}_{\mathrm{C}}\mathrm{i}}\mathrm{t}\mathrm{y} $ of solutions for the following singularly per-turbed semilinear elliptic equations: $(\mathrm{P})_{\epsilon}$ where $\in>0, $ $p\in(2,2^{*})(2^{*} $ denotes the critical exponent of the Sobolev embedding $H^{1}(\Omega)\subset L^{p}(\Omega) $ given by $2^{*}=2\mathit{1}\mathrm{V}/(N-2) $ if $N\geq 3, $ $=+\infty $ if $\mathit{1}\mathrm{V}=1,2 $ ) and $a\in C(\Omega) $ is a function with condition (C) specified below. As for $\Omega $ , we assume that $\bullet$ $\Omega $ is a bounded domain in $\mathbb{R}^{N}(l\mathrm{V}\geq 1) $ , or $\bullet\Omega=1\mathrm{R}^{N}(\mathit{1}\mathrm{V}\geq 1) $. Without loss of generality, we also assume that $0\in \mathrm{i}\mathrm{n}\mathrm{t}\Omega $. When $\Omega=\mathbb{R}^{N} $ , the boundary condition should be understood as $u(x)arrow 0 $ as $|x|arrow\infty $. In order to characterize the topological feature of $a(x) $. we introduce the following condition $(\mathrm{C})_{\delta\delta}0$. for some positive numbers $\delta_{0}, $ $\delta $