A singular perturbed problem with critical Sobolev exponent - Chen, Mengyao; Li, Qi (Topological Methods in Nonlinear Analysis, 2021)
This paper deals with the following nonlinear elliptic problem \begin{equation}\label{eq0.1} -ε^2Δ u+ω V(x)u=u^{p}+u^{2^{*}-1},\quad u> 0\quad\text{in}\ \R^N, \end{equation} where $ω\in\R^{+}$, $N≥ 3$, $p\in (1,2^{*}-1)$ with $2^{*}={2N}/(N-2)$, $ε> 0$ is a small parameter and $V(x)$ is a given function. Under suitable assumptions, we prove that problem (\ref{eq0.1}) has multi-peak solutions by the Lyapunov-Schmidt reduction method for sufficiently small $ε$, which concentrate at local minimum points of potential function $V(x)$. Moreover, we show the local uniqueness of positive multi-peak solutions by using the local Pohozaev identity.