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 equationeq0.1 -ε^2Δ u+ω V(x)u=u^p+u^2^*-1,u> 0in ^N, equation where ω^+, N≥ 3, p(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 (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.