Positive solutions with single and multi-peak for semilinear elliptic equations with nonlinear boundary condition in the half-space

摘要

We consider the existence of single and multi-peak solutions of thefollowing nonlinear elliptic Neumann problem egin{align*} &left{ egin{array}{1} -Delta u + lambda^{2} u = Q(x)|u|^{p-2}u & {m in}quad {mathbb R}^N_+,rac{partial u}{partial n} = f(x,u) & {m on} quad partial {mathbb R}^N_+, end{array} ight. end{align*} where $lambda$ is a large number, $p in (2, rac{2N}{Na??2})$ for $N geq 3, f (x, u)$ is subcritical about $u$ and ${mathcal Q}$ is positive and has some non-degenerate critical points in ${mathbb R}^{N}_{+}$. For $lambda$ large, we can get solutions which have peaks near the non-degenerate critical points of ${mathcal Q}$.