Existence of infinitely many solutions to semilinear elliptic Neumann problems with concave-convex type nonlinearity

Abstract

In this paper, we consider the semilinear elliptic problem $-\Delta u=a(x)|u|^{p-2}u+\lambda b(x)|u|^{q-2}u$ in a bounded domain $\Omega$ with Neumann boundary condition. We show the existence infinitely many solutions by applying critical point theory with a suitable decomposition of the Sobolev space $W^{1,2}(\Omega)$. Also we prove the $C^{\alpha}$ regularity of the solutions.