Infinitely many positive energy solutions for semilinear Neumann equations with critical Sobolev exponent and concave-convex nonlinearity

The authors of Cao and Yan (J Differ Equ 251:1389-1414, 2011) have considered the followi Omega ng semilinear critical Neumann problem-Delta u = |u|2 & lowast;-2u + g(u) in, partial derivative u = 0 on partial derivative, partial derivative Omega where Omega is a bounded domain in RN satisfying some geometric conditions, nu is the outward unit normal of partial derivative, 2 & lowast; := 2N N-2 and g(t) := mu|t| p-2t - t, where p is an element of (2, 2 & lowast;) and mu > 0 are constants. They proved the existence of infinitely many solutions with positive energy for the 2( p+1) above problem if N > max p-1 , 4 . In this present paper, we consider the case where the exponent p is an element of (1, 2) and we show that if N > 2( p+1) p-1 , then the above problem admits an infinite set of solutions with positive energy. Our main result extend that obtained by P. Han in [9] for the case of elliptic problem with Dirichlet boundary conditions.