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 following semilinear critical Neumann problem

$$\begin{aligned} \varvec{-\Delta u=\vert u\vert ^{2^{*}-2} u+g(u) \quad \text{ in } \Omega , \quad \frac{\partial u}{\partial \nu }=0 \quad \text{ on } \partial \Omega ,} \end{aligned}$$

where \(\varvec{\Omega }\) is a bounded domain in \(\varvec{\mathbb {R}^{N}}\) satisfying some geometric conditions, \(\varvec{\nu }\) is the outward unit normal of \(\varvec{\partial \Omega , 2^{*}:=\frac{2 N}{N-2}}\) and \(\varvec{g(t):=\mu \vert t\vert ^{p-2} t-t,}\) where \(\varvec{p \in \left( 2,2^{*}\right) }\) and \(\varvec{\mu >0}\) are constants. They proved the existence of infinitely many solutions with positive energy for the above problem if \(\varvec{N>\max \left( \frac{2(p+1)}{p-1}, 4\right) .}\) In this present paper, we consider the case where the exponent \(\varvec{p \in \left( 1,2\right) }\) and we show that if \(\varvec{N>\frac{2(p+1)}{p-1},}\) 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.