The authors of Cao and Yan (J Differ Equ 251:1389–1414, 2011) have considered the following semilinear critical Neumann problem

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.

Cao 和 Yan (J Differ Equ 251:1389–1414, 2011) 的作者考虑了以下半线性临界诺伊曼问题

其中\(\varvec{\Omega }\)是\(\varvec{\mathbb {R}^{N}}\)中满足某些几何条件的有界域， \(\varvec{\nu }\)是\(\varvec{\partial \Omega , 2^{*}:=\frac{2 N}{N-2}}\) 和 \(\varvec{g(t):=\ mu \)的外向单位法线vert t\vert ^{p-2} tt,}\)其中\(\varvec{p \in \left( 2,2^{*}\right) }\)和\(\varvec{\mu >0 }\)是常数。他们证明了上述问题存在无穷多个具有正能量的解，如果\(\varvec{N>\max \left( \frac{2(p+1)}{p-1}, 4\right) .} \)在本文中，我们考虑指数\(\varvec{p \in \left( 1,2\right) }\)的情况，并且我们证明如果\(\varvec{N>\frac{2 (p+1)}{p-1},}\)那么上述问题有无限组具有正能量的解。我们的主要结果扩展了 P. Han 在 [9] 中针对具有狄利克雷边界条件的椭圆问题的情况所获得的结果。