On the existence of small energy solutions for a sublinear Neumann problem

In this paper, we are concerned with the sublinear problem (0.1) { − Δ u = | u | p − 2 u in Ω , u ν = 0 on ∂ Ω , where Ω ⊂ R N is a bounded domain, and 1 ≤ p 2 . For p = 1 , the nonlinearity | u | p − 2 u will be identified by sgn ( u ) . In contrast to previous work on the Dirichlet problem, some difficulties arise due to the fact that the associated energy functional is not bounded from below. Complementing recent work by Parini and Weth in [15] on least energy solutions, we prove that (0.1) has infinitely many solutions with small negative energy.