On semilinear elliptic equations with bifurcation at zero

Abstract

In this paper, we investigate a semilinear elliptic problem given by(P){−Δu=λu+f(x,u),inΩ,u=0,on∂Ω, where λ>0 represents a parameter and the nonlinearity f(x,u) adheres to suitable conditions ensuring that zero serves as a saddle point for the corresponding energy functional Iλ of the equation (P). Our findings can be seen as a refinement and extension of Rabinowitz et al. (2007) [21].Our main novelties are threefold. Firstly, we prove the existence of multiple solutions to the problem (P), by using the linking geometry of Iλ and the perturbation invariance of Gromoll-Meyer pair. Here we require the parameter λ staying close enough to an eigenvalue of (−Δ,H01(Ω)). We also derive information on energy estimates and critical groups for some of these solutions. Secondly, we improve the existence results in [21] by weakening the assumptions therein. At last, under the same hypothesis as [21, Theorem 1.2], we find a new solution of the problem (P).