Parameter estimates and a uniqueness result for double phase problem with a singular nonlinearity

Abstract

We consider the boundary value problem −Δpuλ−Δquλ=λg(x)uλ−β in Ω, uλ=0 on ∂Ω with uλ>0 in Ω. We assume Ω is a bounded open set in RN with smooth boundary, 1<p<q<∞, β∈[0,1), g is a positive weight function and λ is a positive parameter. We derive an estimate for uλ which describes its exact behavior when the parameter λ is large. In general, by invoking appropriate comparison principles, this estimate can be used as a powerful tool in deducing the existence, non-existence and multiplicity of positive solutions of nonlinear elliptic boundary value problems. Here, as an application of this estimate, we obtain a uniqueness result for a nonlinear elliptic boundary value problem with a singular nonlinearity.