Positive Solutions to Nonhomogeneous Quasilinear Problems with Singular and Supercritical Nonlinearities

Abstract

In recent years, there has been a growing interest in nonlinear singular elliptic PDEs. We study the existence of nonnegative solutions for the following quasilinear and singular elliptic problems with supercritical nonlinearity    − ∆pu − ∆qu = λ h(x) u γ + u θ , u > 0 in Ω, u = 0 on ∂Ω, (1) where Ω is an open, bounded subset of R N (N ≥ 3) with C 2 boundary, h is a positive real-valued function, 1 < p < q < ∞ and λ, θ, γ are positive parameters. Our motivation for this problem is taken from [1], where the authors considered the following problem ( − div (M(x)∇u) = λu−γ + u θ , u > 0 in Ω, u = 0 on ∂Ω. Our objective is to investigate problem (1), focusing on the impact of singular and supercritical nonlinearities on the right-hand side, alongside the nonhomogeneous operator. In particular, for supercritical cases, i.e., θ ≥ q ∗ −1, we prove the existence of solutions in a weak sense. To demonstrate the existence of a weak solution, we utilize the method of sub and supersolution.