Nehari manifold method for singular double phase problem with optimal control on parameter

Abstract

In this paper, we consider the following singular double phase problem −div(|∇u|p−2∇u + a(x)|∇u|q−2∇u) = λf(x)u−γ + g(x)ur−1, u > 0 in Ω and u = 0 on ∂Ω, where Ω⊂RN is an open bounded domain with smooth boundary, dimension N ≥ 2, exponents p < q < r < p* = Np/(N − p) with 1 < p < N, while 0 < γ < 1 and λ > 0 is real parameter. The weight functions f, g are bounded continuous functions which may change sign and the modulating function a is non-negative, continuous and has compact support in Ω. Using fibering map and Nehari manifold method, we show the existence of at least two positive solutions for (0, λ* + ϵ) for some ϵ > 0, where λ* is an extremal parameter, characterized via nonlinear Rayleigh quotient. An estimate on the extremal value λ* is also obtained.