In this paper, we deal with the following double phase problem {-div(|∇u|p-2∇u+a(x)|∇u|q-2∇u)=γ(|u|p-2u|x|p+a(x)|u|q-2u|x|q)+f(x,u)inΩ,u=0in∂Ω,where Ω ⊂ RN is an open, bounded set with Lipschitz boundary, 0 ∈ Ω , N≥ 2 , 1 < p< q< N, weight a(·) ≥ 0 , γ is a real parameter and f is a subcritical function. By variational method, we provide the existence of a non-trivial weak solution on the Musielak-Orlicz-Sobolev space W01,H(Ω), with modular function H(t, x) = tp+ a(x) tq. For this, we first introduce the Hardy inequalities for space W01,H(Ω), under suitable assumptions on a(·).
Fiscella, A. (2022). A Double Phase Problem Involving Hardy Potentials. APPLIED MATHEMATICS AND OPTIMIZATION, 85(3) [10.1007/s00245-022-09847-2].
A Double Phase Problem Involving Hardy Potentials
Fiscella A.
2022
Abstract
In this paper, we deal with the following double phase problem {-div(|∇u|p-2∇u+a(x)|∇u|q-2∇u)=γ(|u|p-2u|x|p+a(x)|u|q-2u|x|q)+f(x,u)inΩ,u=0in∂Ω,where Ω ⊂ RN is an open, bounded set with Lipschitz boundary, 0 ∈ Ω , N≥ 2 , 1 < p< q< N, weight a(·) ≥ 0 , γ is a real parameter and f is a subcritical function. By variational method, we provide the existence of a non-trivial weak solution on the Musielak-Orlicz-Sobolev space W01,H(Ω), with modular function H(t, x) = tp+ a(x) tq. For this, we first introduce the Hardy inequalities for space W01,H(Ω), under suitable assumptions on a(·).
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