Abstract
In this paper, we are concerned with a Kirchhoff problem in the presence of a strongly-singular term perturbed by a discontinuous nonlinearity of the Heaviside type in the setting of Orlicz–Sobolev space. The presence of both strongly-singular and non-continuous terms brings up difficulties in associating a differentiable functional to the problem with finite energy in the whole space W_0^{1,Phi }(Omega ). To overcome this obstacle, we establish an optimal condition for the existence of W_0^{1,Phi }(Omega )-solutions to a strongly-singular problem, which allows us to constrain the energy functional to a subset of W_0^{1,Phi }(Omega ) in order to apply techniques of convex analysis and generalized gradient in the sense of Clarke.
Highlights
In this paper, we are concerned in presenting equivalent conditions for the existence of three solutions for the quasilinear problem (Qλ,μ)−M Ω Φ(|∇u|)dx ΔΦu = μb(x)u−δ +λf (x, u) in Ω, u>0 in Ω, u = 0 on ∂Ω, which are linked to an optimal compatibility condition between (b, δ) for existence of solution to the strongly-singular problem (S)−ΔΦu = b(x)u−δ in Ω, u>0 in Ω, u = 0 on ∂Ω0123456789().: V,vol 68 Page 2 of 28NoDEA with the boundary condition still in the sense of the trace.Here, M : [0, ∞) → [0, ∞) is a continuous function, f : Ω × (0, ∞) → (0, ∞) is of Heaviside type, 0 < b ∈ L1(Ω), δ > 1, λ, μ > 0 are real parameters
−ΔΦu = −div(a(|∇u|)∇u) stands for the Φ-Laplace operator, where a : (0, ∞) → (0, ∞) is a C1-function that defines the increasing homeomorphism φ : R → R given by φ(t) =
The main difficulty in treating strongly-singular problems consists in the fact that the energy functional associated to the equation neither belongs to C1, in the sense of Frechet differentiability, nor is defined in the whole space W01,Φ(Ω)
Summary
We are concerned in presenting equivalent conditions for the existence of three solutions for the quasilinear problem (Qλ,μ).
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