Abstract
We use the Hardy-Sobolev inequality to study existence and non-existence results for a positive solution of the quasilinear elliptic problem -\Delta{p}u − \mu \Delta{q}u = \limda[mp(x)|u|p−2u + \mu mq(x)|u|q−2u] in \Omega driven by nonhomogeneous operator (p, q)-Laplacian with singular weights under the Dirichlet boundary condition. We also prove that in the case where μ > 0 and with 1 < q < p < \infinity the results are completely different from those for the usual eigenvalue for the problem p-Laplacian with singular weight under the Dirichlet boundary condition, which is retrieved when μ = 0. Precisely, we show that when μ > 0 there exists an interval of eigenvalues for our eigenvalue problem.
Highlights
Where D(u) = (|∇u|p−2 + μ|∇u|q−2). This system has a wide range of applications in physics and related sciences like chemical reaction design [2], biophysics [8] and plasma physics [18]
The authors proved the existence of positive solutions in resonant cases
Letting μ → 0+, our problem (Pλ,μ) turns into the (p−1)-homogeneous problem known as the usual weighted eigenvalue problem for the p-Laplacian with singular weight mp: (Pλ,mp )
Summary
We will always assume for r = p, q that mrδτ ∈ La(Ω) with δ(x) = dist(x, ∂Ω) and m+r ≡ 0, where a, r and τ satisfy one of the conditions (H1), (H2), (H3) or (H4). The following lemma concerns the Hardy-Sobolev inequality proved in [10], which characterize the first eigenvalue λ1(r, mr) of problem (Pλ,mr ). This inequality is our main tool in this paper. We give an example of the weight mr such that mrδτ ∈ La(Ω) with m+r ≡ 0, where a, τ and r satisfying the condition (H2). The following theorem guarantees the simplicity and isolation of λ1(r, mr), where r = p, q If in addition one assumes mr ∈ L1(Ω) for r > N or mr ∈ LNǫ(ρ/)p(Ω) for 1 < r ≤ N , the first eigenvalue λ1(r, mr) is simple and any positive eigenvalue other than λ1(r, mr) has no positive eigenfunctions
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