Abstract
In this study, we prove the existence of a positive solution for a p -Kirchhoff-type problem with Sobolev exponent.
Highlights
Introduction and Main ResultsIn this study, we are concerned with the following p-Kirchhoff-type problem:⎧⎨ − a‖u‖(θ− 1)p + bdiv|∇u|p− 2∇u |u|p∗− 2u + λ|u|p− 2u, in Ω, ⎩ Pλ (1)u 0, on zΩ, where Ω ⊂ RN is a bounded domain, 1 < p < N, a, b > 0, θ > 1, ‖.‖ is the usual norm in W10,p(Ω) given by ‖u‖p Ω|∇u|pdx, λ is a parameter, and p∗ pN/(N − p) is the critical Sobolev exponent corresponding to the noncompact embedding of W10,p(Ω) into Lp∗ (Ω).Since equation (Pλ) contains an integral over Ω, it is no longer a pointwise identity; it is often called a nonlocal problem
We prove the existence of a positive solution for a p-Kirchhoff-type problem with Sobolev exponent
U 0, on zΩ, where Ω ⊂ RN is a bounded domain, 1 < p < N, a, b > 0, θ > 1, ‖.‖ is the usual norm in W10,p(Ω) given by ‖u‖p Ω|∇u|pdx, λ is a parameter, and p∗ pN/(N − p) is the critical Sobolev exponent corresponding to the noncompact embedding of W10,p(Ω) into Lp∗ (Ω)
Summary
We are concerned with the following p-Kirchhoff-type problem:. u 0, on zΩ, where Ω ⊂ RN is a bounded domain, 1 < p < N, a, b > 0, θ > 1, ‖.‖ is the usual norm in W10,p(Ω) given by ‖u‖p Ω|∇u|pdx, λ is a parameter, and p∗ pN/(N − p) is the critical Sobolev exponent corresponding to the noncompact embedding of W10,p(Ω) into Lp∗ (Ω). Since equation (Pλ) contains an integral over Ω, it is no longer a pointwise identity; it is often called a nonlocal problem. It is called nondegenerate if b > 0 and a ≥ 0, while it is named degenerate if b 0 and a > 0. Such nonlocal elliptic problem such as (Pλ) is related to the original Kirchhoff’s equation in [1] which was first introduced by Kirchhoff as an extension of the classical. N ≥ 4, they proved the existence of a positive solution for λ ∈ (0, λ1) and no positive solution for λ > λ1 or λ ≤ 0 and Ω is a starshaped domain
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