Abstract
This paper is devoted to the followingp-Kirchhoff type of problems−a+b∫Ω∇updxΔpu=fx,u,x∈Ωu=0,x∈∂Ωwith the Dirichlet boundary value. We show that thep-Kirchhoff type of problems has at least a nontrivial weak solution. The main tools are variational method, critical point theory, and mountain-pass theorem.
Highlights
Introduction and Main ResultsConsider the following p-Kirchhoff type of problems with the Dirichlet boundary value:⎧⎪⎨ − a + b |∇u|pdxΔpu f(x, u), x ∈ Ω, ⎪⎩ Ω (1)u 0, x ∈ z Ω, where Ω is a smooth bounded domain in RN, a > 0 and b ≥ 0 are real constants, Δp denotes the p-Laplacian operator by △pu div(|∇u|p− 2∇u)(1 < p < N), and f(x, u) is continuous on Ω × R.We look for the weak solutions of (1) which are the same as the critical points of the functional I: W10,p(Ω) ⟶ R defined by a I(u) |∇u|pdx + b |∇u|pdx − F(x, u)dx, pΩ
We show that the p-Kirchhoff type of problems has at least a nontrivial weak solution. e main tools are variational method, critical point theory, and mountain-pass theorem
Li and Niu [7] have obtained the existence of nontrivial solutions for the p-Kirchhoff type equations with critical exponents by the Ekeland variational principle and mountain-pass lemma
Summary
Consider the following p-Kirchhoff type of problems with the Dirichlet boundary value:. Li and Niu [7] have obtained the existence of nontrivial solutions for the p-Kirchhoff type equations with critical exponents by the Ekeland variational principle and mountain-pass lemma. In [16, 17], the authors obtained the existence and multiplicity of solutions for the p-Kirchhoff equation by using the genus theory. In [20, 21], the authors used the fountain theorem and concentration-compactness principle to consider multiplicity of solutions for p-Kirchhoff equations. By applying a variant of the mountain-pass theorem and the Ekeland variational principle, Cheng et al [22] obtained the existence of multiple nontrivial solutions for a class of Kirchhoff type problems with concave nonlinearity. Zn ∈ W10,p(Ω) with ‖zn‖ 1. en, there exists a subsequence (denoted by zn) such that zn ⇀ z0 in W10,p(Ω), zn ⟶ z0 in Lr(Ω), where p ≤ r < p∗, (11)
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
