Abstract
This paper deals with a $p$-Kirchhoff type problem involvingsign-changing weight functions. It is shown that under certainconditions, by means of variational methods, the existence ofmultiple nontrivial nonnegative solutions for the problem with thesubcritical exponent are obtained. Moreover, in the case of criticalexponent, we establish the existence of the solutions and prove thatthe elliptic equation possesses at least one nontrivial nonnegativesolution.
Highlights
Introduction and main theoremsThe purpose of this article is to investigate the existence of multiple nontrivial nonnegative solutions to the following nonlocal boundary value problem of the p-Kirchhoff type−M |∇u|pdx ∆pu = λf (x)|u|q−2u + g(x)|u|r−2uΩ u = 0 in Ω, (1)on ∂Ω, where ∆pu = div(|∇u|p−2∇u), Ω is a bounded domain in RN with a smooth boundary ∂Ω, < q p r ≤ p∗ where p∗
With regard to p-Kirchhoff type elliptic problems, Correa and Figueiredo [10] proved a result of existence and multiplicity of solutions by the Krasnoselskii’s genus when the nonlinear term is nonnegative function and satisfies subcritical growth condition
In order to overcome this difficulty and obtain the existence of nontrivial nonnegative solutions, we will adopt a variational method on the Nahari manifold which is similar to the fibering method
Summary
Department of Mathematics, Champlain College Saint-Lambert Quebec, J4P3P2, Canada and Department of Mathematics and Statistics, McGill University Montreal, Quebec, H3A2K6, Canada School of Mathematics and Statistics, Northeast Normal University Changchun 130024, China (Communicated by Yuan Lou)
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