Abstract
We study the existence of positive solutions to the following nonlocal boundary value problem in , on , where , is a Carathéodory function, is a positive continuous function, and is a real parameter. Direct variational methods are used. In particular, the proof of the main result is based on a property of the infimum on certain spheres of the energy functional associated to problem in , .
Highlights
This paper aims to establish the existence of positive solutions in W01,2 Ω to the following problem involving a nonlocal equation of Kirchhoff type:
By a positive solution of Pλ, we mean a positive function u ∈ W01,2 Ω ∩ C0 Ω which is a solution of Pλ in the weak sense, that is such that
Our first main result gives some conditions in order that the energy functional associated to the unperturbed problem Pλ has a unique global minimum
Summary
This paper aims to establish the existence of positive solutions in W01,2 Ω to the following problem involving a nonlocal equation of Kirchhoff type:−K u 2 Δu λus−1 f x, u , in Ω, Pλ u 0 on ∂Ω.Here Ω is an open bounded set in RN with smooth boundary ∂Ω, s ∈ 1, 2 , f : Ω × 0, ∞ →0, ∞ is a Caratheodory function, K : R → R is a positive continuous function, λ is a real parameter, and uΩ |∇u|2dx 1/2 is the standard norm in W01,2 Ω. Condition a5 of 6, Theorem 1 is not satisfied by ts−1. Our aim is to study the existence of positive solution to problem Pλ , where, unlike previous existence results and, in particular, those of the aforementioned papers , no growth condition is required on f.
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