Topological Methods in Nonlinear Analysis

Some nonlocal elliptic problem involving positive parameter

Anmin Mao, Runan Jing, Shixia Luan, Jinling Chu, and Yan Kong

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We consider the following superlinear Kirchhoff type nonlocal problem: $$ \begin{cases} \displaystyle -\bigg(a+b\int_\Omega |\nabla u|^2dx\bigg)\Delta u =\lambda f(x,u) & \text{in } \Omega,\ a > 0, \ b > 0, \ \lambda > 0, \\ u=0 &\text{on } \partial\Omega. \end{cases} $$ Here, $f(x,u)$ does not satisfy the usual superlinear condition, that is, for some $\theta > 0$, $$ 0\leq F(x,u)\overset{\triangle}{=} \int_0^u f(x,s)ds \leq \frac1{2+\theta}f(x,u)u, \quad \text{for all } (x,u)\in \Omega \times \mathbb{R}^+ $$ or the following variant $$ 0\leq F(x,u) \overset{\triangle}{=} \int_0^u f(x,s)ds \leq \frac1{4+\theta}f(x,u)u, \quad \text{for all } (x,u)\in \Omega \times \mathbb{R}^+ $$ which is quiet important and natural. But this superlinear condition is very restrictive eliminating many nonlinearities. The aim of this paper is to discuss how to use the mountain pass theorem to show the existence of non-trivial solution to the present problem when we lose the above superlinear condition. To achieve the result, we first consider the existence of a solution for almost every positive parameter $\lambda$ by varying the parameter $\lambda$. Then, it is considered the continuation of the solutions.

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Topol. Methods Nonlinear Anal., Volume 42, Number 1 (2013), 207-220.

First available in Project Euclid: 21 April 2016

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Mao, Anmin; Jing, Runan; Luan, Shixia; Chu, Jinling; Kong, Yan. Some nonlocal elliptic problem involving positive parameter. Topol. Methods Nonlinear Anal. 42 (2013), no. 1, 207--220.

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