Some nonlocal elliptic problem involving positive parameter. (English) Zbl 1291.35086

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

Topol. Methods Nonlinear Anal. 42, No. 1, 207-220 (2013).

Summary: We consider the following superlinear Kirchhoff type nonlocal problem: \[ \begin{cases} -\biggl(a+b\int_\Omega |\nabla u|^2dx\biggl)\Delta u=\lambda f(x,u)\quad & \text{in }\Omega,\quad a>0,\,b>0,\,\lambda>0,\\ u=0\quad & \text{on }\partial\Omega.\end{cases} \] Here, \(f(x,u)\) does not satisfy he usual superlinear condition, that is, for some \(\theta>0\), \[ 0\leq F(x,u)\triangleq \int_0^u f(x,s)\,ds \leq \frac1{2+\theta}f(x,u)\,u, \quad \text{for all } (x,u)\in \Omega \times \mathbb{R}^+ \] or the following variant \[ 0\leq F(x,u)\triangleq \int_0^u f(x,s)\,ds \leq \frac 1{4+\theta}f(x,u)\,u, \quad \text{for all } (x,u)\in \Omega \times \mathbb{R}^+ \] 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 nontrivial 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.

MSC:

35J65	Nonlinear boundary value problems for linear elliptic equations
35J20	Variational methods for second-order elliptic equations
47J10	Nonlinear spectral theory, nonlinear eigenvalue problems

Keywords:

Kirchhoff type nonlocal problem; mountain pass theorem; Cerami sequence