Some nonlocal elliptic problem involving positive parameter

Abstract

We consider the following superlinear Kirchhoff type nonlocal problem: $$ \cases \displaystyle -\bigg(a+b\int_\Omega |\nabla u|^2dx\bigg)\Delta u =\lambda f(x,u) & \text{in } \Omega,\ a> 0, \ b> 0, \ \lambda > 0, \\ u=0 &\text{on } \partial\Omega. \endcases $$ Here, $f(x,u)$ does not satisfy the 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 $$ 0\leq F(x,u)\triangleq \int_0^u f(x,s)ds \leq \frac1{4+\theta}f(x,u)u, \quad \text{for all } (x,u)\in \Omega \times \mathbb{R}^+ $$ 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.