Some nonlocal elliptic problem involving positive parameter

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

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.

Keywords


Kirchhoff type nonlocal problem; Mountain Pass Theorem; Cerami sequence

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