### 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.

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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