Mahmoud El Ahmadi, Abdesslem Ayoujil, Mohammed Berrajaa

$p(x)$-Kirchhoff Type Problems without (AR)-Condition

abstract:
In this paper, we study the following $p(x)$-Kirchhoff problem
$$ \begin{cases}
-M \bigg( \int\limits_{\Omega} \frac{1}{p(x)}\,|\nabla u|^{p(x)}\,dx\bigg) \Delta _{p(x)} u= g(x,u) & \text{in}\;\Omega,\\
u=0 & \text{on}\;\partial \Omega,
\end{cases} $$
where $ M : \mathbb{R}_+ \to \mathbb{R}_+$ is a continuous function and the nonlinear term $g:\Omega \times \mathbb{R} \to \mathbb{R}$ satisfies the Carathéodory condition. Using the mountain pass theorem with the Cerami condition, we give a result on the existence of at least one nontrivial solution without assuming the $\mathbf{(AR)}$-condition. Next, Employing the fountain theorem, we show the existence of infinitely many solutions of the above problem.

Mathematics Subject Classification: 35A01, 35A15, 35B38, 35J60

Key words and phrases: $p(x)$-Kirchhoff type problems, variational methods, generalized Sobolev spaces, critical point theory, Cerami condition