Noureddine Moujane, Mohamed El Ouaarabi, Chakir Allalou, Said Melliani
abstract:
In this paper, we are interested in the existence of weak solutions for a class
of Kirchhoff-type systems driven by the $(\alpha_1(m),\alpha_2(m))$-Kirchhoff--Laplacian
operator with the Dirichlet boundary conditions as follows:
$$ \begin{cases}
-\mathcal{R}_{1}\bigg(\int\limits_{\mathcal{D}}\mathcal{L}^{\alpha_1}_{\psi}\,dm\bigg)\Big(\Delta_{\alpha_1(m)}\psi-|\psi|^{\alpha_1(m)-2}\psi\Big)+\delta_1|\psi
|^{p(m)-2}\psi=\lambda_1f(m, \psi, \nabla \psi) & \text{in}\;\;\mathcal{D},\\
-\mathcal{R}_{2}\bigg(\int\limits_{\mathcal{D}}\mathcal{L}^{\alpha_2}_{\varphi}\,dm\big)
\Big(\Delta_{\alpha_2(m)}\varphi-|\varphi|^{\alpha_2(m)-2}\varphi\Big)+\delta_2|\varphi|^{q(m)-2}\varphi=\lambda_2g(m,
\varphi, \nabla \varphi) & \text{in}\;\; \mathcal{D}, \\
\psi=\varphi=0 & \text{on}\;\;\partial\mathcal{D},
\end{cases} $$
where $\mathcal{L}^{\alpha_1}_{\psi}$ and $\mathcal{L}^{\alpha_2}_{\varphi}$ are
non-local integro-differential operators, $\mathcal{D}$ is an open bounded
subset of $\mathbb{R}^{N}$ with the Lipshcitz boundary $\partial\mathcal{D}$.
Under some suitable assumptions on the functions $\mathcal{R}_{1}$,
$\mathcal{R}_{2}$, $f$ and $g$, together with the Berkovits topological degree
and the variable exponent Sobolev spaces theory, we discuss the existence of
weak solutions for the above problem on the spaces $W_0^{1,\alpha_{1}(m)}(\mathcal{D})\times
W_0^{1, \alpha_{2}(m)}(\mathcal{D})$.
Mathematics Subject Classification: 35J57, 35B33, 47H11
Key words and phrases: Kirchhoff-type system, $(\alpha_1(m),\alpha_2(m))$-Kirchhoff--Laplacian operator, weak solutions, Berkovits topological degree