Existence and multiplicity of solutions for a Kirchhoff system with critical growth

Marcelo F. Furtado¹ ; Luan D. de Oliveira¹ ; João Pablo P. da Silva²

¹ Universidade de Brasília, Departamento de Matemática
² Universidade Federal do Pará, Departamento de Matemática

Annales Fennici Mathematici, Tome 46 (2021) no. 1, pp. 295-308

Résumé

  We consider the system \[\begin{cases}-m\left(\|u\|^2\right)\Delta u = \lambda F_u(x,u,v)+\frac{1}{2^*}G_u(u,v), &in\ \Omega,\\ -l\left(\|v\|^2\right)\Delta v = \lambda F_v(x,u,v)+\frac{1}{2^*}G_v(u,v), &in\ \Omega,\\ u,v\in H_0^1(\Omega),\end{cases}\] where $\Omega\subset\mathbf{R}^N$, $N\ge 3$, is a bounded smooth domain, $\|\cdot \|^2 = \int_{\Omega}|\nabla \cdot|^2 \,\mathrm{d}x$, $\lambda>0$ is a parameter, the functions $m$ and $l$ are positive and increasing, the function $F$ is superlinear both at origin and at infinity, the function $G$ is $2^*$-homogeneous. In our first result, we obtain a nonzero nonnegative solution for large values of $\lambda$. We also prove that, for any $k\in\mathbf{N}$, there exists $\lambda^*_k>0$ such that the problem has at least $k$ pairs of nonzero solutions if $\lambda\ge\lambda_k^*$.  

Détail

Keywords: Kirchhoff-type problems, multiple solutions, critical nonlinearities