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

Abstract

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 \(\lambda\ge\lambda_k^*\).