Existence of three solutions for a class of \((p_1,\ldots,p_n)\)-biharmonic systems with Navier boundary conditions (Q2880822)

This paper is concerned with the existence of three nontrivial weak solutions for the system \(\Delta(|\Delta u_i|^{p_i-2}\Delta u_i)=\lambda F_{u_i}(x,u_1,u_2,\dots,u_n)\) in \(\Omega\), subject to Navier boundary conditions. Here \(\Omega\) is a smooth and bounded domain in \(\mathbb{R}^N\), \(N\geq 2\), \(p_i\geq 0\), \(\lambda>0\) and \(F:\overline \Omega\times \mathbb{R}\rightarrow \mathbb{R}\) is a continuous function in all variables which is differentiable in the \(u_i\)-variable for all \(1\leq i\leq n\). By means of critical point theory in the spirit of Ricceri, the authors prove the existence of at least three weak solutions in the standard product space NEWLINE\[NEWLINE X=(W^{2,p_1}(\Omega)\cap W_0^{1,p_1}(\Omega))\times (W^{2,p_2}(\Omega)\cap W_0^{1,p_2}(\Omega))\times \dots\times (W^{2,p_n}(\Omega)\cap W_0^{1,p_n}(\Omega)). NEWLINE\]