A multiplicity result for gradient-type systems with non-differentiable term (Q949764)

Given a pair \((u_0,v_0)\in X\times Y\) and a parameter \(\lambda>0\), the problem at hand is that of finding a solution \((u,v)\in X\times Y\) to a system of variational inequalities of the form \[ \begin{aligned} \langle u-u_0,w_1\rangle_X+\lambda\int_\Omega b(x) F_1^\circ (u(x),v(x);-w_1(x))\,dx & \geq 0 \;\forall w_1\in X,\\ \langle v-v_0,w_2\rangle_Y+\lambda\int_\Omega b(x) F_2^\circ (u(x),v(x);-w_2(x))\,dx & \geq 0 \;\forall w_2\in Y, \end{aligned} \] with \(F_1^\circ (u(x),v(x);-w_1(x))\) standing for Clarke's generalized derivative of the function \(F(\cdot,v(x))\) at the point \(u(x)\) in the direction \(-w_1(x)\). As an introduction to the topic, the reader may consult [\textit{B.\,Ricceri}, Proc.\ Am.\ Math.\ Soc.\ 133, No.\,11, 3255--3261 (2005; Zbl 1069.47068)].