biharmonic equation with discontinuous nonlinearities

We study the biharmonic equation with discontinuous nonlinearity and homogeneous Dirichlet type boundary conditions $$\displaylines{ \Delta^2u=H(u-a)q(u) \quad \hbox{in }\Omega,\cr u=0 \quad \hbox{on }\partial\Omega,\cr \frac{\partial u}{\partial n}=0 \quad \hbox{on }\partial\Omega, }$$ where \(\Delta\) is the Laplace operator, \(a> 0\), \(H\) denotes the Heaviside function, \(q\) is a continuous function, and \(\Omega\) is a domain in \(R^N \) with \(N\geq 3\). Adapting the method introduced by Ambrosetti and Badiale (The Dual Variational Principle), which is a modification of Clarke and Ekeland's Dual Action Principle, we prove the existence of nontrivial solutions. This method provides a differentiable functional whose critical points yield solutions despite the discontinuity of \(H(s-a)q(s)\) at \(s=a\). Considering \(\Omega\) of class \(\mathcal{C}^{4,\gamma}\) for some \(\gamma\in(0,1)\), and the function \(q\) constrained under certain conditions, we show the existence of two non-trivial solutions. Furthermore, we prove that the free boundary set \(\Omega_a=\{x\in\Omega:u(x)=a\}\) for the solution obtained through the minimizer has measure zero. For more information see https://ejde.math.txstate.edu/Volumes/2024/15/abstr.html