Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 15, pp. 1-9.

Title: A biharmonic equation with discontinuous nonlinearities

Authors: Eduardo Arias (Escuela Politecnica Nacional, Quito, Ecuador)
         Marco Calahorrano (Escuela Politecnica Nacional, Quito, Ecuador)
         Alfonso Castro (Harvey Mudd College, Claremont, CA, USA)

Abstract:
We study the  biharmonic equation with discontinuous nonlinearity and
homogeneous Dirichlet type boundary conditions
$$\displaylines{
 \Delta^2u=H(u-a)q(u) \quad \text{in }\Omega,\cr
 u=0 \quad \hbox{on }\partial\Omega,\cr
 \frac{\partial u}{\partial n}=0 \quad \hbox{on }\partial\Omega,
}$$
where $\Delta$ is the Laplace operator, $a> 0$, $H$ denotes the Heaviside function,
$q$ is a continuous function, and $\Omega$ is a bounded 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 to \eqref{P-B-ND}. 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 $\Omega_a=\{x\in\Omega:u(x)=a\}$ has measure zero when $u$ is a minimizer 
of the action functional.

Submitted October 15, 2022. Published February 06, 2024.
Math Subject Classifications: 31B30, 35J60, 35J65, 58E05.
Key Words: Biharmonic equation; nonlinear discontinuity; critical point;
           dual variational principle; free boundary problem.