A BIHARMONIC EQUATION WITH DISCONTINUOUS NONLINEARITIES

Abstract:

We study the biharmonic equation with discontinuous nonlinearity and homogeneous Dirichlet type boundary conditions ^A2u = H(u — a)q(u) in Ω; u = 0 on ∂Ω; <inf>(1)</inf> ∂u — = 0 on ∂Ω; ∂n where A is the Laplace operator, a > 0, H denotes the Heaviside function, q is a continuous function, and Q is a bounded domain in R^N with N 3. Adapting the method introduced by Ambrosetti and Badiale (The Dual Variational Principle), which is a modication of Clarke and Ekeland's Dual Action Principle, we prove the existence of nontrivial solutions to (1). This method provides a dierentiable functional whose critical points yield solutions to (1) despite the discontinuity of H(s — a)q(s) at s = a. Considering Q of class C^4; for some 2 (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 Qa = fx 2 Q: u(x) = ag has measure zero when u is a minimizer of the action functional.

Año de publicación:

2024

Keywords:

• Biharmonic equation
• Critical point
• dual variational principle
• Free boundary problem
• nonlinear discontinuity

Fuente:

scopus