Ecuación biarmónica con discontinuidades no lineales

Abstract

Critical point theory and nonlinear analysis are used to study from another approach the existence of solution or solutions (in a certain sense) of the biarmonic equation with nonlinear discontinuities, considering homogeneous conditions on the Dirichlet boundary. More precisely we will consider the problem (PBDN):

𝚫𝟐𝒖=𝑯(𝒖−𝒂)𝒒(𝒖) 𝒆𝒏 𝛀, 𝒖=𝟎 𝒔𝒐𝒃𝒓𝒆 𝛛𝛀, 𝛛𝒖𝛛𝒏=𝟎 𝒔𝒐𝒃𝒓𝒆 𝛛𝛀 where 𝚫 is the Laplace operator, 𝐚>𝟎, 𝐇 denotes the Heaviside function and 𝐪∈𝓒(ℝ). The idea is to find solutions to the previous problem by adapting the method introduced by Ambrosetti and Badiale (1989), which is a modification of Clarke's Dual Action Principle shown by Clarke and Ekeland (1980). The most notable thing about this method is the smoothing it generates, in the sense that it will allow finding solutions to the problem (PBDN) as critical points of a functional that, despite the discontinuity of H(s−a)q(s) en s=𝑎, is 𝒞1.