Existence and multiplicity of solutions to elliptic problems with discontinuities and free boundary conditions

• Main Authors: Sabri Bensid, Sidi Mohammed Bouguima
• Format: Article
• Language: English
• Published: Texas State University 2010-04-01
• Series: Electronic Journal of Differential Equations
• Subjects: Green function; maximum principle; bifurcation; free boundary problem
• Online Access: http://ejde.math.txstate.edu/Volumes/2010/56/abstr.html
• ISSN: 1072-6691

We study the nonlinear elliptic problem with discontinuous nonlinearity $$displaylines{ -Delta u = f(u)H(u-mu ) quadhbox{in } Omega, cr u =h quad hbox{on }partial Omega, }$$ where $H$ is the Heaviside unit function, $f,h$ are given functions and $mu$ is a positive real parameter. The domain $Omega$ is the unit ball in $mathbb{R}^n$ with $ngeq 3$. We show the existence of a positive solution $u$ and a hypersurface separating the region where $-Delta u=0$ from the region where $-Delta u=f(u)$. Our method relies on the implicit function theorem and bifurcation analysis.