Elliptic equations with discontinuous nonlinearities

Degenerate elliptic equations with singular nonlinearities

The behavior of the “minimal branch” is investigated for quasilinear eigenvalue problems involving the p-Laplace operator, considered in a smooth bounded domain of RN , and compactness holds below a critical dimension N #. The nonlinearity f (u) lies in a very general class and the results we present are new even for p = 2. Due to the degeneracy of p-Laplace operator, for p = 2 it is crucial to...

Semilinear Elliptic Equations with Generalized Cubic Nonlinearities

A semilinear elliptic equation with generalized cubic nonlinearity is studied. Global bifurcation diagrams and the existence of multiple solutions are obtained and in certain cases, exact multiplicity is proved.

Quasilinear Elliptic Problems with Critical Exponents and Discontinuous Nonlinearities

Using a recent fixed point theorem in ordered Banach spaces by S. Carl and S. Heikkilä, we study the existence of weak solutions to nonlinear elliptic problems −diva(x,∇u) = f (x,u) in a bounded domain Ω ⊂ Rn with Dirichlet boundary condition. In particular, we prove that for some suitable function g , which may be discontinuous, and δ small enough, the p -Laplace equation −div(|∇u|p−2∇u) = |u|...

Oscillation Theory for Semilinear Elliptic Equations with Arbitrary Nonlinearities

Discontinuous Solutions of Linear, Degenerate Elliptic Equations

We give examples of discontinuous solutions of linear, degenerate elliptic equations with divergence structure. These solve positively conjectures of De Giorgi.