Nonnegative entire solutions of a class of degenerate semilinear elliptic equations

Existence Results for a Class of Semilinear Degenerate Elliptic Equations

We prove existence results for the Dirichlet problem associated with an elliptic semilinear second-order equation of divergence form. Degeneracy in the ellipticity condition is allowed.

Asymptotic Spatial Patterns and Entire Solutions of Semilinear Elliptic Equations

(2) ε∆uε + f(uε) = 0, x ∈ Ω, Bu = 0, x ∈ ∂Ω, where ε > 0 is a small parameter, Ω is a smooth bounded domain in Rn, and Bu is an appropriate boundary condition. The connection of (1) and (2) are made by a typical technique called blowup method. Suppose that {uε} is a family of solutions of (2). The simplest setup of the blowup method is to choose Pε ∈ Ω, and define vε(y) = uε(εy + Pε), for y ∈ Ω...

Entire Solutions to a Class of Fully Nonlinear Elliptic Equations

We study nonlinear elliptic equations of the form F (Du) = f(u) where the main assumption on F and f is that there exists a one dimensional solution which solves the equation in all the directions ξ ∈ Rn. We show that entire monotone solutions u are one dimensional if their 0 level set is assumed to be Lipschitz, flat or bounded from one side by a hyperplane.

Entire Bounded Solutions for a Class of Quasilinear Elliptic Equations

Multi–peak Solutions for a Class of Degenerate Elliptic Equations

By means of a penalization argument due to del Pino and Felmer, we prove the existence of multi–spike solutions for a class of quasilinear elliptic equations under natural growth conditions. Compared with the semilinear case some difficulties arise, mainly concerning the properties of the limit equation. The study of concentration of the solutions requires a somewhat involved analysis in which ...