Comparison principle for singular degenerate elliptic equations on unbounded domains

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...

Elliptic Equations in Weighted Sobolev Spaces on Unbounded Domains

Singular perturbation problems for nonlinear elliptic equations in degenerate settings

Here N ≥ 1, g(s) ∈ C(R,R) is a function with a subcritical growth, V (x) ∈ C(R ,R) is a positive function and 0 < ε 1. Among solutions of (0.1)ε, we are interested in concentrating families (uε) of solutions, which have the following behavior: (i) uε(x) has a local maximum at xε ∈ R and xε converges to some x0 ∈ R as ε → 0. (ii) rescaled function vε(y) = uε(εy + xε) converges as ε → 0 to a solu...

Liouville type results for semilinear elliptic equations in unbounded domains

This paper is devoted to the study of some class of semilinear elliptic equations in the whole space: −aij(x)∂iju(x)− qi(x)∂iu(x) = f(x, u(x)), x ∈ R . The aim is to prove uniqueness of positive bounded solutions Liouville type theorems. Along the way, we establish also various existence results. We first derive a sufficient condition, directly expressed in terms of the coefficients of the line...

Attractors for Modulation Equations on Unbounded Domains { Existence and Comparison {

We are interested in the long{time behavior of nonlinear parabolic PDEs deened on unbounded cylindrical domains. For dissipative systems deened on bounded domains, the long{time behavior can often be described by the dynamics in their nite{dimensional attractors. For systems deened on the innnite line, very little is known at present, since the lack of compactness prevents application of the st...