Semilinear elliptic singular perturbation problems with nonuniform interior behavior

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...

Singular Solutions for some Semilinear Elliptic Equations

We are concerned with the behavior of u near x = O. There are two distinct cases: 1) When p >= N / ( N -2) and (N ~ 3) it has been shown by BR~ZIS & V~RON [9] that u must be smooth at 0 (See also BARAS & PIERRE [1] for a different proof). In other words, isolated singularities are removable. 2) When 1-< p < N / ( N 2) there are solutions of (1) with a singularity at x ---0. Moreover all singula...

On a singular nonlinear semilinear elliptic problem

where K(x)μC2,b(V9 ), a, pμ(0, 1) and l is a real parameter. Such singular elliptic problems arise in the contexts of chemical heterogeneous catalysts, nonNewtonian fluids and also the theory of heat conduction in electrically conducting materials, see [3, 5, 8, 9] for a detailed discussion. Obviously (1.1) cannot have a solution uμC2(V9 ) if K(x) is not vanishing near ∂V. However, under variou...

Existence and Nonexistence of Positive Singular Solutions for Semilinear Elliptic Problems with Applications in Astrophysics

A C0 Interior Penalty Method for a Fourth Order Elliptic Singular Perturbation Problem

Abstract. In this paper, we develop a C0 interior penalty method for a fourth order singular perturbation elliptic problem in two dimensions on polygonal domains. Using some a posteriori error analysis techniques, we are able to show that the method converges in the energy norm uniformly with respect to the perturbation parameter under minimal regularity assumptions. In addition, we analyze the...