Singular limits for 4-dimensional semilinear elliptic problems with exponential nonlinearity

Using some nonlinear domain decomposition method, we prove the existence of singular limits for solution of semilinear elliptic problems with exponential nonlinearity.Comment: 29 page

Singular limit solutions for 4-dimensional stationary Kuramoto-Sivashinsky equations with exponential nonlinearity

Let $\Omega$ be a bounded domain in $\mathbb{R}^4$ with smooth boundary, and let $x_1, x_2, \dots, x_m$ be points in $\Omega$. We are concerned with the singular stationary non-homogenous Kuramoto-Sivashinsky equation $$\Delta^2 u -\gamma\Delta u- \lambda|\nabla u|^2 = \rho^4f(u),$$ where $f$ is a function that depends only the spatial variable. We use a nonlinear domain decomposition method to give sufficient conditions for the existence of a positive weak solution satisfying the Dirichlet-like boundary conditions $u =\Delta u =0$, and being singular at each $x_i$ as the parameters $\lambda, \gamma$ and $\rho$ tend to $0$. An analogous problem in two-dimensions was considered in [2] under condition (A1) below. However we do not assume that condition

Singular limiting solutions for elliptic problem involving exponentially dominated nonlinearity and convection term

Abstract Given &#937; bounded open regular set of &#8477;2 and x1, x2, ..., xm &#8712; &#937;, we give a sufficient condition for the problem to have a positive weak solution in &#937; with u = 0 on &#8706;&#937;, which is singular at each xi as the parameters &#961;, &#955; &gt; 0 tend to 0 and where f(u) is dominated exponential nonlinearities functions. 2000 Mathematics Subject Classification: 35J60; 53C21; 58J05.</p