Construction of singular limits for four-dimensional elliptic problems with exponentially dominated nonlinearity

AbstractThe authors consider the existence of singular limit solution for a family of nonlinear elliptic problems with exponentially dominated nonlinearity and Navier boundary condition

Bifurcation for elliptic forth-order problems with quasilinear source term

We study the bifurcations of the semilinear elliptic forth-order problem with Navier boundary conditions \displaylines{ \Delta^2 u - \hbox{div} ( c(x) \nabla u ) = \lambda f(u) \quad \text{in }\Omega, \cr \Delta u = u = 0 \quad\text{on } \partial \Omega. } Where $\Omega \subset \mathbb{R}^n$, $n \geq 2$ is a smooth bounded domain, f is a positive, increasing and convex source term and $c(x)$ is a smooth positive function on $\overline{\Omega}$ such that the $L^\infty$-norm of its gradient is small enough. We prove the existence, uniqueness and stability of positive solutions. We also show the existence of critical value $\lambda^*$ and the uniqueness of its extremal solutions

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