Craig Cowan

Liouville theorems for stable Lane-Emden systems and biharmonic problems

We examine the elliptic system given by for 1?<?p???? and the fourth order scalar equation where 1?<??. We prove various Liouville type theorems for positive stable solutions. For instance we show there are no positive stable solutions of (1) (respectively, (2)) provided N???10 and 2???p???? (respectively, N???10 and 1?<??). Results for higher dimensions are also obtained. These results regarding stable solutions on the full space imply various Liouville theorems for positive (possibly unstable) bounded solutions of with u?=?v?=?0 on . In particular there is no positive bounded solution of (3) for any 2???p???? if N???11. Higher dimensional results are also obtained.

On stable entire solutions of semi-linear elliptic equations with weights

We are interested in the existence versus non-existence of nontrivial stable suband super-solutions of (0.1) −div(ω1∇u) = ω2f(u) in R , with positive smooth weights ω1(x), ω2(x). We consider the cases f(u) = eu, up where p > 1 and −u−p where p > 0. We obtain various non-existence results which depend on the dimension N and also on p and the behaviour of ω1, ω2 near infinity. Also the monotonicity of ω1 is involved in some results. Our methods here are the methods developed by Farina. We examine a specific class of weights ω1(x) = (|x|2 + 1) α 2 and ω2(x) = (|x|2 + 1) β 2 g(x), where g(x) is a positive function with a finite limit at ∞. For this class of weights, non-existence results are optimal. To show the optimality we use various generalized Hardy inequalities.

Regularity of the extremal solutions in a Gelfand system problem

Abstract We examine the elliptic system given by where λ, γ are positive parameters and where Ω is a smooth bounded domain in ℝN. Let U denote the parameter region (λ, γ) of strictly positive parameters where (P)λ,γ has a smooth solution and let Υ denote the boundary of U. We show that the extremal solution (u∗, v∗) associated with (λ∗, γ∗) ∈ Υ is smooth provided that 3 ≤ N ≤ 9 and

ESTIMATES ON PULL-IN DISTANCES IN MICROELECTROMECHANICAL SYSTEMS MODELS AND OTHER NONLINEAR EIGENVALUE PROBLEMS

Motivated by certain mathematical models for microelectromechanical systems (MEMS), we give upper and lower

A Liouville Theorem for a Fourth Order H´ enon Equation

L^\infty

A short remark regarding Pohozaev-type results on general domains assuming finite Morse index

estimates for the minimal solutions of nonlinear eigenvalue problems of the form

The Critical Dimension for a Fourth Order Elliptic Problem with Singular Nonlinearity

-\Delta u=\lambda f(x)F(u)

Regularity of extremal solutions in fourth order nonlinear eigenvalue problems on general domains

on a smooth bounded domain

Optimal Hardy inequalities for general elliptic operators with improvements

\Omega

Regularity of semi-stable solutions to fourth order nonlinear eigenvalue problems on general domains

in