Sixth Mississippi State Conference on Differential Equations and
Computational Simulations.
Electronic Journal of Differential Equations,
Conference 15 (2007), pp. 107-126. 

Title: Positive solutions for elliptic problems with critical
       indefinite nonlinearity in bounded domains

Authors: Jacques Giacomoni (Manufacture des Tabacs, Toulouse, France)
         Jyotshana V. Prajapat (Tata Inst. of Fundamental Research,India)
	 Mythily Ramaswamy (TIFR Center, Bangalore, India)
 
Abstract: 
 In this paper, we study the semilinear elliptic problem with critical
 nonlinearity and an indefinite weight function, namely
 $$
 - \Delta u =\lambda u + h (x) u^{(n+2)/(n-2)}
 $$
 in  a smooth open  bounded domain $\Omega\subseteq \mathbb{R}^n$, 
 $n > 4 $
 with Dirichlet boundary conditions and for $\lambda  \geq 0 $.
 Under suitable assumptions on the weight function, we obtain
 the positive solution branch, bifurcating from the first
 eigenvalue $\lambda_1(\Omega)$. For $n=2$, we get similar results
 for $-\Delta u =\lambda u + h (x)\phi(u)e^u$ where
 $\phi$ is bounded and superlinear near zero.

Published February 28, 2007.
Math Subject Classifications: 35J60, 35B45, 35B33, 35B32.
Key Words: Critical indefinite nonlinearity; bifurcation; a priori estimates.