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.