Sixth Mississippi State Conference on Differential Equations and Computational Simulations.
Electron. J. Diff. Eqns., Conference 15 (2007), pp. 107-126.

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

Jacques Giacomoni, Jyotshana V. Prajapat, Mythily Ramaswamy

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 $\Omega\subseteq \mathbb{R}^n$, $n$ greater than 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.

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  Jacques Giacomoni
MIP-CEREMATH/bat C, Manufacture des Tabacs
Allée de Brienne 21
31000 Toulouse, France
Jyotshana V. Prajapat
School of Mathematics
Tata Institute of Fundamental Research
Homi Bhabha Road, Mumbai 400 005, India
Mythily Ramaswamy
TIFR Center, IISc. Campus
Bangalore 560 012, India

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