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

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 $\Omega\subseteq \mathbb{R}^n$, $n$ 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
email: giacomo4@yahoo.fr
Jyotshana V. Prajapat
School of Mathematics
Tata Institute of Fundamental Research
Homi Bhabha Road, Mumbai 400 005, India
email: jyotsna@math.tifr.res.in
Mythily Ramaswamy
TIFR Center, IISc. Campus
Bangalore 560 012, India
email: mythily@math.tifrbng.res.in

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