July  1997, 3(3): 333-340. doi: 10.3934/dcds.1997.3.333

Indefinite elliptic problems in a domain

1. 

Department of Mathematics, Southwest Missouri State University, United States

2. 

Department of Applied Mathematics, University of Colorado at Boulder

Received  January 1997 Published  April 1997

In this paper, we study the elliptic boundary value problem in a bounded domain $\Omega$ in $R^n$, with smooth boundary:

$-\Delta u = R(x) u^p \quad \quad u > 0 x \in \Omega$

$u(x) = 0 \quad \quad x \in \partial \Omega.$

where $R(x)$ is a smooth function that may change signs. In [2], using a blowing up argument, Berestycki, Dolcetta, and Nirenberg, obtained a priori estimates and hence the existence of solutions for the problem when the exponent $1 < p < {n+2}/{n-1}$. Inspired by their result, in this article, we use the method of moving planes to fill the gap between ${n+2}/{n-1}$ and the critical Sobolev exponent ${n+2}/{n-2}$. We obtain a priori estimates for the solutions for all $1 < p < {n+2}/{n-2}$.

Citation: Wenxiong Chen, Congming Li. Indefinite elliptic problems in a domain. Discrete and Continuous Dynamical Systems, 1997, 3 (3) : 333-340. doi: 10.3934/dcds.1997.3.333
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