American Institute of Mathematical Sciences

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 & Continuous Dynamical Systems - A, 1997, 3 (3) : 333-340. doi: 10.3934/dcds.1997.3.333
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