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Indefinite elliptic problems in a domain

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  • 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 $1 < p < {n+2}/{n-2}$.

    Mathematics Subject Classification: 35J25, 35J60.


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