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May  2010, 9(3): 751-760. doi: 10.3934/cpaa.2010.9.751

Estimates for extremal values of $-\Delta u= h(x) u^{q}+\lambda W(x) u^{p}$

1. 

Department of Mathematics, Graduate University of Chinese Academy of Sciences, Beijing 100049, China

Received  May 2009 Revised  November 2009 Published  January 2010

In this paper we provide uniform estimates for $\lambda^{*}(N, \Omega, q, p, h, W)$ of nonlinear elliptic equations $-\Delta u= h(x) u^{q}+\lambda W(x) u^{p}$ where $W$ may change sign. We use a variational technique. Still few general results are known for this type of estimates except [6] of Gazzola and Malchiodi, which provide uniform estimates for the extremal value in case $-\Delta u=\lambda (1+u)^{p}$.
Citation: Yijing Sun. Estimates for extremal values of $-\Delta u= h(x) u^{q}+\lambda W(x) u^{p}$. Communications on Pure & Applied Analysis, 2010, 9 (3) : 751-760. doi: 10.3934/cpaa.2010.9.751
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