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

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

Yijing Sun 1,

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 $-\Delta u=\lambda (1+u)^{p}$.
Keywords: variational methods., extremal values, Semilinear elliptic equations.
Mathematics Subject Classification: Primary: 35J60, 35B30; Secondary: 35B4.
Citation: Yijing Sun. Estimates for extremal values of $-\Delta u= h(x) u^{q}+\lambda W(x) u^{p}$. Communications on Pure and Applied Analysis, 2010, 9 (3) : 751-760. doi: 10.3934/cpaa.2010.9.751
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