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

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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

Abstract
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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 & Applied Analysis, 2010, 9 (3) : 751-760. doi: 10.3934/cpaa.2010.9.751

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