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

Yijing Sun

doi:10.3934/cpaa.2010.9.751

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2010, Volume 9, Issue 3: 751-760. Doi: 10.3934/cpaa.2010.9.751

This issue Previous Article A note on coupled nonlinear Schrödinger systems under the effect of general nonlinearities Next Article Arbitrarily many solutions for an elliptic Neumann problem with sub- or supercritical nonlinearity

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

Received: April 30, 2009

Revised: October 31, 2009

Published: April 2010

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Abstract

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 $-\Delta u=\lambda (1+u)^{p}$.

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Mathematics Subject Classification: Primary: 35J60, 35B30; Secondary: 35B40.

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

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