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

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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 $-\Delta u=\lambda (1+u)^{p}$.
    Mathematics Subject Classification: Primary: 35J60, 35B30; Secondary: 35B40.


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