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On the elliptic equations \(\Delta u=K(x)u^{\sigma}\) and \(\Delta u=K(x)e^{2u}\). (English) Zbl 0635.35027
The authors present several sufficient conditions on the “average” \(\bar K\) which imply the nonexistence of positive solutions for the equations in the title, where \(\sigma >1\) and \(K(x)\geq 0\) is a bounded Hölder continuous function in \(R^ n\). These equations arise in a geometrical context by J. L. Kazdan and F. W. Warner [J. Differ. Geom. 10, 113-134 (1975; Zbl 0296.53037)] and existence results have been obtained by W.-M. Ni [Indiana Univ. Math. J. 31, 493-529 (1982; Zbl 0496.35036)] and others. The proofs are by contradiction and are essentially equivalent to the technique of J. B. Keller [Commun. Pure Appl. Math. 10, 503-510 (1957; Zbl 0090.318)].
Reviewer: P.W.Schaefer

35J60 Nonlinear elliptic equations
35J15 Second-order elliptic equations
45G10 Other nonlinear integral equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
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