On positive solutions of semilinear elliptic equations

Dancer, E. N.; Schmitt, Klaus

doi:10.1090/S0002-9939-1987-0908646-2

On positive solutions of semilinear elliptic equations
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Proc. Amer. Math. Soc. 101 (1987), 445-452 Request permission

Abstract:

This paper is concerned with necessary conditions for the existence of positive solutions of the semilinear problem $\Delta u + f(u) = 0,x \in \Omega ,u = 0,x \in \partial \Omega$, whose supremum norm bears a certain relationship to zeros of the nonlinearity $f$. We first discuss the smooth case (i.e., $f$ and $\partial \Omega$ smooth) and then show how to obtain similar results in the nonsmooth case.

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