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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Positive solutions of semilinear equations in cones


Author: Henrik Egnell
Journal: Trans. Amer. Math. Soc. 330 (1992), 191-201
MSC: Primary 35B05; Secondary 35J65
DOI: https://doi.org/10.1090/S0002-9947-1992-1034662-5
MathSciNet review: 1034662
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Abstract: In this paper we consider the problem of finding a positive solution of the equation $ \Delta u + \vert x{\vert^\nu }{u^{(n + 2 + 2\nu)/(n - 2)}} = 0$ in a cone $ \mathcal{C}$, with zero boundary data. We are only interested in solutions that are regular at infinity (i.e. such that $ u(x) = o(\vert x{\vert^{2 - n}})$, as $ \mathcal{C} \ni x \to \infty $). We will always assume that $ \nu > - 2$.

We show that the existence of a solution depends on the sign of $ \nu $ and also on the shape of the cone $ \mathcal{C}$.


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DOI: https://doi.org/10.1090/S0002-9947-1992-1034662-5
Article copyright: © Copyright 1992 American Mathematical Society