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Transactions of the American Mathematical Society

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



Radially symmetric solutions to a Dirichlet problem involving critical exponents

Authors: Alfonso Castro and Alexandra Kurepa
Journal: Trans. Amer. Math. Soc. 343 (1994), 907-926
MSC: Primary 35B05; Secondary 35J65
MathSciNet review: 1207581
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Abstract: In this paper we answer, for $ N = 3,4$, the question raised in [1] on the number of radially symmetric solutions to the boundary value problem $ - \Delta u(x) = \lambda u(x) + u(x)\vert u(x){\vert^{4/(N - 2)}}$, $ x \in B: = \{ x \in {R^N}:\left\Vert x \right\Vert < 1\} $, $ u(x) = 0$, $ x \in \partial B$, where $ \Delta $ is the Laplacean operator and $ \lambda > 0$. Indeed, we prove that if $ N = 3,4$, then for any $ \lambda > 0$ this problem has only finitely many radial solutions. For $ N = 3,4,5$ we show that, for each $ \lambda > 0$, the set of radially symmetric solutions is bounded. Moreover, we establish geometric properties of the branches of solutions bifurcating from zero and from infinity.

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Keywords: Critical exponent, radially symmetric solutions, Dirichlet problem, nodal curves, bifurcation
Article copyright: © Copyright 1994 American Mathematical Society

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