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Existence of decaying entire solutions of a class of semilinear elliptic equations


Authors: Takaŝi Kusano, Ezzat S. Noussair and Charles A. Swanson
Journal: Proc. Amer. Math. Soc. 104 (1988), 1141-1147
MSC: Primary 35J60; Secondary 35B40
DOI: https://doi.org/10.1090/S0002-9939-1988-0929405-1
MathSciNet review: 929405
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Abstract: The main result establishes the existence of a nontrivial nonnegative radial solution $ u \in C_{\operatorname{loc} }^2({{\mathbf{R}}^N})$ of a semilinear elliptic eigenvalue problem in $ {{\mathbf{R}}^N},N \geq 3$, such that $ u(\vert x\vert)$ has uniform limit zero as $ \vert x\vert \to \infty $. Asymptotic decay estimates and necessary conditions are obtained. Since such solutions do not exist in the space $ W_0^{1,2}({{\mathbf{R}}^N})$, a considerable departure from standard procedures is required.


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DOI: https://doi.org/10.1090/S0002-9939-1988-0929405-1
Keywords: Semilinear elliptic equation, bounded nonlinearity, entire nonnegative solution, asymptotic decay law
Article copyright: © Copyright 1988 American Mathematical Society

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