On the uniqueness of bound state solutions of a semilinear equation with weights

Carmen Cortázar; Marta García-Huidobro; Pilar Herreros

doi:10.3934/dcds.2019294

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2019, Volume 39, Issue 11: 6761-6784. Doi: 10.3934/dcds.2019294

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On the uniqueness of bound state solutions of a semilinear equation with weights

Departamento de Matemática, Pontificia Universidad Católica de Chile, Casilla 306, Correo 22, Santiago, Chile

Received: April 30, 2019

Revised: April 30, 2019

Published: August 2019

This research was supported by FONDECYT-1190102 for the first and second author, FONDECYT-1160540 for the second author and FONDECYT-1170665 for third author

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Abstract

We consider radial solutions of a general elliptic equation involving a weighted Laplace operator. We establish the uniqueness of the radial bound state solutions to

$ \mbox{div}\big(\mathsf A\, \nabla v\big)+\mathsf B\, f(v) = 0\, , \quad\lim\limits_{|x|\to+\infty}v(x) = 0, \quad x\in\mathbb R^n, ~~~~{(P)} $

$ n>2 $, where $ \mathsf A $ and $ \mathsf B $ are two positive, radial, smooth functions defined on $ \mathbb R^n\setminus\{0\} $. We assume that the nonlinearity $ f\in C(-c, c) $, $ 0<c\le\infty $ is an odd function satisfying some convexity and growth conditions, and has a zero at $ b>0 $, is non positive and not identically 0 in $ (0, b) $, positive in $ (b, c) $, and is differentiable in $ (0, c) $.

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Mathematics Subject Classification: 35J61, 35A02.

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References

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