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Volume 30, Issue 4
Existence and Nonexistence for Semilinear Equations on Exterior Domains

Joseph A. Iaia

DOI: 10.4208/jpde.v30.n4.2

J. Part. Diff. Eq., 30 (2017), pp. 299-316.

Published online: 2017-11

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  • Abstract

In this paper we prove the existence of an infinite number of radial solutions of $\Delta u + K(r)f(u)= 0$ on the exterior of the ball of radius $R>0$ centered at the origin in ${\mathbb R}^{N}$ where $f$ is odd with $f<0$ on $(0, \beta) $, $f>0$ on $(\beta, \delta),$  $f\equiv 0$ for $u> \delta$, and where the function $K(r)$ is assumed to be positive and $K(r)\to 0$ as $r \to \infty.$ The primitive $F(u)  = \int_{0}^{u} f(s) \, {\rm d}s$ has a  ``hilltop'' at $u=\delta$ which allows one to use the shooting method to prove the existence of solutions.

  • Keywords

Semilinear hilltop

  • AMS Subject Headings

34B40, 35B05

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

iaia@unt.edu (Joseph A. Iaia)

  • BibTex
  • RIS
  • TXT
@Article{JPDE-30-299, author = {Iaia , Joseph A.}, title = {Existence and Nonexistence for Semilinear Equations on Exterior Domains}, journal = {Journal of Partial Differential Equations}, year = {2017}, volume = {30}, number = {4}, pages = {299--316}, abstract = {

In this paper we prove the existence of an infinite number of radial solutions of $\Delta u + K(r)f(u)= 0$ on the exterior of the ball of radius $R>0$ centered at the origin in ${\mathbb R}^{N}$ where $f$ is odd with $f<0$ on $(0, \beta) $, $f>0$ on $(\beta, \delta),$  $f\equiv 0$ for $u> \delta$, and where the function $K(r)$ is assumed to be positive and $K(r)\to 0$ as $r \to \infty.$ The primitive $F(u)  = \int_{0}^{u} f(s) \, {\rm d}s$ has a  ``hilltop'' at $u=\delta$ which allows one to use the shooting method to prove the existence of solutions.

}, issn = {2079-732X}, doi = {https://doi.org/10.4208/jpde.v30.n4.2}, url = {http://global-sci.org/intro/article_detail/jpde/10676.html} }
TY - JOUR T1 - Existence and Nonexistence for Semilinear Equations on Exterior Domains AU - Iaia , Joseph A. JO - Journal of Partial Differential Equations VL - 4 SP - 299 EP - 316 PY - 2017 DA - 2017/11 SN - 30 DO - http://doi.org/10.4208/jpde.v30.n4.2 UR - https://global-sci.org/intro/article_detail/jpde/10676.html KW - Semilinear KW - hilltop AB -

In this paper we prove the existence of an infinite number of radial solutions of $\Delta u + K(r)f(u)= 0$ on the exterior of the ball of radius $R>0$ centered at the origin in ${\mathbb R}^{N}$ where $f$ is odd with $f<0$ on $(0, \beta) $, $f>0$ on $(\beta, \delta),$  $f\equiv 0$ for $u> \delta$, and where the function $K(r)$ is assumed to be positive and $K(r)\to 0$ as $r \to \infty.$ The primitive $F(u)  = \int_{0}^{u} f(s) \, {\rm d}s$ has a  ``hilltop'' at $u=\delta$ which allows one to use the shooting method to prove the existence of solutions.

Iaia , Joseph A.. (2017). Existence and Nonexistence for Semilinear Equations on Exterior Domains. Journal of Partial Differential Equations. 30 (4). 299-316. doi:10.4208/jpde.v30.n4.2
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