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

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

34B40, 35B05

iaia@unt.edu (Joseph A. Iaia)

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@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.

Joseph A. Iaia. (2019). 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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