Volume 23, Issue 4
Nonradial Entire Large Solutions of Semilinear Elliptic Equations

J. Part. Diff. Eq., 23 (2010), pp. 366-373.

Published online: 2010-11

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• Abstract

We consider the problem of whether the equation $Δu = p(x) f (u)$ on $R^N, N ≥ 3$, has a positive solution for which $lim_{|x|→∞} u(x)=∞$ where f is locally Lipschitz continuous, positive, and nondecreasing on (0,∞) and satisfies $∫^∞_1[F(t)]^{-1/2}dt=∞$ where $F(t)=∫^t_0f(s)ds$. The nonnegative function p is assumed to be asymptotically radial in a certain sense. We show that a sufficient condition to ensure such a solution u exists is that p satisfies $∫^∞_0r\min_{|x|=r}p(x)dr=∞$. Conversely, we show that a necessary condition for the solution to exist is that p satisfies $∫^∞_0r^{1+ε}\min_{|x|=r}p(x)dr=∞$ for all $ε > 0$.

• Keywords

Large solution elliptic equation semilinear equation

35J61 35J25

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@Article{JPDE-23-366, author = {}, title = {Nonradial Entire Large Solutions of Semilinear Elliptic Equations}, journal = {Journal of Partial Differential Equations}, year = {2010}, volume = {23}, number = {4}, pages = {366--373}, abstract = {

We consider the problem of whether the equation $Δu = p(x) f (u)$ on $R^N, N ≥ 3$, has a positive solution for which $lim_{|x|→∞} u(x)=∞$ where f is locally Lipschitz continuous, positive, and nondecreasing on (0,∞) and satisfies $∫^∞_1[F(t)]^{-1/2}dt=∞$ where $F(t)=∫^t_0f(s)ds$. The nonnegative function p is assumed to be asymptotically radial in a certain sense. We show that a sufficient condition to ensure such a solution u exists is that p satisfies $∫^∞_0r\min_{|x|=r}p(x)dr=∞$. Conversely, we show that a necessary condition for the solution to exist is that p satisfies $∫^∞_0r^{1+ε}\min_{|x|=r}p(x)dr=∞$ for all $ε > 0$.

}, issn = {2079-732X}, doi = {https://doi.org/10.4208/jpde.v23.n4.4}, url = {http://global-sci.org/intro/article_detail/jpde/5239.html} }
TY - JOUR T1 - Nonradial Entire Large Solutions of Semilinear Elliptic Equations JO - Journal of Partial Differential Equations VL - 4 SP - 366 EP - 373 PY - 2010 DA - 2010/11 SN - 23 DO - http://dor.org/10.4208/jpde.v23.n4.4 UR - https://global-sci.org/intro/jpde/5239.html KW - Large solution KW - elliptic equation KW - semilinear equation AB -

We consider the problem of whether the equation $Δu = p(x) f (u)$ on $R^N, N ≥ 3$, has a positive solution for which $lim_{|x|→∞} u(x)=∞$ where f is locally Lipschitz continuous, positive, and nondecreasing on (0,∞) and satisfies $∫^∞_1[F(t)]^{-1/2}dt=∞$ where $F(t)=∫^t_0f(s)ds$. The nonnegative function p is assumed to be asymptotically radial in a certain sense. We show that a sufficient condition to ensure such a solution u exists is that p satisfies $∫^∞_0r\min_{|x|=r}p(x)dr=∞$. Conversely, we show that a necessary condition for the solution to exist is that p satisfies $∫^∞_0r^{1+ε}\min_{|x|=r}p(x)dr=∞$ for all $ε > 0$.

Alan V. Lair . (2019). Nonradial Entire Large Solutions of Semilinear Elliptic Equations. Journal of Partial Differential Equations. 23 (4). 366-373. doi:10.4208/jpde.v23.n4.4
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