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Volume 32, Issue 1
Entire Large Solutions to Semilinear Elliptic Systems of Competitive Type

Alan V. Lair

DOI: 10.4208/jpde.v32.n1.4

J. Part. Diff. Eq., 32 (2019), pp. 52-65.

Published online: 2019-04

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  • Abstract

We consider the elliptic system $\Delta u = p(|x|)u^av^b$, $\Delta v = q(|x|)u^cv^d$ on ${\bf R}^n$ ($n \geq 3$) where $a$, $b$, $c$, $d$ are nonnegative constants with $\max\{a,d\} \leq 1$, and the functions $p$ and $q$ are nonnegative, continuous, and the support of $\min\{p(r),q(r)\}$ is not compact.  We establish conditions on $p$ and $q$, along with the exponents $a$, $b$, $c$, $d$, which ensure the existence of a positive entire solution satisfying $\lim_{|x|\rightarrow \infty}u(x) = \lim_{|x| \rightarrow \infty}v(x) = \infty$.

  • Keywords

Large solution entire solution semilinear elliptic system.

  • AMS Subject Headings

35J47, 35B44

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

alan.lair@afit.edu (Alan V. Lair)

  • BibTex
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  • TXT
@Article{JPDE-32-52, author = {Lair , Alan V.}, title = {Entire Large Solutions to Semilinear Elliptic Systems of Competitive Type}, journal = {Journal of Partial Differential Equations}, year = {2019}, volume = {32}, number = {1}, pages = {52--65}, abstract = {

We consider the elliptic system $\Delta u = p(|x|)u^av^b$, $\Delta v = q(|x|)u^cv^d$ on ${\bf R}^n$ ($n \geq 3$) where $a$, $b$, $c$, $d$ are nonnegative constants with $\max\{a,d\} \leq 1$, and the functions $p$ and $q$ are nonnegative, continuous, and the support of $\min\{p(r),q(r)\}$ is not compact.  We establish conditions on $p$ and $q$, along with the exponents $a$, $b$, $c$, $d$, which ensure the existence of a positive entire solution satisfying $\lim_{|x|\rightarrow \infty}u(x) = \lim_{|x| \rightarrow \infty}v(x) = \infty$.

}, issn = {2079-732X}, doi = {https://doi.org/10.4208/jpde.v32.n1.4}, url = {http://global-sci.org/intro/article_detail/jpde/13122.html} }
TY - JOUR T1 - Entire Large Solutions to Semilinear Elliptic Systems of Competitive Type AU - Lair , Alan V. JO - Journal of Partial Differential Equations VL - 1 SP - 52 EP - 65 PY - 2019 DA - 2019/04 SN - 32 DO - http://doi.org/10.4208/jpde.v32.n1.4 UR - https://global-sci.org/intro/article_detail/jpde/13122.html KW - Large solution KW - entire solution KW - semilinear KW - elliptic system. AB -

We consider the elliptic system $\Delta u = p(|x|)u^av^b$, $\Delta v = q(|x|)u^cv^d$ on ${\bf R}^n$ ($n \geq 3$) where $a$, $b$, $c$, $d$ are nonnegative constants with $\max\{a,d\} \leq 1$, and the functions $p$ and $q$ are nonnegative, continuous, and the support of $\min\{p(r),q(r)\}$ is not compact.  We establish conditions on $p$ and $q$, along with the exponents $a$, $b$, $c$, $d$, which ensure the existence of a positive entire solution satisfying $\lim_{|x|\rightarrow \infty}u(x) = \lim_{|x| \rightarrow \infty}v(x) = \infty$.

Alan V. Lair. (2019). Entire Large Solutions to Semilinear Elliptic Systems of Competitive Type. Journal of Partial Differential Equations. 32 (1). 52-65. doi:10.4208/jpde.v32.n1.4
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