Differential and Integral Equations

Positive solutions for infinite semipositone problems on exterior domains

Eun Kyoung Lee, Lakshmi Sankar, and R. Shivaji

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We study positive radial solutions to the problem \begin{equation} \label{maineqn1} \begin{cases} -\Delta u = \lambda K(\left|x \right|) f(u) \quad & x \in \Omega, \\ \hskip 15pt u = 0 & \mbox {if } \left|x \right| = r_0, \\ \hskip 15pt u \rightarrow0 & \mbox{as } \left|x \right|\rightarrow \infty , \end{cases} \end{equation} where $\lambda$ is a positive parameter, $\Delta u=\mbox{div} \big(\nabla u\big)$ is the Laplacian of $u$, $\Omega=\{x\in\ \mathbb{R} ^{n},n>2 : \left|x \right|>r_0\}$ is an exterior domain and $f:(0,\infty)\rightarrow \mathbb{R}$ belongs to a class of sublinear functions at $\infty$ such that they are continuous and $f(0^+)= \lim_{s \to 0^+} f(s) < 0$. In particular we also study infinite semipositone problems where $\lim_{s\rightarrow 0^{+}} f(s) =-\infty$. Here $K: [r_0,\infty)\rightarrow(0,\infty)$ belongs to a class of continuous functions such that $\lim_{r\rightarrow \infty}K(r)=0$. We establish various existence results for such boundary value problems and also extend our results to classes of systems. We prove our results by the method of sub-/supersolutions.

Article information

Differential Integral Equations, Volume 24, Number 9/10 (2011), 861-875.

First available in Project Euclid: 20 December 2012

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J25: Boundary value problems for second-order elliptic equations


Lee, Eun Kyoung; Sankar, Lakshmi; Shivaji, R. Positive solutions for infinite semipositone problems on exterior domains. Differential Integral Equations 24 (2011), no. 9/10, 861--875. https://projecteuclid.org/euclid.die/1356012889

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