Differential and Integral Equations

An existence result for superlinear semipositone $p$-Laplacian systems on the exterior of a ball

Maya Chhetri, Lakshmi Sankar, R. Shivaji, and Byungjae Son

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We study the existence of positive radial solutions to the problem \begin{equation*} \left\{ \begin{aligned} -\Delta_p u &= \lambda K_1(|x|) f(v) \hspace{.3in}\mbox{in } \Omega_e,\\ -\Delta_p v &= \lambda K_2(|x|) g(u) \hspace{.31in}\mbox{in } \Omega_e, \\u &= v=0 \hspace{.7in} \mbox{ if } |x|=r_0, \\u(x)&\rightarrow 0,v(x)\rightarrow 0 \hspace{.4in} \mbox{as }\left|x \right|\rightarrow\infty, \end{aligned} \right. \end{equation*} where $\Delta_p w:=\mbox{div}(|\nabla w|^{p-2}\nabla w)$, $1 < p < n$, $\lambda$ is a positive parameter, $r_0>0$ and $\Omega_e:=\{x\in\mathbb{R}^n|~|x|>r_0\}$. Here, $K_i:[r_0,\infty)\rightarrow (0,\infty)$, $i=1,2$ are continuous functions such that $\lim_{r \rightarrow \infty} K_i(r)=0$, and $f, g:[0,\infty)\rightarrow \mathbb{R}$ are continuous functions which are negative at the origin and have a superlinear growth at infinity. We establish the existence of a positive radial solution for small values of $\lambda$ via degree theory and rescaling arguments.

Article information

Differential Integral Equations, Volume 31, Number 7/8 (2018), 643-656.

First available in Project Euclid: 11 May 2018

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 34B16: Singular nonlinear boundary value problems 34B18: Positive solutions of nonlinear boundary value problems 35J57: Boundary value problems for second-order elliptic systems


Chhetri, Maya; Sankar, Lakshmi; Shivaji, R.; Son, Byungjae. An existence result for superlinear semipositone $p$-Laplacian systems on the exterior of a ball. Differential Integral Equations 31 (2018), no. 7/8, 643--656. https://projecteuclid.org/euclid.die/1526004034

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