Rendiconti del Seminario Matematico della Università di Padova


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Volume 132, 2014, pp. 25–32
DOI: 10.4171/RSMUP/132-2

Published online: 2014-11-04

Positive solutions for a semipositone problem involving nonlocal operator

Ghasem A. Afrouzi[1], Nguyen Thanh Chung[2] and S. Shakeri[3]

(1) University of Mazandaran, Babolsar, Iran
(2) Quang Binh University, Vietnam
(3) University of Mazandaran, Babolsar, Iran

In this article, we are interested in the existence of positive solutions for the following Kirchhoff type problems $$\cases{ -M\Big (\int\limits _ {{\Omega} }\vert \nabla u\vert ^p\,dx\Big){\rm div}\big (\vert \nabla u\vert ^{p-2}\nabla u\big) = {\lambda} a(x)\,f(u)-{\mu}{\ in\ } {\Omega},\cr u =0 {\ on\ } x \in \partial {\Omega},}$$ where $ {\Omega} $ is a bounded smooth domain of $ R^N, 1< p< N , M: R^+_ 0\to R^+$ is a continuous and increasing function, $ {\lambda}, {\mu} $ are two positive parameters, $ a\in C(\overline {\Omega} ) , a(x)\geq a_ 0> 0$ , and $ f$ is a $ C^1([0,\infty ))$ function such that $ f(0)=0 , f(t)> 0$ for all $ 0< t< t_ 0$ and $f(t)\leq 0$ for all $ t \geq t_ 0$ , where $ t_ 0> 0$ .

Keywords: Kirchhoff type problems, semipositone, positive solution, sub and supersolutions

Afrouzi Ghasem, Chung Nguyen Thanh, Shakeri S.: Positive solutions for a semipositone problem involving nonlocal operator. Rend. Sem. Mat. Univ. Padova 132 (2014), 25-32. doi: 10.4171/RSMUP/132-2