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Positive solutions for a semipositone problem involving nonlocal operator. (English) Zbl 1304.35276

Summary: In this article, we are interested in the existence of positive solutions for the following Kirchhoff type problems \[ \begin{cases} -M \left(\int\limits _ {{\Omega} }| \nabla u| ^p\,dx\right){\mathrm {div}}\big (| \nabla u| ^{p-2}\nabla u\big) = {\lambda} a(x)\,f(u)-{\mu}\text{ in } {\Omega},\\ u =0\text{ on } x \in \partial {\Omega}, \end{cases} \] where \( {\Omega} \) is a bounded smooth domain of \(\mathbb R^N\), \(1<p< N\), \(M:\mathbb R^+_ 0\to\mathbb R^+\) 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 \( t_ 0> 0\).

MSC:

35J60 Nonlinear elliptic equations
35B09 Positive solutions to PDEs
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References:

[1] G.A. AFROUZI - N.T. CHUNG - S. SHAKERI, Existence of positive solutions for Kirchhoff type equations, Electron. J. Differential Equations 2013, no. 180, 8 pp. · Zbl 1306.35030
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