| Existence of Positive Solutions for a Class of Kirchhoff Type Systems |
| Received:August 09, 2017 Revised:March 01, 2018 |
| Key Words:
positive solutions existence Kirchhoff type systems
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| Fund Project:Supported by the National Natural Science Foundation of China (Grant Nos.11571093; 11471164). |
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| Abstract: |
| In this paper, we are interested in the existence of positive solutions for the Kirchhoff type problems $$\left\{\begin{array}{ll}-(a_1+b_1M_1(\int_\Omega |\nabla u|^p\d x))\triangle_pu=\lambda f(u,v),&\mbox{in}\ \Omega,\\ -(a_2+b_2M_2(\int_\Omega |\nabla v|^q\d x))\triangle_qv=\lambda g(u,v), &\mbox{in}\ \Omega,\\ u=v=0, &\mbox{on}\ \partial\Omega,\end{array}\right.$$ where $1< p,q < N, Mi : R^+_0 \rightarrow R^+~(i = 1,2)$ are continuous and increasing functions. $\lambda$ is a parameter, $f, g\in C^1((0,\infty)\times(0, \infty))\times C([0,\infty)\times[0, \infty))$ are monotone functions such that $f_s,f_t, g_s, g_t\geq 0$, and $f(0,0) < 0, g(0,0) < 0$ (semipositone). Our proof is based on the sub- and super-solutions techniques. |
| Citation: |
| DOI:10.3770/j.issn:2095-2651.2018.04.008 |
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