Existence of Positive Solutions for a Class of Kirchhoff Type Systems
Received:August 09, 2017  Revised:March 01, 2018
Key Word: positive solutions   existence   Kirchhoff type systems
Fund ProjectL:Supported by the National Natural Science Foundation of China (Grant Nos.11571093; 11471164).
 Author Name Affiliation Xiaomin LIU Institute of Mathematics, School of Mathematics Science, Nanjing Normal University, Jiangsu 210023, P. R. China Zuodong YANG School of Teacher Education, Nanjing Normal University, Jiangsu 210097, P. R. China
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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.