一类Kirchhoff型系统解的存在性

Existence of Positive Solutions for a Class of Kirchhoff Type Systems

英文关键词:positive solutions  existence  Kirchhoff type systems

中文摘要:

      

英文摘要:

      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.