一类Kirchhoff型系统解的存在性

In this paper,we are interested in the existence of positive solutions for the Kirchhoff type problems{-(a_1 + b_1M_1(∫_?|▽u|~pdx))△_(_pu) = λf(u,v),in ?,-(a_2 + b_2M_2(∫?|▽v|~qdx))△_(_qv) = λg(u,v),in ?,u = v = 0,on ??,where 1 p,q N,M i:R_0~+→ R~+(i = 1,2) are continuous and increasing functions.λ is a parameter,f,g ∈ C~1((0,∞) ×(0,∞)) × C(0,∞) × 0,∞)) are monotone functions such that f_s,f_t,g_s,g_t ≥ 0,and f(0,0) 0,g(0,0) 0(semipositone).Our proof is based on the sub-and super-solutions techniques.

Existence of Positive Solutions for a Class of Kirchhoff Type Systems

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, $f, g\in C^1((0,\infty)\times(0, \infty))\times C(0,\infty)\times0, \infty))$ 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.