Absence of solutions for a system of ordinary differential inequalities
The authors prove nonexistence theorems of non-negative non-trivial solutions for a system of ordinary differential inequalities in bounded domains with singular points on the boundary of the form \[ \begin{cases} (|u'|^{p-2}u')'\geq a(x)v^{q_{1}}, \,x\in (0,x_{0}] \\ ( |v'|^{q-2}v')'\geq b(x)u^{p_{1}}, \,x\in(0,x_{0}] \\ u(x)\geq 0,v(x)\geq 0,\, x\in(0,x_{0}], \\ u'(x_{0})< 0, \,v'(x_{0})< 0, \end{cases} \tag{1} \] where \(x_{0}> 0\), \(p,q > 1\), \[ p_{1}> p-1, \;q_{1}> q-1 \tag{2} \] and the functions \(a,b\in C((0,x_{0}])\) satisfy the estimates \[ a(x)\geq cx^{-\alpha}, \;b(x)\geq dx^{-\beta} \tag{3} \] with some constants \(\alpha,\beta\in \mathbb{R}\) and \(c,d > 0\). Solutions to problem (1) are understood in the sense of distributions. The authors define the class of admissible solutions to problem (1) as \[ X((0,x_{0}]):= \{ (u,v):(0,x_{0}]\rightarrow \mathbb{R}_{+}\times \mathbb{R}_{+}, \,av^{q_{1}},bu^{p_{1}},|u'|^{p},|v'|^{q}\in L^{1}_{loc}((0,x_{0}])\}. \] In the following their main result is stated: Theorem. Suppose that the functions \(a,b\in C((0,x_{0}])\) are non-negative and satisfy inequality (3). Let (2) hold and \(\min \{ \theta_{1},\theta_{2}\}\geq 0\), where \(\theta_{1}=(\alpha-1)(q-1)+ q_{1}(\beta-q-p_{1})\) and \(\theta_{2}=(\beta-1)(p-1)+p_{1}(\alpha-p-q_{1})\). Then the problem (1) has no nontrivial non-negative solutions in \(X((0,x_{0}])\).
system of ordinary differential inequalities
scientific article; zbMATH DE number 7369736
scientific article