Existence of a solution to the Dirichlet problem associated to a second-order differential equation with singularities: the method of lower and upper functions (Q2854191)

This paper treats the solvability of the boundary value problem NEWLINE\[NEWLINEu''=f(t,u)+g(t,u)u',NEWLINE\]NEWLINE NEWLINE\[NEWLINEu(a+)=0,\;u(b-)=1,NEWLINE\]NEWLINE where \(f,g:(a,b)\times(0,1)\to\mathbb R\) satisfy the local Carathéodory conditions and may be singular at \(t=a\), \(t=b\), \(u=0\) and \(u=1\).NEWLINENEWLINEAssuming that the considered equation has lower and upper functions \(\sigma_1\) and \(\sigma_2\) with the properties \(0<\sigma_1(t)\leq\sigma_2(t)<1\) for \(t\in(a,b)\), \(\sigma_1(a+)=0\), \(\sigma_2(a+)<1\), \(\sigma_1(b-)>0\) and \(\sigma_2(b-)=1\) as well as NEWLINE\[NEWLINE|f(t,x)|\leq p_{\eta}(t)\quad\text{and}\quad g(t,x)\text{sgn}(t-c)\leq{{\lambda_{\eta}}\over{(b-t)(t-a)}}+q_{\eta}(t)NEWLINE\]NEWLINE for every \(\eta\in(0,1/2)\) and for a.e. \(t\in(a,b)\) and \(\sigma_{1\eta}(t)\leq x\leq \sigma_{2\eta}(t)\), where \(\sigma_{1\eta}(t)=\max\{\eta,\sigma_{1}(t)\}\), \(\sigma_{2\eta}(t)=\min\{1-\eta,\sigma_{2}(t)\}\) on \((a,b)\), \(c\in(a,b)\), \(\lambda_{\eta}\in[0,b-a)\), \(q_{\eta}\in L([a,b];\mathbb R_+)\), \(p_{\eta}\in L_{\text{loc}}((a,b);\mathbb R_+)\) with \(\int_a^b(b-s)(s-a)p_{\eta}(s)ds<+\infty,\) the authors show that the considered problem has at least one solution \(u\in AC_{\text{loc}}^1((a,b);\mathbb R)\) such that \(\sigma_1(t)\leq u(t)\leq\sigma_2(t)\) for \(t\in(a,b).\)NEWLINENEWLINEA similar result is given under the assumption that NEWLINE\[NEWLINEf(t,x)\text{sgn}(t-c)\leq p_{\eta}(t)\quad\text{and}\quad|g(t,x)|\leq{{\lambda_{\eta}}\over{(b-t)(t-a)}}+q_{\eta}(t)NEWLINE\]NEWLINE for every \(\eta\in(0,1/2)\) and for a.e. \(t\in(a,b)\) and \(\sigma_{1\eta}(t)\leq x\leq \sigma_{2\eta}(t),\) where \(c, \lambda_{\eta}, p_{\eta}, q_{\eta}, \sigma_{1\eta}\) and \(\sigma_{2\eta}\) are as above.