Lower and upper solutions for singular derivative dependent Dirichlet problem (Q2745655)

The author develops the lower and upper solution method to work out the Dirichlet problem NEWLINE\[NEWLINE u''+f(t,u,u')=0,\quad u(a)=0,\;u(b)=0, NEWLINE\]NEWLINE where \(f\) can be singular for \(t=a\), \(t=b\) and \(u=0\). Solutions are supposed to be in \(W^{1,1}(a,b)\cap W^{2,1}_{loc}(a,b)\). Here, the main difficulty concerns the derivative dependence of \(f\) and the obtaining of a priori bounds on this derivative. Classical Nagumo conditions lead to a bound on \(\|u'\|_{\infty}\) which rules out simple singularities for \(t=a\) and \(t=b\). The basic idea in the paper is to modify the Nagumo condition to obtain a time dependent bound \(|u'(t)|\leq h(t)\), with \(h\in L^1(a,b)\). The framework used by the author allows in particular singularities in \(t=a\) and \(t=b\) such that the function \(f\) is uniformly bounded on compact \((u,u')\)-sets by some \(h\in L^1_{loc}(a,b)\) with \(\int_a^b(t-a)(b-t)h(t) dt<\infty\). Singularity in \(u=0\) is also allowed but is appropriately lower bounded such that there exist small lower solutions.NEWLINENEWLINENEWLINEA first section concerns the lower and upper solution method for regular problems with \(L^p\)-nonlinearities. A result is obtained from a classical modification technique without assuming that the lower and upper solutions have bounded derivatives. The singular problem is studied with great care in a second section. Generalizations to general bounding functions using diagonal functions are considered. The whole paper is illustrated by examples which show the interest of the various generalizations. In the last section, the author considers an application where the nonlinearity is lower bounded near the origin and upper bounded far away from the origin by linear functions. This corresponds to nonlinearities with slope larger to the first eigenvalue when \(u\) goes to zero and smaller than that same eigenvalue when \(u\) goes to infinity.