Existence of positive solutions in the superlinear case via coincidence degree: the Neumann and the periodic boundary value problems. (English) Zbl 1345.34031

Feltrin, Guglielmo; Zanolin, Fabio

Adv. Differ. Equ. 20, No. 9-10, 937-982 (2015).

The authors consider the positive solutions of the second order nonlinear boundary value problem \[ u'' + a(x)g(u) = 0, \;\; {\mathcal B}(u,u') = 0, \] where \(g : \mathbb R^+ \to \mathbb R^+\) is continuous, \(g(0) = 0\), \(g(s) > 0\) for \(s > 0\), \(a \in L^1(0,T)\) and \({\mathcal B}(u) = (u'(0),u'(T))\) or \((u(T)-u(0),u'(T)-u'(0))\). A necessary conditions for the existence of a positive solution is that \(a\) changes sign and has negative mean value. Using coincidence degree together with delicate estimates, the authors prove the existence of a positive solution to the above problems when, in addition to the conditions above, the set on which \(a\) is positive is made of a finite number of pairwise disjoint intervals \(J_k\), \(g\) is superlinear at \(0\) and \(\liminf_{s \to + \infty}g(s)/s\) is larger that all the first eigenvalues of the Dirichlet problem with weight \(a\) on the intervals \(J_k\). This is in particular the case when \(g\) is superlinear at infinity. The sharpness of those conditions is discussed.

Reviewer: Jean Mawhin (Louvain-La-Neuve)

MSC:

34B18	Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B15	Nonlinear boundary value problems for ordinary differential equations
34C25	Periodic solutions to ordinary differential equations
47H11	Degree theory for nonlinear operators

Keywords:

positive solutions; Neumann boundary conditions; periodic boundary conditions; superlinear case; coincidence degree