TY - JOUR T1 - Existence of positive solutions in the superlinear case via coincidence degree: the Neumann and the periodic boundary value problems JF - Adv. Differential Equations 20 (2015), 937–982. Y1 - 2015 A1 - Guglielmo Feltrin A1 - Fabio Zanolin AB -

We prove the existence of positive periodic solutions for the second order nonlinear equation u'' + a(x) g(u) = 0, where g(u) has superlinear growth at zero and at infinity. The weight function a(x) is allowed to change its sign. Necessary and sufficient conditions for the existence of nontrivial solutions are obtained. The proof is based on Mawhin's coincidence degree and applies also to Neumann boundary conditions. Applications are given to the search of positive solutions for a nonlinear PDE in annular domains and for a periodic problem associated to a non-Hamiltonian equation.

PB - Khayyam Publishing UR - http://projecteuclid.org/euclid.ade/1435064518 N1 - AMS Subject Classification: 34B18, 34B15, 34C25, 47H11. U1 - 35388 U2 - Mathematics U4 - 1 U5 - MAT/05 ER - TY - JOUR T1 - Existence of positive solutions of a superlinear boundary value problem with indefinite weight JF - Conference Publications Y1 - 2015 A1 - Guglielmo Feltrin KW - boundary value problem KW - indefi nite weight KW - Positive solution; existence result. KW - superlinear equation AB -

We deal with the existence of positive solutions for a two-point boundary value problem associated with the nonlinear second order equation $u''+a(x)g(u)=0$. The weight $a(x)$ is allowed to change sign. We assume that the function $g\colon\mathopen[0,+∞\mathclose[\to\mathbb{R}$ is continuous, $g(0)=0$ and satisfies suitable growth conditions, including the superlinear case $g(s)=s^p$, with $p>1$. In particular we suppose that $g(s)/s$ is large near infinity, but we do not require that $g(s)$ is non-negative in a neighborhood of zero. Using a topological approach based on the Leray-Schauder degree we obtain a result of existence of at least a positive solution that improves previous existence theorems.

VL - 2015 UR - http://aimsciences.org//article/id/b3c1c765-e8f5-416e-8130-05cc48478026 ER -