We prove the existence of positive periodic solutions for the second order nonlinear equation u{\textquoteright}{\textquoteright} + 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{\textquoteright}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.

}, url = {http://projecteuclid.org/euclid.ade/1435064518}, author = {Guglielmo Feltrin and Fabio Zanolin} } @article {0133-0189_2015_special_436, title = {Existence of positive solutions of a superlinear boundary value problem with indefinite weight}, journal = {Conference Publications}, volume = {2015}, number = {0133-0189_2015_special_43}, year = {2015}, pages = {436}, abstract = {We deal with the existence of positive solutions for a two-point boundary value problem associated with the nonlinear second order equation $u{\textquoteright}{\textquoteright}+a(x)g(u)=0$. The weight $a(x)$ is allowed to change sign. We assume that the function $g\colon\mathopen[0,+$\infty$\mathclose[\to\mathbb{R}$ 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.

}, keywords = {boundary value problem, indefinite weight, Positive solution; existence result., superlinear equation}, issn = {0133-0189}, doi = {10.3934/proc.2015.0436}, url = {http://aimsciences.org//article/id/b3c1c765-e8f5-416e-8130-05cc48478026}, author = {Guglielmo Feltrin} }