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.

1 aFeltrin, Guglielmo1 aZanolin, Fabio uhttp://projecteuclid.org/euclid.ade/143506451801364nas a2200181 4500008004100000022001400041245009900055210006900154300000800223490000900231520072700240653002700967653002300994653004101017653002501058100002301083856007601106 2015 eng d a0133-018900aExistence of positive solutions of a superlinear boundary value problem with indefinite weight0 aExistence of positive solutions of a superlinear boundary value a4360 v20153 aWe 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.

10aboundary value problem10aindefinite weight10aPositive solution; existence result.10asuperlinear equation1 aFeltrin, Guglielmo uhttp://aimsciences.org//article/id/b3c1c765-e8f5-416e-8130-05cc48478026