Positive solutions to indefinite Neumann problems when the weight has positive average

We deal with positive solutions for the Neumann boundary value problem associated with the scalar second order ODE $$u" + q(t)g(u) = 0, \quad t \in [0, T],$$ where $g: [0, +\infty[\, \to \mathbb{R}$ is positive on $\,]0, +\infty[\,$ and $q(t)$ is an indefinite weight. Complementary to previous investigations in the case $\int_0^T q(t) < 0$, we provide existence results for a suitable class of weights having (small) positive mean, when $g'(x) < 0$ at infinity. Our proof relies on a shooting argument for a suitable equivalent planar system of the type $$x' = y, \qquad y' = h(x)y^2 + q(t),$$ with $h(x)$ a continuous function defined on the whole real line.Comment: 17 pages, 3 figure

Pairs of positive periodic solutions of nonlinear ODEs with indefinite weight: a topological degree approach for the super-sublinear case

We study the periodic and the Neumann boundary value problems associated with the second order nonlinear differential equation \begin{equation*} u'' + c u' + \lambda a(t) g(u) = 0, \end{equation*} where $g \colon \mathopen{[}0,+\infty\mathclose{[}\to \mathopen{[}0,+\infty\mathclose{[}$ is a sublinear function at infinity having superlinear growth at zero. We prove the existence of two positive solutions when $\int_{0}^{T} a(t) \!dt < 0$ and $\lambda > 0$ is sufficiently large. Our approach is based on Mawhin's coincidence degree theory and index computations.Comment: 26 page

The similarity problem for indefinite Sturm-Liouville operators and the HELP inequality

We study two problems. The first one is the similarity problem for the indefinite Sturm-Liouville operator A=-(\sgn\, x)\frac{d}{wdx}\frac{d}{rdx} acting in $L^2_{w}(-b,b)$. It is assumed that w,r\in L^1_{\loc}(-b,b) are even and positive a.e. on $(-b,b)$. The second object is the so-called HELP inequality $$(\int_{0}^b\frac{1}{\tilde{r}}|f'|\, dx)^2 \le K^2 \int_{0}^b|f|^2\tilde{w}\,dx\int_{0}^b\Big|\frac{1}{\tilde{w}}\big(\frac{1}{\tilde{r}}f'\big)'\Big|^2\tilde{w}\, dx,$$ where the coefficients \tilde{w},\tilde{r}\in L^1_{\loc}[0,b) are positive a.e. on $(0,b)$. Both problems are well understood when the corresponding Sturm-Liouville differential expression is regular. The main objective of the present paper is to give criteria for both the validity of the HELP inequality and the similarity to a self-adjoint operator in the singular case. Namely, we establish new criteria formulated in terms of the behavior of the corresponding Weyl-Titchmarsh $m$-functions at 0 and at $\infty$. As a biproduct of this result we show that both problems are closely connected. Namely, the operator $A$ is similar to a self-adjoint one precisely if the HELP inequality with $\tilde{w}=r$ and $\tilde{r}=w$ is valid. Next we characterize the behavior of $m$-functions in terms of coefficients and then these results enable us to reformulate the obtained criteria in terms of coefficients. Finally, we apply these results for the study of the two-way diffusion equation, also known as the time-independent Fokker-Plank equation.Comment: 42 page