[Paper Review] Existence of positive solutions in the superlinear case via coincidence degree: the Neumann and the periodic boundary value problems

Guglielmo Feltrin, Fabio Zanolin|arXiv (Cornell University)|Mar 17, 2015

Nonlinear Differential Equations AnalysisMathematics32 references19 citations

TL;DR

This paper establishes the existence of positive periodic and Neumann solutions for second-order superlinear indefinite problems of the form $ u'' + a(x)g(u) = 0 $, where $ g(u) $ exhibits superlinear growth at both zero and infinity, and $ a(x) $ is sign-changing. Using Mawhin’s coincidence degree theory, the authors derive necessary and sufficient conditions for nontrivial positive solutions, extending prior results from Dirichlet to Neumann and periodic settings.

ABSTRACT

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.

Motivation & Objective

• To establish existence criteria for positive solutions in superlinear indefinite boundary value problems with Neumann and periodic boundary conditions.
• To extend existing results—previously limited to Dirichlet problems—toward Neumann and periodic settings using topological methods.
• To analyze the role of sign-changing weight functions $ a(x) $ with negative mean value in enabling positive solutions despite superlinear nonlinearity.
• To provide necessary and sufficient conditions for the existence of nontrivial positive solutions in the superlinear case.
• To apply the results to nonlinear PDEs in annular domains and non-Hamiltonian periodic problems.

Proposed method

• Employ Mawhin’s coincidence degree theory to analyze the existence of solutions in the context of Neumann and periodic boundary value problems.
• Use a priori bounds and topological degree arguments to handle the superlinear nonlinearity $ g(u) $, which satisfies $ g(0) = 0 $, $ g(s) > 0 $ for $ s > 0 $, and has superlinear growth at both 0 and $ +\infty $.
• Apply Sturm comparison theorems and eigenvalue estimates to control the behavior of solutions in subintervals where $ a(x) > 0 $, particularly focusing on intervals $ J $ where $ a(x) $ is positive.
• Construct a sequence of approximate solutions $ \tilde{u}_n $ and use concavity and comparison techniques to derive contradiction estimates via integration against a positive eigenfunction $ \varphi $.
• Use the Carathéodory condition on $ h(x,s) $ and integrability assumptions to apply the dominated convergence theorem in the limit argument.
• Establish a contradiction by showing that a nonnegative quantity must be positive, thus proving the nonexistence of large solutions under certain assumptions, which implies the existence of a priori bounds.

Experimental results

Research questions

• RQ1 What are the necessary and sufficient conditions for the existence of positive solutions in superlinear indefinite Neumann and periodic boundary value problems?
• RQ2 How does the sign-changing nature of $ a(x) $, particularly with negative mean value, affect the existence of positive solutions in the superlinear case?
• RQ3 Can Mawhin’s coincidence degree method be extended from Dirichlet to Neumann and periodic problems for superlinear indefinite equations?
• RQ4 What role does the superlinear growth of $ g(u) $ at both zero and infinity play in enabling positive solutions despite the indefinite weight?
• RQ5 How can the topological approach be adapted to handle the lack of a first eigenvalue $ \lambda_0 > 0 $ in Neumann/periodic problems?

Key findings

• The paper proves the existence of at least one positive solution for the Neumann and periodic boundary value problems under superlinear growth conditions on $ g(u) $, even when $ a(x) $ changes sign.
• A necessary condition for existence is $ \int_0^T a(x)\,dx < 0 $, which arises from integration by parts and the positivity of $ g'(s) $, ensuring the weight cannot be non-negative.
• The authors establish that the first eigenvalue $ \lambda_0 $ in the Neumann/periodic case is zero, which invalidates the classical approach used in Dirichlet problems, necessitating a new method.
• By constructing a sequence of solutions $ \tilde{u}_n $ and using Sturm comparison with a positive eigenfunction $ \varphi $, a contradiction is derived if solutions become unbounded, proving a priori bounds.
• The contradiction arises from showing that a quantity bounded below by $ \varepsilon N \int_J q_N(x)\varphi(x)\,dx > 0 $ must be $ \leq 0 $, which is impossible.
• The method applies to both Neumann and periodic problems, as the boundary conditions do not affect the core contradiction argument, which relies only on non-negativity and concavity of solutions.