Luxemburg Norm Localisation for Nonlocal Differential Equations in Variable Exponent Lebesgue Spaces
Christopher S. Goodrich, Gabriel Nakhl
TL;DR
The paper addresses a nonlocal ODE with variable growth in a Kirchhoff-type form and analyzes it in variable exponent Lebesgue spaces using the Luxemburg norm. It introduces a Luxemburg-driven hybrid cone and a fixed-point index approach to obtain existence and precise localisation of positive solutions within a Luxemburg-norm annulus, while relaxing traditional nonlocal coefficient restrictions. Key contributions include sharp Luxemburg-based localisation bounds, embedding results that ensure compactness, and an explicit example showing significant improvements over L∞-based methods. Overall, the framework provides tighter control over solutions and broader applicability to nonlocal problems with variable exponents.
Abstract
We investigate a class of variable growth nonlocal differential equations of Kirchhoff-type having the general form \(-A\!\left(\int_0^1 b(1-s)\,\big(u(s)\big)^{p(s)}\,ds\right)\,u''(t) = λ\,f(t,u(t))\) for \(t\in(0,1)\), where \(A\) is a possibly sign-changing function. Our analysis is carried out in the variable-exponent Lebesgue space \(L^{p(\cdot)}([0,1])\) under the standing hypothesis \(p(t)>1\). We demonstrate that using the Luxemburg norm allows for a much sharper localisation of the solution to the nonlocal problem. Moreover, the conditions imposed on both \(λ\) and \(f\) are appreciably weakened when the problem is analysed within the Luxemburg norm framework. An example explicitly demonstrates both the qualitative and quantitative advantages over earlier techniques.