On Kirchhoff-type p(.)-Laplacian problems withsandwich-type and arbitrary growth

TL;DR

This paper proves the existence of positive bounded weak solutions for a class of Kirchhoff-type p(.)-Laplacian problems with arbitrary and sandwich-type growth, addressing significant analytical challenges.

Contribution

It introduces a novel approach combining truncation and a priori estimates to handle complex growth conditions in Kirchhoff-type problems.

Findings

Existence of positive bounded weak solutions established.

Method effectively manages arbitrary and sandwich-type growth conditions.

Analytical framework addresses variational difficulties in nonlinear PDEs.

Abstract

We establish the existence of a positive bounded weak solution for a class of Kirchhoff-type $p(\cdot)$-Laplacian problems involving an arbitrary growth and a sandwich-type growth $s(\cdot)\in (\inf p,\sup p)$. This setting leads to substantial analytical difficulties in the variational analysis of the associated energy functional. By combining truncation arguments with a priori estimates, we prove the existence result under suitable assumptions on the data.