Elliptic Systems with Unbalanced Growth and Logarithmic Perturbation: Existence, Uniqueness and Boundedness

Abstract

In this paper, we study elliptic systems driven by logarithmic double phase operators with unbalanced growth and gradient dependent nonlinearities. The system couples two logarithmic double phase equations and is subject to homogeneous Dirichlet boundary conditions. Due to the presence of convection type terms, the problem is nonvariational in nature and classical critical point methods are not applicable. Under suitable growth and coercivity assumptions on the nonlinearities, we establish the existence of nontrivial weak solutions by means of pseudomonotonicity arguments and eigenvalue estimates for the pi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_i$$\end{document}-Laplacians for i=1,2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i=1,2$$\end{document}. For a special case, we further derive a uniqueness result based on a specific structural condition imposed on the convection terms. In addition, under stronger subcritical growth assumptions, we prove boundedness of weak solutions by adapting a Moser iteration scheme to the logarithmic double phase setting. To the best of our knowledge, these results provide the first existence, uniqueness, and boundedness theory for elliptic systems involving logarithmic double phase operators.