Bounded Solutions for Nonhomogeneous Double Phase Problems With Gradient Dependence

Abstract

This paper investigates the existence of bounded weak solutions to a class of nonhomogeneous double phase problems involving -Laplacian and -Laplacian operators with solution-dependent weights. The problem is set on a bounded domain and features a nonlinear right-hand side that depends on both the solution and its gradient. We establish uniform bounds for the solution set through a Moser iteration technique and prove existence results using truncation methods and pseudomonotone operator theory. Our work extends previous results by considering more general weight structures and gradient-dependent nonlinearities under minimal regularity assumptions. The analysis combines Sobolev space theory, variational methods, and careful energy estimates to handle the interplay between the different growth conditions and degeneracies in the problem.