Multiple Solutions for the p-Laplacian Equation with Convection

This paper investigates a class of nonlinear elliptic problems involving a p -Laplacian operator and a gradient-dependent reaction term, which introduces a convective component and disrupts the variational structure of the problem. Motivated by recent advances in the field, we establish the existence of at least two weak solutions with sign information—one positive and one negative—without relying on asymptotic conditions near zero. Our approach relies on variational techniques and the Leray–Schauder alternative principle, extending previous results and pertaining to a broader class of nonlinearities. Our existence results are supported by examples that illustrate the applicability of our framework.