Variational analysis and numerical approximations for a class of quasilinear elliptic equations

This paper studies the existence, conditional uniqueness, and numerical approximation of weak solutions for a class of quasilinear elliptic equations. The problem involves a nonlinear principal operator with nonstandard growth, together with a superlinear reaction term. We demonstrate the existence of a nontrivial weak solution and prove uniqueness within the class of weak solutions with a common L^∞ bound and bounded gradient, based on a variant of the Minimum Principle, established through a weak form of the Cerami–Palais–Smale condition. Under some general growth assumptions on the nonlinearity, combined with a non-resonance condition at zero involving the first eigenvalue of the p-Laplacian, we derive our main result. Some numerical simulations based on Physics-Informed Neural Networks (PINNs) are illustrated to support the theoretical existence results and investigate the influence of exponents p and s on the solution behaviour.