Classical Sobolev approach for a critical fourth-order Leray-Lions type problem: existence and multiplicity of solutions

TL;DR

This paper investigates a critical fourth-order elliptic problem of Leray-Lions type, establishing the existence and multiplicity of solutions using variational methods and Sobolev space analysis, considering different nonlinear growth regimes.

Contribution

It introduces a Sobolev space framework for a critical fourth-order Leray-Lions problem, proving solution existence and multiplicity under sublinear and superlinear conditions.

Findings

Infinitely many solutions in the sublinear case.

At least one solution in the superlinear case.

Application to Hamiltonian systems.

Abstract

A fourth-order elliptic problem of Leray-Lions type is considered for combined nonlinearities and Sobolev-critical growth with Navier and Dirichlet boundary conditions. By combining variational methods and critical point theory, the existence and multiplicity of weak solutions are established in the setting of classical Sobolev spaces. Two distinct asymptotic regimes are considered for the perturbation term: sublinear and superlinear. In the sublinear case, the existence of infinitely many solutions is proved by using topological tools such as Krasnosel'skii's genus and Clark's deformation lemma. In the superlinear case, the existence of at least one nontrivial solution is obtained via the Mountain Pass Theorem. Furthermore, the applicability of the main results is illustrated in the context of Hamiltonian systems.