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

Angelo Guimarães, Edcarlos Domingos da Silva, Eduardo. H. Gomes Tavares, Jin-Yun Yuan

TL;DR

The paper studies a fourth-order Leray–Lions type equation with Sobolev-critical growth under Navier or Dirichlet boundaries, recasting it in a classical $E_p$ Sobolev setting and formulating a $C^1$ energy functional $I_F$. It develops a localized Palais–Smale framework to overcome Sobolev-critical compactness, then treats sublinear perturbations with a truncated functional to obtain infinitely many weak solutions via genus and Clark deformation, and treats the superlinear case via Mountain Pass theory to obtain at least one weak solution. The results hinge on precise PS level thresholds and concentration-compactness arguments, providing existence and multiplicity results in the sublinear regime and a Mountain Pass existence result in the superlinear regime, for both Navier and Dirichlet problems. A key contribution is connecting these variational results to Hamiltonian systems through the transformation $v=-f( riangle u)$, yielding insights into regularity and qualitative properties of solutions in this class of problems. Overall, the work extends the theory of critical growth fourth-order PDEs in classical Sobolev spaces and strengthens the link between elliptic PDE theory and Hamiltonian dynamics.

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.

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

TL;DR

The paper studies a fourth-order Leray–Lions type equation with Sobolev-critical growth under Navier or Dirichlet boundaries, recasting it in a classical Sobolev setting and formulating a energy functional . It develops a localized Palais–Smale framework to overcome Sobolev-critical compactness, then treats sublinear perturbations with a truncated functional to obtain infinitely many weak solutions via genus and Clark deformation, and treats the superlinear case via Mountain Pass theory to obtain at least one weak solution. The results hinge on precise PS level thresholds and concentration-compactness arguments, providing existence and multiplicity results in the sublinear regime and a Mountain Pass existence result in the superlinear regime, for both Navier and Dirichlet problems. A key contribution is connecting these variational results to Hamiltonian systems through the transformation , yielding insights into regularity and qualitative properties of solutions in this class of problems. Overall, the work extends the theory of critical growth fourth-order PDEs in classical Sobolev spaces and strengthens the link between elliptic PDE theory and Hamiltonian dynamics.

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.

Paper Structure

This paper contains 17 sections, 20 theorems, 140 equations.

Table of Contents

  1. Introduction
  2. State of the art
  3. Main goal
  4. Contributions and organization of the work
  5. Functional setting
  6. The $(PS)_c$ condition
  7. A localizing result
  8. Convergence results
  9. Proof of Proposition \ref{['propcompacidade']}
  10. The sublinear case: multiplicity of weak solutions
  11. The truncated functional
  12. Proof of Theorem \ref{['theo1']}
  13. The superlinear case: existence of weak solutions
  14. The Geometric Condition
  15. Upper bound for the Mountain Pass Level
  16. ...and 2 more sections

Key Result

Lemma 2.1

Under assumptions $(f_1)$-$(f_4)$, the functional $I_F(u)$ given by functional satisfies with where $C_p:=c_q\cdot c_p,~C_r:=\frac{r}{r+1}\cdot c_r$, $C_{g,s}=\Lambda_s^{-1}|g|_\infty$, and $S:=\Lambda_q$.

Theorems & Definitions (38)

  • Lemma 2.1
  • proof
  • Definition 3.1
  • Proposition 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Corollary 3.4
  • proof
  • ...and 28 more