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Anisotropic Quasilinear Elliptic Systems with Homogeneous Critical Nonlinearities

Mathew Gluck

Abstract

In this work we consider a system of quasilinear elliptic equations driven by an anisotropic $p$-Laplacian. The lower-order nonlinearities are in potential form and exhibit critical Sobolev growth. We exhibit conditions on the coefficients of the differential operator, the domain of the unknown function, and the lower-order nonlinearities under which nontrivial solutions are guaranteed to exist and conditions on these objects under which a nontrivial solution does not exist.

Anisotropic Quasilinear Elliptic Systems with Homogeneous Critical Nonlinearities

Abstract

In this work we consider a system of quasilinear elliptic equations driven by an anisotropic -Laplacian. The lower-order nonlinearities are in potential form and exhibit critical Sobolev growth. We exhibit conditions on the coefficients of the differential operator, the domain of the unknown function, and the lower-order nonlinearities under which nontrivial solutions are guaranteed to exist and conditions on these objects under which a nontrivial solution does not exist.
Paper Structure (11 sections, 25 theorems, 172 equations)

This paper contains 11 sections, 25 theorems, 172 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Variational Problem
  4. Inequalities of Sobolev Type
  5. Sufficient Condition for Existence of a Minimizer
  6. Proofs of Existence Theorems
  7. Construction of Test Function for $\gamma\leq p$
  8. Construction of Test Function for $\gamma>p$
  9. Proof of the Non-Existence Result
  10. Appendix
  11. Regularity

Key Result

Theorem A

Let $n\geq 3$ and let $\Omega \subset \mathbb R^n$ be a bounded open set.

Theorems & Definitions (48)

  • Theorem A
  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 1.4
  • Example 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Remark 1.8
  • Proposition 2.1
  • ...and 38 more