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Existence of weak solutions for the anisotropic $p(x)$-Laplacian via degree theory

Pablo Ochoa, Analía Silva, Federico Valverde

Abstract

In this paper, we consider Dirichlet boundary value problem involving the anisotropic $p(x)$-Laplacian, where $p(x)= (p_1(x), ..., p_n(x))$, with $p_i(x)> 1$ in $\overlineΩ$. Using the topological degree constructed by Berkovits, we prove, under appropriate assumptions on the data, the existence of weak solutions for the given problem. An important contribution is that we are considering the degenerate and the singular cases in the discussion. Finally, according to the compact embedding for anisotropic Sobolev spaces, we point out that the considered boundaru value problem may be critical in some region of $Ω$.

Existence of weak solutions for the anisotropic $p(x)$-Laplacian via degree theory

Abstract

In this paper, we consider Dirichlet boundary value problem involving the anisotropic -Laplacian, where , with in . Using the topological degree constructed by Berkovits, we prove, under appropriate assumptions on the data, the existence of weak solutions for the given problem. An important contribution is that we are considering the degenerate and the singular cases in the discussion. Finally, according to the compact embedding for anisotropic Sobolev spaces, we point out that the considered boundaru value problem may be critical in some region of .

Paper Structure

This paper contains 6 sections, 9 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Scalar variable exponent spaces
  4. Anisotropic variable exponent spaces
  5. Preliminaries in Degree Theory
  6. Proof of Theorem \ref{['existence']}

Key Result

Theorem 1.1

Suppose that $p_i \in \mathcal{C}(\overline{\Omega})$ and $p^{-}_i> 1$ for all $i=1, ..., n$. Assume $(f1)$ and $(f2)$. Then the Dirichlet problem has a weak solution $u\in W_0^{1, {\textbf{p}}(x)}(\Omega)$.

Theorems & Definitions (18)

  • Theorem 1.1
  • Theorem 2.1: Hölder's inequality
  • Proposition 2.2
  • Lemma 2.3
  • Remark 2.4
  • Theorem 2.5
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Lemma 3.1
  • ...and 8 more