Description

• Title:
  Existence of a weak solution for the fractional $p$-Laplacian equations with discontinuous nonlinearities via the Berkovits-Tienari degree theory
• Creator:
  Kim, Yun-Ho
• Description:
  We are concerned with the following nonlinear elliptic equations of the fractional $p$-Laplace type: \begin{equation*} \begin{cases} (-\Delta)_p^su \in \lambda[\underline{f}(x,u(x)), \overline{f}(x,u(x))] &\textmd{in } \Omega,\\ u= 0 &\text{on } \mathbb{R}^N\setminus\Omega, \end{cases} \end{equation*} where $(-\Delta)_p^s$ is the fractional $p$-Laplacian operator, $\lambda$ is a parameter, $0< s< 1< p< +\infty$, $sp< N$, and the measurable functions $\underline{f}$, $ \overline{f}$ are induced by a possibly discontinuous at the second variable function $f\colon \Omega\times\mathbb R \to \mathbb R$. By using the Berkovits-Tienari degree theory for upper semicontinuous set-valued operators of type (S$_+)$ in reflexive Banach spaces, we show that our problem with the discontinuous nonlinearity $f$ possesses at least one nontrivial weak solution. In addition, we show the existence of two nontrivial weak solutions in which one has negative energy and another has positive energy.
• Object availability:
  free access
• Rights:
  Prawa autorskie (c) 2018 Topological Methods in Nonlinear Analysis
• Date:
  2018.03.18
• Type:
  info:eu-repo/semantics/article;
  info:eu-repo/semantics/publishedVersion;
  Recenzowany artykuł
• Language:
  English
• Source:
  Topological Methods in Nonlinear Analysis; Vol 51, No 2 (June 2018); 371 - 388;
  1230-3429
• Publisher:
  Nicolaus Copernicus University in Toruń
• Relation: https://apcz.umk.pl/TMNA/article/view/TMNA.2017.064/14354 ;https://apcz.umk.pl/TMNA/article/view/TMNA.2017.064/14355
• Subject:
  Fractional p-Laplacian;
  weak solution;
  critical point;
  degree theory;
  Primary;
  35R11;
  35J60;
  Secondary;
  47H11
• Identifier: https://apcz.umk.pl/TMNA/article/view/TMNA.2017.064
• Data provider:
  Akademicka Platforma Czasopism
• Can I use it?:
  seek permission
• Type:
  other