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Multiplicity of Positive Solutions for a Quasilinear Schrödinger Equation with an Almost Critical Nonlinearity

Authors
  • Figueiredo, Giovany M.1
  • Severo, Uberlandio B.2
  • Siciliano, Gaetano3
  • 1 Universidade de Brasília, 70910-900, DF , (Brazil)
  • 2 Universidade Federal da Paraíba, 58051-900, PB , (Brazil)
  • 3 Universidade de São Paulo, Instituto de Matemática e Estatística, Rua do Matão 1010, 05508-090, SP , (Brazil)
Type
Published Article
Journal
Advanced Nonlinear Studies
Publisher
De Gruyter
Publication Date
Aug 06, 2020
Volume
20
Issue
4
Pages
933–963
Identifiers
DOI: 10.1515/ans-2020-2105
Source
De Gruyter
Keywords
License
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Abstract

In this paper we prove an existence result of multiple positive solutions for the following quasilinear problem: {-Δ⁢u-Δ⁢(u2)⁢u=|u|p-2⁢uin ⁢Ω,u=0on ⁢∂⁡Ω,\left\{\begin{aligned} \displaystyle-\Delta u-\Delta(u^{2})u&\displaystyle=|u|% ^{p-2}u&&\displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\phantom{}\text{on }\partial% \Omega,\end{aligned}\right. where Ω is a smooth and bounded domain in ℝN,N≥3{\mathbb{R}^{N},N\geq 3}. More specifically we prove that, for p near the critical exponent 22*=4⁢N/(N-2){22^{*}=4N/(N-2)}, the number of positive solutions is estimated below by topological invariants of the domain Ω: the Ljusternick–Schnirelmann category and the Poincaré polynomial. With respect to the case involving semilinear equations, many difficulties appear here and the classical procedure does not apply immediately. We obtain also en passant some new results concerning the critical case.

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