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

Author(s)

Figueiredo, Giovany M. ^[1] ; Severo, Uberlandio B. ^[2] ; Siciliano, Gaetano ^[3]

Affiliation

^[1] Univ Brasilia, Dept Matemat, BR-70910900 Brasilia, DF - Brazil

^[2] Univ Fed Paraiba, Dept Matemat, BR-58051900 Joao Pessoa, PB - Brazil

^[3] Univ Sao Paulo, Inst Matemat & Estat, Dept Matemat, Rua Matao 1010, BR-05508090 Sao Paulo, SP - Brazil

• Document type: Journal article
• Source: ADVANCED NONLINEAR STUDIES; v. 20, n. 4, p. 933-963, NOV 2020.
• Web of Science Citations: 0

Abstract

In this paper we prove an existence result of multiple positive solutions for the following quasilinear problem: [-Delta u - Delta(u(2))u = vertical bar u vertical bar(p-2)u in Omega, u = 0 on partial derivative Omega, where Omega is a smooth and bounded domain in R-N, N >= 3. More specifically we prove that, for p near the critical exponent 22{*} = 4N/(N - 2), the number of positive solutions is estimated below by topological invariants of the domain Omega: the Ljusternick-Schnirelmann category and the Poincare 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. (AU)

• FAPESP's process: 18/17264-4 - Existence of solutions for nonlinear elliptic equations
• Grantee: Gaetano Siciliano
• Support Opportunities: Regular Research Grants