scholarly journals Arbitrary Number of Positive Solutions For an Elliptic Problem with Critical Nonlinearity

2005 ◽  
pp. 449-476 ◽  
Author(s):  
Olivier Rey ◽  
JUNCHENG WEI
Keyword(s):  
Positive Solutions ◽  
Elliptic Problem ◽  
Arbitrary Number ◽  
Critical Nonlinearity

Positive Solutions of a Nonlocal and Nonvariational Elliptic Problem

2021 ◽  
Vol 41 (5) ◽  
pp. 1764-1776
Author(s):  
Lingjun Liu ◽  
Feilin Shi
Keyword(s):  
Positive Solutions ◽  
Elliptic Problem

Positive solutions to a supercritical elliptic problem that concentrate along a thin spherical hole

2015 ◽  
Vol 126 (1) ◽  
pp. 341-357 ◽  
Author(s):  
Mónica Clapp ◽  
Jorge Faya ◽  
Angela Pistoia
Keyword(s):  
Positive Solutions ◽  
Elliptic Problem

NONNEGATIVE SOLUTIONS FOR INDEFINITE SUBLINEAR ELLIPTIC PROBLEMS

2012 ◽  
Vol 14 (03) ◽  
pp. 1250021 ◽  
Author(s):  
FRANCISCO ODAIR DE PAIVA
Keyword(s):  
Bounded Domain ◽  
Positive Solutions ◽  
Elliptic Problem ◽  
Solution Method ◽  
Mountain Pass Theorem ◽  
Semilinear Elliptic Problem ◽  
Semilinear Elliptic ◽  
Nonnegative Solutions ◽  
Carathéodory Function ◽  
Multiplicity Of Positive Solutions

This paper is devoted to the study of existence, nonexistence and multiplicity of positive solutions for the semilinear elliptic problem [Formula: see text] where Ω is a bounded domain of ℝN, λ ∈ ℝ and g(x, u) is a Carathéodory function. The obtained results apply to the following classes of nonlinearities: a(x)uq + b(x)up and c(x)(1 + u)p (0 ≤ q < 1 < p). The proofs rely on the sub-super solution method and the mountain pass theorem.

MULTIPLE POSITIVE SOLUTIONS OF NONLINEAR EIGENVALUE PROBLEMS ASSOCIATED TO A CLASS OF p-LAPLACIAN LIKE OPERATORS

2003 ◽  
Vol 05 (05) ◽  
pp. 737-759 ◽  
Author(s):  
NOBUYOSHI FUKAGAI ◽  
KIMIAKI NARUKAWA
Keyword(s):  
Positive Solutions ◽  
Elliptic Problem ◽  
Eigenvalue Problems ◽  
Multiple Positive Solutions ◽  
Combined Effects ◽  
Nonlinear Eigenvalue Problems ◽  
Quasilinear Elliptic Problem ◽  
Quasilinear Elliptic ◽  
Nonlinear Eigenvalue

This paper deals with positive solutions of a class of nonlinear eigenvalue problems. For a quasilinear elliptic problem (#) - div (ϕ(|∇u|)∇u) = λf(x,u) in Ω, u = 0 on ∂Ω, we assume asymptotic conditions on ϕ and f such as ϕ(t) ~ tp0-2, f(x,t) ~ tq0-1as t → +0 and ϕ(t) ~ tp1-2, f(x,t) ~ tq1-1as t → ∞. The combined effects of sub-nonlinearity (p0> q0) and super-nonlinearity (p1< q1) with the subcritical term f(x,u) imply the existence of at least two positive solutions of (#) for 0 < λ < Λ.

scholarly journals On an Elliptic Problem with Critical Nonlinearity in Expanding Annuli

1999 ◽  
Vol 163 (1) ◽  
pp. 29-62 ◽  
Author(s):  
Mohameden O. Ahmedou ◽  
Khalil O. El Mehdi
Keyword(s):  
Elliptic Problem ◽  
Critical Nonlinearity

scholarly journals On the number of positive solutions of a quasilinear elliptic problem

2006 ◽  
Vol 55 (6) ◽  
pp. 1835-1856 ◽  
Author(s):  
Marcelo Furtado ◽  
Giovany Malcher Figueiredo
Keyword(s):  
Positive Solutions ◽  
Elliptic Problem ◽  
Quasilinear Elliptic Problem ◽  
Quasilinear Elliptic

Multiple Positive Solutions to a Kirchhoff Type Problem Involving a Critical Nonlinearity in ℝ3

2017 ◽  
Vol 17 (4) ◽  
pp. 661-676 ◽  
Author(s):  
Xiao-Jing Zhong ◽  
Chun-Lei Tang
Keyword(s):  
Positive Solutions ◽  
Principal Eigenvalue ◽  
Nehari Manifold ◽  
Multiple Positive Solutions ◽  
Critical Nonlinearity ◽  
Kirchhoff Type ◽  
Concentration Compactness Principle ◽  
Kirchhoff Type Problem ◽  
The Nehari Manifold ◽  
Kirchhoff Type Problems

AbstractIn this paper, we investigate a class of Kirchhoff type problems in {\mathbb{R}^{3}} involving a critical nonlinearity, namely,-\biggl{(}1+b\int_{\mathbb{R}^{3}}\lvert\nabla u|^{2}\,dx\biggr{)}\triangle u=% \lambda f(x)u+|u|^{4}u,\quad u\in D^{1,2}(\mathbb{R}^{3}),where {b>0}, {\lambda>\lambda_{1}} and {\lambda_{1}} is the principal eigenvalue of {-\triangle u=\lambda f(x)u}, {u\in D^{1,2}(\mathbb{R}^{3})}. We prove that there exists {\delta>0} such that the above problem has at least two positive solutions for {\lambda_{1}<\lambda<\lambda_{1}+\delta}. Furthermore, we obtain the existence of ground state solutions. Our tools are the Nehari manifold and the concentration compactness principle. This paper can be regarded as an extension of Naimen’s work [21].

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