Multiplicity of positive solutions to boundary blow-up problem with variable exponent and sign-changing weights

2016 ◽  
Vol 40 (6) ◽  
pp. 2057-2070
Author(s):  
Juping Ji ◽  
Bo Li ◽  
Guanghui Zhang
Keyword(s):  
Positive Solutions ◽  
Variable Exponent ◽  
Blow Up ◽  
Multiplicity Of Positive Solutions

scholarly journals New multiplicity of positive solutions for some class of nonlocal problems

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Zhigao Shi ◽  
Xiaotao Qian
Keyword(s):  
Asymptotic Behavior ◽  
Variational Method ◽  
Positive Solutions ◽  
Blow Up ◽  
Nonlocal Problem ◽  
Nehari Manifold ◽  
Multiple Positive Solutions ◽  
Nonlocal Problems ◽  
Smooth Bounded Domain ◽  
Multiplicity Of Positive Solutions

AbstractIn this paper, we study the following nonlocal problem: $$ \textstyle\begin{cases} - (a-b \int _{\Omega } \vert \nabla u \vert ^{2}\,dx ) \Delta u= \lambda \vert u \vert ^{q-2}u, & x\in \Omega , \\ u=0, & x\in \partial \Omega , \end{cases} $$ { − ( a − b ∫ Ω | ∇ u | 2 d x ) Δ u = λ | u | q − 2 u , x ∈ Ω , u = 0 , x ∈ ∂ Ω , where Ω is a smooth bounded domain in $\mathbb{R}^{N}$ R N with $N\ge 3$ N ≥ 3 , $a,b>0$ a , b > 0 , $1< q<2$ 1 < q < 2 and $\lambda >0$ λ > 0 is a parameter. By virtue of the variational method and Nehari manifold, we prove the existence of multiple positive solutions for the nonlocal problem. As a co-product of our arguments, we also obtain the blow-up and the asymptotic behavior of these solutions as $b\searrow 0$ b ↘ 0 .

Positive solutions for a quasilinear system Via blow up

1993 ◽  
Vol 18 (12) ◽  
pp. 2071-2106
Author(s):  
Philippe Clément ◽  
Raúl Manásevich ◽  
Enzo Mitidieri
Keyword(s):  
Positive Solutions ◽  
Blow Up ◽  
Quasilinear System

scholarly journals Positive Solutions for a Class of Nonlinear Singular Fractional Differential Systems with Riemann–Stieltjes Coupled Integral Boundary Value Conditions

Symmetry ◽  
2021 ◽  
Vol 13 (1) ◽  
pp. 107
Author(s):  
Daliang Zhao ◽  
Juan Mao
Keyword(s):  
Positive Solutions ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Main Tool ◽  
Integral Boundary ◽  
Differential Systems ◽  
Fractional Differential Systems ◽  
Existence And Multiplicity ◽  
Fractional Differential ◽  
Multiplicity Of Positive Solutions

In this paper, sufficient conditions ensuring existence and multiplicity of positive solutions for a class of nonlinear singular fractional differential systems are derived with Riemann–Stieltjes coupled integral boundary value conditions in Banach Spaces. Nonlinear functions f(t,u,v) and g(t,u,v) in the considered systems are allowed to be singular at every variable. The boundary conditions here are coupled forms with Riemann–Stieltjes integrals. In order to overcome the difficulties arising from the singularity, a suitable cone is constructed through the properties of Green’s functions associated with the systems. The main tool used in the present paper is the fixed point theorem on cone. Lastly, an example is offered to show the effectiveness of our obtained new results.

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.

On the blow-up of solutions of a convective reaction diffusion equation

1993 ◽  
Vol 123 (3) ◽  
pp. 433-460 ◽  
Author(s):  
J. Aguirre ◽  
M. Escobedo
Keyword(s):  
Positive Solutions ◽  
Finite Time ◽  
Blow Up ◽  
Global Solutions ◽  
Reaction Diffusion ◽  
Scalar Function ◽  
Reaction Diffusion Equation ◽  
Semilinear Parabolic Equation ◽  
The Cauchy Problem ◽  
Image Position

SynopsisWe study the blow-up of positive solutions of the Cauchy problem for the semilinear parabolic equationwhere u is a scalar function of the spatial variable x ∈ ℝN and time t > 0, a ∈ ℝV, a ≠ 0, 1 < p and 1 ≦ q. We show that: (a) if p > 1 and 1 ≦ q ≦ p, there always exist solutions which blow up in finite time; (b) if 1 < q ≦ p ≦ min {1 + 2/N, 1 + 2q/(N + 1)} or if q = 1 and 1 < p ≦ l + 2/N, then all positive solutions blow up in finite time; (c) if q > 1 and p > min {1 + 2/N, 1 + 2q/N + 1)}, then global solutions exist; (d) if q = 1 and p > 1 + 2/N, then global solutions exist.

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