Positive solutions for a degenerate Kirchhoff problem

2021 ◽  
pp. 1-14
Author(s):  
David Arcoya ◽  
João R. Santos Júnior ◽  
Antonio Suárez
Keyword(s):  
Positive Solutions ◽  
Nonlocal Term ◽  
Reaction Term ◽  
Concentration Phenomena ◽  
Kirchhoff Problem

Abstract By assuming that the Kirchhoff term has $K$ degeneracy points and other appropriated conditions, we have proved the existence of at least $K$ positive solutions other than those obtained in Santos Júnior and Siciliano [Positive solutions for a Kirchhoff problem with vanishing nonlocal term, J. Differ. Equ. 265 (2018), 2034–2043], which also have ordered $H_{0}^{1}(\Omega )$ -norms. A concentration phenomena of these solutions is verified as a parameter related to the area of a region under the graph of the reaction term goes to zero.

Positive solutions for a Kirchhoff problem with vanishing nonlocal term

2018 ◽  
Vol 265 (5) ◽  
pp. 2034-2043 ◽  
Author(s):  
João R. Santos Júnior ◽  
Gaetano Siciliano
Keyword(s):  
Positive Solutions ◽  
Nonlocal Term ◽  
Kirchhoff Problem

scholarly journals Multi-peak positive solutions to a class of Kirchhoff equations

2018 ◽  
Vol 149 (04) ◽  
pp. 1097-1122 ◽  
Author(s):  
Peng Luo ◽  
Shuangjie Peng ◽  
Chunhua Wang ◽  
Chang-Lin Xiang
Keyword(s):  
Singular Perturbation ◽  
Reduction Method ◽  
Singularly Perturbed ◽  
Nonlocal Term ◽  
Kirchhoff Equations ◽  
Unperturbed Problem ◽  
New Feature ◽  
Image Position ◽  
Perturbation Problems ◽  
Kirchhoff Problem

In the present paper, we consider the nonlocal Kirchhoff problem$$-\left(\epsilon^2a+\epsilon b\int_{{\open R}^{3}}\vert \nabla u \vert^{2}\right)\Delta u+V(x)u=u^{p}, \quad u \gt 0 \quad {\rm in} {\open R}^{3},$$ where a, b>0, 1<p<5 are constants, ϵ>0 is a parameter. Under some mild assumptions on the function V, we obtain multi-peak solutions for ϵ sufficiently small by Lyapunov–Schmidt reduction method. Even though many results on single peak solutions to singularly perturbed Kirchhoff problems have been derived in the literature by various methods, there exist no results on multi-peak solutions before this paper, due to some difficulties caused by the nonlocal term $\left(\int_{{\open R}^{3}} \vert \nabla u \vert^{2}\right)\Delta u$. A remarkable new feature of this problem is that the corresponding unperturbed problem turns out to be a system of partial differential equations, but not a single Kirchhoff equation, which is quite different from most of the elliptic singular perturbation problems.

scholarly journals Existence of Multiple Solutions for a -Kirchhoff Problem with Nonlinear Boundary Conditions

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Zonghu Xiu ◽  
Caisheng Chen
Keyword(s):  
Variational Method ◽  
Boundary Conditions ◽  
Positive Solutions ◽  
Elliptic Problem ◽  
Multiple Solutions ◽  
Nonlinear Boundary ◽  
Nonlinear Boundary Conditions ◽  
Nehari Manifold ◽  
The Nehari Manifold ◽  
Kirchhoff Problem

The paper considers the existence of multiple solutions of the singular nonlocal elliptic problem , ,   = , on , where , . By the variational method on the Nehari manifold, we prove that the problem has at least two positive solutions when some conditions are satisfied.

scholarly journals Isolated singularities of positive solutions for Choquard equations in sublinear case

2018 ◽  
Vol 20 (04) ◽  
pp. 1750040 ◽  
Author(s):  
Huyuan Chen ◽  
Feng Zhou
Keyword(s):  
Positive Solutions ◽  
Riesz Potential ◽  
Singular Solutions ◽  
Nonlocal Term ◽  
Choquard Equation ◽  
Qualitative Properties ◽  
Isolated Singularities ◽  
Nonexistence And Existence

Our purpose of this paper is to study the isolated singularities of positive solutions to Choquard equation in the sublinear case [Formula: see text] [Formula: see text] where [Formula: see text] and [Formula: see text] is the Riesz potential, which appears as a nonlocal term in the equation. We investigate the nonexistence and existence of isolated singular solutions of Choquard equation under different range of the pair of exponent [Formula: see text]. Furthermore, we obtain qualitative properties for the minimal singular solutions of the equation.

A CLASS OF CRITICAL KIRCHHOFF PROBLEM ON THE HYPERBOLIC SPACE

2019 ◽  
Vol 62 (1) ◽  
pp. 109-122
Author(s):  
PAULO CESAR CARRIÃO ◽  
AUGUSTO CÉSAR DOS REIS COSTA ◽  
OLIMPIO HIROSHI MIYAGAKI
Keyword(s):  
Nontrivial Solution ◽  
Hyperbolic Space ◽  
Stereographic Projection ◽  
Mountain Pass Theorem ◽  
Singular Problem ◽  
Nonlocal Term ◽  
Open Ball ◽  
Kirchhoff Type Problems ◽  
The Hardy Inequality ◽  
Kirchhoff Problem

AbstractWe investigate questions on the existence of nontrivial solution for a class of the critical Kirchhoff-type problems in Hyperbolic space. By the use of the stereographic projection the problem becomes a singular problem on the boundary of the open ball $B_1(0)\subset \mathbb{R}^n$ Combining a version of the Hardy inequality, due to Brezis–Marcus, with the mountain pass theorem due to Ambrosetti–Rabinowitz are used to obtain the nontrivial solution. One of the difficulties is to find a range where the Palais Smale converges, because our equation involves a nonlocal term coming from the Kirchhoff term.

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