scholarly journals Rabinowitz Alternative for Non-cooperative Elliptic Systems on Geodesic Balls

2018 ◽  
Vol 18 (4) ◽  
pp. 845-862 ◽  
Author(s):  
Sławomir Rybicki ◽  
Naoki Shioji ◽  
Piotr Stefaniak
Keyword(s):  
Degree Theory ◽  
Elliptic Systems ◽  
Geodesic Ball ◽  
Global Symmetry ◽  
Nontrivial Solutions ◽  
Strongly Indefinite Functionals ◽  
Nonlinear Anal ◽  
Connected Sets ◽  
Equivariant Bifurcation Theory ◽  
Closed Connected Sets

AbstractThe purpose of this paper is to study properties of continua (closed connected sets) of nontrivial solutions of non-cooperative elliptic systems considered on geodesic balls in {S^{n}}. In particular, we show that if the geodesic ball is a hemisphere, then all these continua are unbounded. It is also shown that the phenomenon of global symmetry-breaking bifurcation of such solutions occurs. Since the problem is variational and {\operatorname{SO}(n)}-symmetric, we apply the techniques of equivariant bifurcation theory to prove the main results of this article. As the topological tool, we use the degree theory for {\operatorname{SO}(n)}-invariant strongly indefinite functionals defined in [A. Gołȩbiewska and S. A. Rybicki, Global bifurcations of critical orbits of G-invariant strongly indefinite functionals, Nonlinear Anal. 74 2011, 5, 1823–1834].

Existence of three nontrivial solutions for elliptic systems with critical exponents and weights

2008 ◽  
Vol 69 (10) ◽  
pp. 3537-3548 ◽  
Author(s):  
Xiyou Cheng ◽  
Shan Ma
Keyword(s):  
Critical Exponents ◽  
Elliptic Systems ◽  
Nontrivial Solutions

scholarly journals Nontrivial Solutions for a Boundary Value Problem of nth-Order Impulsive Differential Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Jiafa Xu ◽  
Zhongli Wei
Keyword(s):  
Differential Equation ◽  
Boundary Value Problem ◽  
Impulsive Differential Equation ◽  
Nonlinear Term ◽  
Boundary Value ◽  
Degree Theory ◽  
Impulsive Effects ◽  
Nontrivial Solutions

We study the existence of nontrivial solutions for nth-order boundary value problem with impulsive effects. We utilize Leray-Schauder degree theory to establish our main results. Furthermore, our nonlinear term f is allowed to grow superlinearly and sublinearly.

scholarly journals Closed connected sets which remain connected upon the removal of certain, connected subsets

1924 ◽  
Vol 5 ◽  
pp. 3-10 ◽  
Author(s):  
John Kline
Keyword(s):  
Connected Sets ◽  
Closed Connected Sets

Existence and nonexistence results for a class of Hamiltonian Choquard-type elliptic systems with lower critical growth on ℝ2

2021 ◽  
pp. 1-28
Author(s):  
B. B. V. Maia ◽  
O. H. Miyagaki
Keyword(s):  
Riesz Potential ◽  
Elliptic Systems ◽  
Ground State Solution ◽  
Nehari Manifold ◽  
Critical Growth ◽  
Nontrivial Solutions ◽  
Existence And Nonexistence ◽  
Exponential Critical Growth ◽  
Beta 2 ◽  
Image Position

In this paper, we investigate the existence and nonexistence of results for a class of Hamiltonian-Choquard-type elliptic systems. We show the nonexistence of classical nontrivial solutions for the problem \[ \begin{cases} -\Delta u + u= ( I_{\alpha} \ast |v|^{p} )v^{p-1} \text{ in } \mathbb{R}^{N},\\ -\Delta v + v= ( I_{\beta} \ast |u|^{q} )u^{q-1} \text{ in } \mathbb{R}^{N}, \\ u(x),v(x) \rightarrow 0 \text{ when } |x|\rightarrow \infty, \end{cases} \] when $(N+\alpha )/p + (N+\beta )/q \leq 2(N-2)$ (if $N\geq 3$ ) and $(N+\alpha )/p + (N+\beta )/q \geq 2N$ (if $N=2$ ), where $I_{\alpha }$ and $I_{\beta }$ denote the Riesz potential. Second, via variational methods and the generalized Nehari manifold, we show the existence of a nontrivial non-negative solution or a Nehari-type ground state solution for the problem \[ \begin{cases} -\Delta u + u= (I_{\alpha} \ast |v|^{\frac{\alpha}{2}+1})|v|^{\frac{\alpha}{2}-1}v + g(v) \hbox{ in } \mathbb{R}^{2},\\ - \Delta v + v= (I_{\beta} \ast |u|^{\frac{\beta}{2}+1})|u|^{\frac{\beta}{2}-1}u + f(u), \hbox{ in } \mathbb{R}^{2},\\ u,v \in H^{1}(\mathbb{R}^{2}), \end{cases} \] where $\alpha ,\,\beta \in (0,\,2)$ and $f,\,g$ have exponential critical growth in the Trudinger–Moser sense.

scholarly journals Strongly indefinite functionals and multiple solutions of elliptic systems

2003 ◽  
Vol 355 (7) ◽  
pp. 2973-2989 ◽  
Author(s):  
D. G. De Figueiredo ◽  
Y. H. Ding
Keyword(s):  
Multiple Solutions ◽  
Elliptic Systems ◽  
Strongly Indefinite Functionals

Nontrivial solutions for resonant cooperative elliptic systems via computations of the critical groups

2010 ◽  
Vol 73 (12) ◽  
pp. 3856-3872 ◽  
Author(s):  
Shiwang Ma
Keyword(s):  
Elliptic Systems ◽  
Critical Groups ◽  
Nontrivial Solutions

scholarly journals Nontrivial solutions for critical potential elliptic systems

2011 ◽  
Vol 250 (8) ◽  
pp. 3398-3417 ◽  
Author(s):  
Ezequiel R. Barbosa ◽  
Marcos Montenegro
Keyword(s):  
Elliptic Systems ◽  
Nontrivial Solutions ◽  
Critical Potential
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