scholarly journals A Complement to the Fredholm Theory of Elliptic Systems on Bounded Domains

2009 ◽  
Vol 2009 ◽  
pp. 1-9 ◽  
Author(s):  
Patrick J. Rabier
Keyword(s):  
Elliptic Systems ◽  
Fredholm Theory ◽  
Bounded Domains

scholarly journals Symmetry Results for Nonvariational Quasi-Linear Elliptic Systems

2010 ◽  
Vol 10 (4) ◽  
Author(s):  
Luigi Montoro ◽  
Berardino Sciunzi ◽  
Marco Squassina
Keyword(s):  
Half Space ◽  
Comparison Principle ◽  
Axial Symmetry ◽  
Elliptic Systems ◽  
Radial Symmetry ◽  
Bounded Domains ◽  
Two Dimensional ◽  
Regular Solutions ◽  
Variational Form ◽  
Weak Comparison Principle

AbstractBy virtue of a weak comparison principle in small domains we prove axial symmetry in convex and symmetric smooth bounded domains as well as radial symmetry in balls for regular solutions of a class of quasi-linear elliptic systems in non-variational form. Moreover, in the two dimensional case, we study the system when set in a half-space.

scholarly journals On the existence and multiplicity of solutions to fractional Lane-Emden elliptic systems involving measures

2020 ◽  
Vol 9 (1) ◽  
pp. 1480-1503
Author(s):  
Mousomi Bhakta ◽  
Phuoc-Tai Nguyen
Keyword(s):  
Positive Solution ◽  
Positive Solutions ◽  
Total Mass ◽  
Elliptic Systems ◽  
Linking Theorem ◽  
Bounded Domains ◽  
Multiplicity Of Solutions ◽  
Existence And Multiplicity ◽  
Smallness Condition

Abstract We study positive solutions to the fractional Lane-Emden system $$\begin{array}{} \displaystyle \left\{ \begin{aligned} (-{\it\Delta})^s u &= v^p+\mu \quad &\text{in } {\it\Omega} \\(-{\it\Delta})^s v &= u^q+\nu \quad &\text{in } {\it\Omega}\\u = v &= 0 \quad &&\!\!\!\!\!\!\!\!\!\!\!\!\text{in } {\it\Omega}^c={\mathbb R}^N \setminus {\it\Omega}, \end{aligned} \right. \end{array}$$(S) where Ω is a C2 bounded domains in ℝN, s ∈ (0, 1), N > 2s, p > 0, q > 0 and μ, ν are positive measures in Ω. We prove the existence of the minimal positive solution of (S) under a smallness condition on the total mass of μ and ν. Furthermore, if p, q ∈ $\begin{array}{} (1,\frac{N+s}{N-s}) \end{array}$ and 0 ≤ μ, ν ∈ Lr(Ω), for some r > $\begin{array}{} \frac{N}{2s}, \end{array}$ we show the existence of at least two positive solutions of (S). The novelty lies at the construction of the second solution, which is based on a highly nontrivial adaptation of Linking theorem. We also discuss the regularity of the solutions.

Multiplicity of Positive Solutions for Critical Singular Elliptic Systems With Concave-Convex Nonlinearities

2009 ◽  
Vol 9 (2) ◽  
Author(s):  
Tsing-San Hsu
Keyword(s):  
Variational Methods ◽  
Elliptic System ◽  
Positive Solutions ◽  
Elliptic Systems ◽  
Critical Growth ◽  
Bounded Domains ◽  
Multiplicity Results ◽  
Existence And Multiplicity ◽  
Multiplicity Of Positive Solutions

AbstractIn this paper, we consider a singular elliptic system with both concave-convex nonlinearities and critical growth terms in bounded domains. The existence and multiplicity results of positive solutions are obtained by variational methods.

Sign in / Sign up

Export Citation Format

Share Document