scholarly journals On one-homogeneous solutions to elliptic systems with spatial variable dependence in two dimensions

2010 ◽  
Vol 140 (3) ◽  
pp. 449-475 ◽  
Author(s):  
Jonathan J. Bevan
Keyword(s):  
Variational Method ◽  
Elliptic System ◽  
Homogeneous Solution ◽  
Elliptic Systems ◽  
Divergence Form ◽  
Two Dimensions ◽  
Spatial Variable ◽  
Original Result ◽  
Homogeneous Solutions ◽  
Special Case

We extend a result from Phillips by showing that one-homogeneous solutions of certain elliptic systems in divergence form either do not exist or must be affine. The result is novel in two ways. Firstly, the system is allowed to depend (in a sufficiently smooth way) on the spatial variable x. Secondly, Phillips's original result is shown to apply to W one-homogeneous solutions, from which his treatment of Lipschitz solutions follows as a special case. A singular one-homogeneous solution to an elliptic system violating the hypotheses of the main theorem is constructed using a variational method.

scholarly journals Quasilinear elliptic systems in divergence form associated to general nonlinearities

2018 ◽  
Vol 7 (4) ◽  
pp. 425-447 ◽  
Author(s):  
Lorenzo D’Ambrosio ◽  
Enzo Mitidieri
Keyword(s):  
Positive Solutions ◽  
A Priori Estimates ◽  
A Priori ◽  
Elliptic Systems ◽  
Divergence Form ◽  
Systems Of Equations ◽  
Quasilinear Elliptic Systems ◽  
Open Set ◽  
Priori Estimates ◽  
Quasilinear Elliptic

AbstractThe paper is concerned with a priori estimates of positive solutions of quasilinear elliptic systems of equations or inequalities in an open set of {\Omega\subset\mathbb{R}^{N}} associated to general continuous nonlinearities satisfying a local assumption near zero. As a consequence, in the case {\Omega=\mathbb{R}^{N}}, we obtain nonexistence theorems of positive solutions. No hypotheses on the solutions at infinity are assumed.

Quasilinear degenerated elliptic system in divergence form with mild monotonicity in weighted sobolev spaces

2019 ◽  
Vol 30 (7-8) ◽  
pp. 1153-1168
Author(s):  
El Houcine Rami ◽  
Elhoussine Azroul ◽  
Meryem El Lekhlifi
Keyword(s):  
Elliptic System ◽  
Sobolev Spaces ◽  
Weighted Sobolev Spaces ◽  
Divergence Form

Ground state solutions of Hamiltonian elliptic systems in dimension two

2019 ◽  
Vol 150 (4) ◽  
pp. 1737-1768 ◽  
Author(s):  
Djairo G. de Figueiredo ◽  
João Marcos do Ó ◽  
Jianjun Zhang
Keyword(s):  
Elliptic System ◽  
Ground State ◽  
Elliptic Systems ◽  
Nehari Manifold ◽  
Existence Results ◽  
Ground State Solutions ◽  
Variational Framework ◽  
Hamiltonian Elliptic System ◽  
Schrödinger Systems ◽  
Image Position

AbstractThe aim of this paper is to study Hamiltonian elliptic system of the form 0.1$$\left\{ {\matrix{ {-\Delta u = g(v)} & {{\rm in}\;\Omega,} \cr {-\Delta v = f(u)} & {{\rm in}\;\Omega,} \cr {u = 0,v = 0} & {{\rm on}\;\partial \Omega,} \cr } } \right.$$ where Ω ⊂ ℝ2 is a bounded domain. In the second place, we present existence results for the following stationary Schrödinger systems defined in the whole plane 0.2$$\left\{ {\matrix{ {-\Delta u + u = g(v)\;\;\;{\rm in}\;{\open R}^2,} \cr {-\Delta v + v = f(u)\;\;\;{\rm in}\;{\open R}^2.} \cr } } \right.$$We assume that the nonlinearities f, g have critical growth in the sense of Trudinger–Moser. By using a suitable variational framework based on the generalized Nehari manifold method, we obtain the existence of ground state solutions of both systems (0.1) and (0.2).

scholarly journals A Variational Method for Multivalued Boundary Value Problems

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Droh Arsène Béhi ◽  
Assohoun Adjé
Keyword(s):  
Variational Method ◽  
Boundary Value Problems ◽  
Critical Point ◽  
Critical Point Theory ◽  
Point Theory ◽  
Existence Of Solution ◽  
Laplacian Operator ◽  
Differential Systems ◽  
Lagrange Functional ◽  
Special Case

In this paper, we investigate the existence of solution for differential systems involving a ϕ−Laplacian operator which incorporates as a special case the well-known p−Laplacian operator. In this purpose, we use a variational method which relies on Szulkin’s critical point theory. We obtain the existence of solution when the corresponding Euler–Lagrange functional is coercive.

Stokes-multiplier expansion in an inhomogeneous differential equation with a small parameter

2005 ◽  
Vol 461 (2059) ◽  
pp. 2243-2256 ◽  
Author(s):  
E.I Ólafsdóttir ◽  
A.B Olde Daalhuis ◽  
J Vanneste
Keyword(s):  
Small Parameter ◽  
Gravity Waves ◽  
Homogeneous Solution ◽  
Boussinesq Equations ◽  
Inhomogeneous Equation ◽  
Stokes Phenomenon ◽  
Homogeneous Solutions ◽  
Inhomogeneous Solution ◽  
Type Construction ◽  
Special Solutions

Accurate approximations to the solutions of a second-order inhomogeneous equation with a small parameter ϵ are derived using exponential asymptotics. The subdominant homogeneous solutions that are switched on by an inhomogeneous solution through a Stokes phenomenon are computed. The computation relies on a resurgence relation, and it provides the ϵ -dependent Stokes multiplier in the form of a power series. The ϵ -dependence of the Stokes multiplier is related to constants of integration that can be chosen arbitrarily in the WKB-type construction of the homogeneous solution. The equation under study governs the evolution of special solutions of the Boussinesq equations for rapidly rotating, strongly stratified fluids. In this context, the switching on of subdominant homogeneous solutions is interpreted as the generation of exponentially small inertia–gravity waves.

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