scholarly journals Multiple solutions for some strongly degenerate second order elliptic equations

2021 ◽  
pp. 1-12
Author(s):  
João R. Santos ◽  
Gaetano Siciliano
Keyword(s):  
Boundary Value Problem ◽  
Bounded Domain ◽  
Sobolev Spaces ◽  
Elliptic Equations ◽  
Multiple Solutions ◽  
Existence Of Solutions ◽  
Weighted Sobolev Spaces ◽  
Degenerate Operator ◽  
Second Order Elliptic Equations ◽  
Strongly Degenerate

We consider a boundary value problem in a bounded domain involving a degenerate operator of the form L ( u ) = − div ( a ( x ) ∇ u ) and a suitable nonlinearity f. The function a vanishes on smooth 1-codimensional submanifolds of Ω where it is not allowed to be C 2 . By using weighted Sobolev spaces we are still able to find existence of solutions which vanish, in the trace sense, on the set where a vanishes.

Existence and non-existence of solutions to a Hamiltonian strongly degenerate elliptic system

2017 ◽  
Vol 8 (1) ◽  
pp. 661-678 ◽  
Author(s):  
Cung The Anh ◽  
Bui Kim My
Keyword(s):  
Elliptic System ◽  
Bounded Domain ◽  
Existence Of Solutions ◽  
Smooth Boundary ◽  
Infinitely Many Solutions ◽  
Degenerate Elliptic System ◽  
Degenerate Elliptic ◽  
Degenerate Operator ◽  
Strongly Degenerate

Abstract We study the non-existence and existence of infinitely many solutions to the semilinear degenerate elliptic system \left\{\begin{aligned} &\displaystyle{-}\Delta_{\lambda}u=\lvert v\rvert^{p-1}% v&&\displaystyle\phantom{}\text{in }\Omega,\\ &\displaystyle{-}\Delta_{\lambda}v=\lvert u\rvert^{q-1}u&&\displaystyle% \phantom{}\text{in }\Omega,\\ &\displaystyle u=v=0&&\displaystyle\phantom{}\text{on }\partial\Omega,\end{% aligned}\right. in a bounded domain {\Omega\subset\mathbb{R}^{N}} with smooth boundary {\partial\Omega} . Here {p,q>1} , and {\Delta_{\lambda}} is the strongly degenerate operator of the form \Delta_{\lambda}u=\sum^{N}_{j=1}\frac{\partial}{\partial x_{j}}\Bigl{(}\lambda% _{j}^{2}(x)\frac{\partial u}{\partial x_{j}}\Bigr{)}, where {\lambda(x)=(\lambda_{1}(x),\dots,\lambda_{N}(x))} satisfies certain conditions.

Existence of Solutions in Weighted Sobolev Spaces for Some Degenerate Quasilinear Elliptic Equations

2008 ◽  
Vol 15 (4) ◽  
pp. 627-634
Author(s):  
Albo Carlos Cavalheiro
Keyword(s):  
Dirichlet Problem ◽  
Sobolev Spaces ◽  
Elliptic Equations ◽  
Existence Result ◽  
Existence Of Solutions ◽  
Weighted Sobolev Spaces ◽  
Quasilinear Elliptic Equations ◽  
Open Set ◽  
Quasilinear Elliptic

Abstract We prove an existence result for the Dirichlet problem associated to some degenerate quasilinear elliptic equations in a bounded open set Ω in in the setting of weighted Sobolev spaces .

scholarly journals -Solutions for Some Nonlinear Degenerate Elliptic Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-17
Author(s):  
Albo Carlos Cavalheiro
Keyword(s):  
Dirichlet Problem ◽  
Sobolev Spaces ◽  
Elliptic Equations ◽  
Existence Of Solutions ◽  
Weighted Sobolev Spaces ◽  
Quasilinear Elliptic Equations ◽  
Degenerate Elliptic Equations ◽  
Degenerate Elliptic ◽  
Quasilinear Elliptic ◽  
Nonlinear Degenerate

We are interested in the existence of solutions for Dirichlet problem associated to the degenerate quasilinear elliptic equations in the setting of the weighted Sobolev spaces .

scholarly journals SOLUTIONS OF THE DIVERGENCE AND ANALYSIS OF THE STOKES EQUATIONS IN PLANAR HÖLDER-α DOMAINS

2010 ◽  
Vol 20 (01) ◽  
pp. 95-120 ◽  
Author(s):  
RICARDO G. DURÁN ◽  
FERNANDO LÓPEZ GARCÍA
Keyword(s):  
Sobolev Spaces ◽  
Existence Of Solutions ◽  
Stokes Equations ◽  
Basic Result ◽  
Weighted Sobolev Spaces ◽  
Mean Value ◽  
Well Posedness ◽  
Existence Of A Solution ◽  
Simply Connected ◽  
Distance To The Boundary

If Ω ⊂ ℝn is a bounded domain, the existence of solutions [Formula: see text] of div u = f for f ∈ L2(Ω) with vanishing mean value, is a basic result in the analysis of the Stokes equations. In particular, it allows to show the existence of a solution [Formula: see text], where u is the velocity and p the pressure. It is known that the above-mentioned result holds when Ω is a Lipschitz domain and that it is not valid for arbitrary Hölder-α domains. In this paper we prove that if Ω is a planar simply connected Hölder-α domain, there exist solutions of div u = f in appropriate weighted Sobolev spaces, where the weights are powers of the distance to the boundary. Moreover, we show that the powers of the distance in the results obtained are optimal. For some particular domains with an external cusp, we apply our results to show the well-posedness of the Stokes equations in appropriate weighted Sobolev spaces obtaining as a consequence the existence of a solution [Formula: see text] for some r < 2 depending on the power of the cusp.

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