Asymptotics of the spectra of variational problems on the solutions of a homogeneous elliptic equation, under the presence of a constraint on part of the boundary

1986 ◽  
Vol 35 (1) ◽  
pp. 2212-2221
Author(s):  
T. A. Suslina
Keyword(s):  
Elliptic Equation ◽  
Variational Problems ◽  
Homogeneous Elliptic Equation

Multiple solutions for a non-homogeneous elliptic equation at the critical exponent

2004 ◽  
Vol 134 (1) ◽  
pp. 69-87 ◽  
Author(s):  
Mónica Clapp ◽  
Manuel Del Pino ◽  
Monica Musso
Keyword(s):  
Elliptic Equation ◽  
Boundary Conditions ◽  
Critical Exponent ◽  
Bounded Domain ◽  
Multiple Solutions ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Sign Changing Solutions ◽  
Homogeneous Elliptic Equation

We consider the equation−Δu = |u|4/(N−2)u + εf(x) under zero Dirichlet boundary conditions in a bounded domain Ω in RN exhibiting certain symmetries, with f ≥ 0, f ≠ 0. In particular, we find that the number of sign-changing solutions goes to infinity for radially symmetric f, as ε → 0 if Ω is a ball. The same is true for the number of negative solutions if Ω is an annulus and the support of f is compact in Ω.

Non-homogeneous elliptic equations in exterior domains

2006 ◽  
Vol 136 (1) ◽  
pp. 139-147 ◽  
Author(s):  
J. do Ó ◽  
S. Lorca ◽  
J. Sánchez ◽  
P. Ubilla
Keyword(s):  
Elliptic Equation ◽  
Positive Solutions ◽  
Elliptic Equations ◽  
Exterior Domains ◽  
Existence And Multiplicity ◽  
Negative Function ◽  
Multiplicity Of Positive Solutions ◽  
Image Position ◽  
Typical Model ◽  
Homogeneous Elliptic Equation

We study the existence and multiplicity of positive solutions of the non-homogeneous elliptic equation where N ≥ 3, the nonlinearity f is superlinear at both zero and infinity, q is a non-trivial, non-negative function, and a and b are non-negative parameters. A typical model is given by f(u) = up, with p ≥ 1.

Space of Solutions of Homogeneous Elliptic Equations

1971 ◽  
Vol 14 (1) ◽  
pp. 17-24
Author(s):  
E. Dubinsky ◽  
T. Husain
Keyword(s):  
Elliptic Equation ◽  
Infinite Series ◽  
Elliptic Equations ◽  
Specific Form ◽  
Common Solution ◽  
Linearly Independent ◽  
All Solutions ◽  
Homogeneous Elliptic Equation

This is the continuation of our paper [1] and includes the results promised there. As in [1], we consider a homogeneous elliptic equation in two variables. In [1] we showed that all solutions of such equations can be written in a specific form, viz. in the form of an infinite series in certain specific polynomials. Here we first establish that a common solution of any two positive powers of any two linearly independent, linear elliptic polynomials can be expressed as a polynomial (Lemma 2).

Regularity of the Very Weak Solutions for Double Obstacle Problems

2010 ◽  
Vol 143-144 ◽  
pp. 1396-1400
Author(s):  
Xu Juan Xu ◽  
Xiao Na Lu ◽  
Yu Xia Tong
Keyword(s):  
Elliptic Equation ◽  
Weak Solutions ◽  
Minkowski Inequality ◽  
Young Inequality ◽  
Obstacle Problems ◽  
Very Weak Solutions ◽  
Local Integrability ◽  
Regularity Results ◽  
Harmonic Equation ◽  
Homogeneous Elliptic Equation

The apply of harmonic eauation is know to all. Our interesting is to get the regularity os their solution, recently for obstacle problems. Many interesting results have been obtained for the solutions of harmonic equation ande their obstacle problems, however the double obstacle problems about the definition and regularity results for non-homogeneous elliptic equation. In this paper , the basic tool for the Young inequality,Hölder inequality, Minkowski inequality, Poincaré inequality and a basic inequality. The definition of very weak solutions for double obstacle problems associated with non-homogeneous elliptic equation is given, and the local integrability result is obtained by using the technique of Hodge decomposition.

scholarly journals Cosmic Analogues of Classic Variational Problems

Universe ◽  
2020 ◽  
Vol 6 (6) ◽  
pp. 71 ◽  
Author(s):  
Valerio Faraoni
Keyword(s):  
Relativistic Particle ◽  
Friedmann Equation ◽  
Variational Calculus ◽  
Variational Problems ◽  
Particle Mechanics ◽  
One Dimensional ◽  
Spatially Homogeneous

Several classic one-dimensional problems of variational calculus originating in non-relativistic particle mechanics have solutions that are analogues of spatially homogeneous and isotropic universes. They are ruled by an equation which is formally a Friedmann equation for a suitable cosmic fluid. These problems are revisited and their cosmic analogues are pointed out. Some correspond to the main solutions of cosmology, while others are analogous to exotic cosmologies with phantom fluids and finite future singularities.

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