scholarly journals Remarks on the pure critical exponent problem

1998 ◽  
Vol 58 (2) ◽  
pp. 333-344 ◽  
Author(s):  
E.N. Dancer
Keyword(s):  
Boundary Conditions ◽  
Critical Exponent ◽  
Positive Solutions ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Analytic Methods ◽  
Existence Of Positive Solutions ◽  
Shaped Domain

In this paper, we use geometric and analytic methods to study the existence of positive solutions of the pure critical exponent problem with Dirichlet boundary conditions. In particular we prove that there is no solution for domains which are nearly star-shaped and we show that being conformal to a star-shaped domain does not characterise the domains for which the problem has no solution. We also answer some questions of Rodriguez et al.

scholarly journals The existence of positive solutions for an elliptic boundary value problem

2005 ◽  
Vol 2005 (13) ◽  
pp. 2005-2010
Author(s):  
G. A. Afrouzi
Keyword(s):  
Boundary Conditions ◽  
Positive Solution ◽  
Positive Solutions ◽  
Dirichlet Boundary ◽  
Elliptic Boundary ◽  
Dirichlet Boundary Conditions ◽  
Mountain Pass Lemma ◽  
First Eigenvalue ◽  
Elliptic Boundary Value ◽  
Existence Of Positive Solutions

By using the mountain pass lemma, we study the existence of positive solutions for the equation−Δu(x)=λ(u|u|+u)(x)forx∈Ωtogether with Dirichlet boundary conditions and show that for everyλ<λ1, whereλ1is the first eigenvalue of−Δu=λuinΩwith the Dirichlet boundary conditions, the equation has a positive solution while no positive solution exists forλ≥λ1.

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.

ESTIMATES ON THE GREEN FUNCTION AND EXISTENCE OF POSITIVE SOLUTIONS FOR SOME NONLINEAR POLYHARMONIC PROBLEMS OUTSIDE THE UNIT BALL

2008 ◽  
Vol 06 (02) ◽  
pp. 121-150 ◽  
Author(s):  
IMED BACHAR ◽  
HABIB MÂAGLI ◽  
NOUREDDINE ZEDDINI
Keyword(s):  
Green Function ◽  
Positive Solutions ◽  
Dirichlet Boundary ◽  
Closed Ball ◽  
Dirichlet Boundary Conditions ◽  
Kato Class ◽  
Nonlinear Elliptic ◽  
Existence Of Positive Solutions ◽  
The Green Function ◽  
Class Of Functions

Let [Formula: see text] be the Green function of (-Δ)m, m ≥ 1, on the complementary D of the unit closed ball in ℝn, n ≥ 2, with Dirichlet boundary conditions [Formula: see text], 0 ≤ j ≤ m - 1. We establish some estimates on [Formula: see text] including the 3G-Inequality given by (1.3). Next, we introduce a polyharmonic Kato class of functions [Formula: see text] and we exploit the properties of this class to study the existence of positive solutions of some polyharmonic nonlinear elliptic problems.

scholarly journals EXISTENCE OF POSITIVE SOLUTION FOR A QUASI-LINEAR PROBLEM WITH CRITICAL GROWTH IN N+

2009 ◽  
Vol 51 (2) ◽  
pp. 367-383 ◽  
Author(s):  
CLAUDIANOR O. ALVES ◽  
ANGELO R. F. DE HOLANDA ◽  
JOSÉ A. FERNANDES
Keyword(s):  
Boundary Conditions ◽  
Critical Exponent ◽  
Positive Solution ◽  
Half Space ◽  
Positive Solutions ◽  
Linear Problem ◽  
Neumann Boundary ◽  
Neumann Boundary Conditions ◽  
Existence Of Positive Solutions ◽  
Linear Problems

AbstractIn this paper we show existence of positive solutions for a class of quasi-linear problems with Neumann boundary conditions defined in a half-space and involving the critical exponent.

scholarly journals A positivity preserving property result for the biharmonic operator under partially hinged boundary conditions

2020 ◽  
Author(s):  
Elvise Berchio ◽  
Alessio Falocchi
Keyword(s):  
Green Function ◽  
Boundary Conditions ◽  
Positive Solutions ◽  
Elliptic Equations ◽  
Fourier Expansion ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Biharmonic Operator ◽  
Positivity Preserving ◽  
Striking Contrast

AbstractIt is well known that for higher order elliptic equations, the positivity preserving property (PPP) may fail. In striking contrast to what happens under Dirichlet boundary conditions, we prove that the PPP holds for the biharmonic operator on rectangular domains under partially hinged boundary conditions, i.e., nonnegative loads yield positive solutions. The result follows by fine estimates of the Fourier expansion of the corresponding Green function.

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