scholarly journals Existence of positive solutions for some nonlinear elliptic problems in unbounded domains ofℝn

2006 ◽  
Vol 2006 ◽  
pp. 1-13
Author(s):  
Noureddine Zeddini
Keyword(s):  
Positive Solutions ◽  
Nonlinear Term ◽  
Nonlinear Elliptic Equations ◽  
Kato Class ◽  
Nonlinear Elliptic ◽  
Existence Of Positive Solutions ◽  
Nonempty Compact ◽  
The Green Function ◽  
Green Function Approach ◽  
Compact Boundary

This paper deals with a class of nonlinear elliptic equations in an unbounded domainDofℝn,n≥3, with a nonempty compact boundary, where the nonlinear term satisfies some appropriate conditions related to a certain Kato classK∞(D). Our purpose is to give some existence results and asymptotic behaviour for positive solutions by using the Green function approach and the Schauder fixed point theorem.

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.

Estimates on the Green Function and Existence of Positive Solutions of Nonlinear Singular Elliptic Equations

2003 ◽  
Vol 05 (03) ◽  
pp. 401-434 ◽  
Author(s):  
Imed Bachar ◽  
Habib Maâgli ◽  
Noureddine Zeddini
Keyword(s):  
Elliptic Equations ◽  
Continuous Solution ◽  
Regular Domain ◽  
Kato Class ◽  
New Class ◽  
Singular Elliptic Equations ◽  
Existence Of Positive Solutions ◽  
The Green Function ◽  
Borel Measurable ◽  
Compact Boundary

We establish a 3G-Theorem for the Green's function for an unbounded regular domain D in ℝn(n ≥ 3), with compact boundary. We exploit this result to introduce a new class of potentials K(D) that properly contains the classical Kato class [Formula: see text]. Next, we study the existence and the uniqueness of a positive continuous solution u in [Formula: see text] of the following nonlinear singular elliptic problem [Formula: see text] where φ is a nonnegative Borel measurable function in D × (0, ∞), that belongs to a convex cone which contains, in particular, all functions φ(x, t) = q(x)t-σ, σ ≥ 0 with q ∈ K(D). We give also some estimates on the solution u.

scholarly journals On the existence of positive solutions for a nonlinear elliptic class of equations in R2 and R3

2021 ◽  
Vol 20 ◽  
pp. 188-210
Author(s):  
Jose Quintero
Keyword(s):  
Positive Solutions ◽  
Water Waves ◽  
Nonlinear Term ◽  
Homogeneous Function ◽  
Concentration Compactness ◽  
Nonlinear Water Waves ◽  
Weakly Nonlinear ◽  
Nonlinear Elliptic ◽  
Existence Of Positive Solutions ◽  
Localized Waves

We study the existence of positive solutions for an elliptic equation in RN for N = 2, 3 which is related with the existence of standing (localized) waves and the existence of the ground state solutions for some physical model or systems in fluid mechanics to describe the evolution of weakly nonlinear water waves. We use a variational approach and the well-known principle of concentration-compactness due to P. Lions to obtain the existence of this type of solutions, even in the case that the nonlinear term g is a non-homogeneous function or an operator defined in H1(RN) with values in R.

Sign in / Sign up

Export Citation Format

Share Document