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Estimates for Solutions of Semilinear Elliptic Equation in Two Dimensions | OMICS International
ISSN: 2168-9679
Journal of Applied & Computational Mathematics
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Estimates for Solutions of Semilinear Elliptic Equation in Two Dimensions

Hichar S*, Guerfi A and Meftah MT

Department of Mathematics, Faculty of Mathematics and Science, Kasdi Merbah University, Algeria

*Corresponding Author:
Hichar S
Department of Mathematics, Faculty of Mathematics and Science
Kasdi Merbah University, P.O.Box 9004 Algeria
Tel: 331456848
E-mail: [email protected]

Received August 06, 2014; Accepted September 25, 2014; Published September 30, 2014

Citation: Hichar S, Guerfi A, Meftah MT (2014) Estimates for Solutions of Semilinear Elliptic Equation in Two Dimensions. J Appl Computat Math 3:186. doi: 10.4172/2168-9679.1000186

Copyright: © 2014 Hichar S, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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Abstract

In this paper we study a family of nonlinear Elliptic problems in two dimensions, we give some estimates for the solutions of this problem, and we decompose it on two problems, the first is the Poisson's equation and the second is the Liouville equation.

Keywords

Liouville equation; Nonlinear problem; Elliptic equation

Introduction

In this paper we study the problem

(1)

Where Ω is a bounded domain, λ is a positive parameter, and for some q>1.

Equation of Liouville-type is used in this study, it has the form:

For some q>1.

This related equation has been received much attention in the recent years. On the one hand, this is due to the wide range of application of this equation: it used in astrophysics [1] and combution theory [2], it is also related to the prescribed Gaussian curvature problem in Riemannian geometry [3], to the mean _led limit of vortices in Euler ows [4], to onsager's formulation in statistical mechanics [5], to the Keller-Siegel system of chemotaxis [6], to the Chern-Simon-Higgs gauge theory [7,8], and it has many other physical applications. On the other hand, Liouville equation is mathematically appealing since it has an interesting solution structure [9-16].

Preliminaries

Assume is a bounded domain and let u be a solution of

(2)

With

Theorem 2.1 For every we have

(3)

Proof: Let so that for some ball of radius R. Extend f to be zero outside Ω and set, for

so that

note that since it follows from the maximum principle that and thus

using Jensen in quality

With

We obtain

but, for we have

and the estimate (3) follows.

Corollary 2.2 Let u be a solution of (2) with Then for every constant k>0

Proof: Let We may split f as f=f1+f2 with Write u = u1 + u2 where ui are the solutions of

choosing, for example , δ = (4π −1) in Theorem (2.1) we find

And thus

The conclusion follows since

And

Statement of the Results

Let u satisfy the nonlinear equation (Liouville Equation)

(4)

Where Ωis a bounded domain in and v(x) a given function on Ω

Corollary 3.1 suppose u is a solution of (4) with and for some . Then

Proof: By corrollary (2:2), we know that

i.e.,

It follows that if Standard elliptic estimates imply that

Resolution of the equation

The Corollary (3:1) still holds for a solution u of

(5)

With is a bounded domain and for some q>1.

Let w be a solution of

(6)

So that

The function k=u-w satisfies:

(7)

The solution k is of the problem of liouville (7). Thus the solution of the problem (5) is u=k+w with w the solution of the problem (6) and k the solution of the problem (7).

Remark 3.2

Corollary (3:1) is not valid for p=1 for that we have this example.

Example 3.3 Let 0 < a < 1. The function with satisfies

(8)

With

Note that and nevertheless since . The same function u with a<0 provides an example where u satisfies (8) with and nevertheless since

References

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