Printer Friendly

On positive solutions of a Superlinear Dirichlet problem with a weight function.

Abstract

We use variational method to study the existence of a positive solution of the equation -[DELTA]u(x) = [lambda]g(x)u(x)(1 + u(x)) with Dirichlet boundary condition to the case where g is positive. We present a numerical approach for finding positive solutions and we will show that in which range of [lambda], this problem achieves a numerical solution and what is the behavior of the branch of this solution.

AMS Subject Classification: 35J60, 35B30, 35B40.

Keywords: Superlinear problems, positive weight function, Dirichlet boundary value problem.

1. Introduction

In studying some problems in selection migration model in population genetics [4], biology and other fields, we often meet the nonlinear problem of this type:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1.1)

where [DELTA] is the standard Laplacian operator, [OMEGA] is a smooth bounded region in [R.sup.N], g [member of] [L.sup.[infinity]] and g : [bar.[OMEGA]] : [right arrow] R is smooth function which is positive in [bar.[OMEGA]]. Let 0 < [[lambda].sub.1] < [[lambda].sub.2] [less than or equal to] [[lambda].sub.3] < ... [infinity] be the eigenvalues of -[DELTA] with zero Dirichlet boundary condition in [partial derivative][OMEGA]. Let H be the Sobolev space [H.sup.1,2.sub.0] ([OMEGA]) with inner product(see [1])

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

It is obvious that u is a positive solution of (1) if and only if u is a positive solution of the following problem

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], (1.2)

so we study (2) in place of (1).

We define J : H [right arrow] R by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

By regularity theory for elliptic boundary value problems u is a solution to (2) if and only if u is a critical point of the action functional J. We prove following result:

Theorem 1.1. Let [[lambda].sub.1](g) denote the positive principal eigenvalue of the problem

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1.3)

then there exists a positive solution to (2) whenever 0 < [lambda] < [[lambda].sub.1](g).

2. Preliminary Lemmas

Our assumptions imply that J [member of] [C.sup.2](H,R) and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

for all v [member of] H

Define [gamma] : H [right arrow] R by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

and compute

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

When J is bounded below on H, J has a minimizer on H which is a critical point of J. In many cases such as (2) J is not bounded below on H, but is bounded below on an appropriate subset of H and a minimizer on this set (if it exists) may give rise to the solution of the corresponding differential equation. A good candidate for an appropriate subset of H is

S = {u [member of] H - {0} : <[nabla]J(u), u> = 0>

which is known as the Nehari manifold (see [6]). It is clear that all critical points of J(u) must lie in S.

Lemma 2.1. u [member of] S is a positive critical point of J on H if and only if u is a positive critical point of [J.sub.|S].

Proof. We mention that forward implication is clear. If u is a critical point for J on S, then u is a solution of the optimization problem so [gamma](u) = 0 where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Hence, by the theory of Lagrange multipliers, there exists [mu] [member of] R such that J'(u) = [mu][gamma]'(u) thus <[nabla]J(u), u> = [mu]<[nabla][gamma](u), u>. Since u [member of] S, <[nabla] J(u), u> = 0 and so

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Hence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

It follows [mu] = 0.

Lemma 2.2. If u is a positive function in H - {0} then there exists unique [bar.t] = [bar.t](u) [member of] (0,1) such that [bar.t]u [member of] S. Moreover [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. If [gamma](u) < 0 then [bar.t] < 1, and if [gamma](u) > 0, then [bar.t] > 1 and J(u) > 0. Also 0 is a local minimum of J.

Proof. Let u [member of] H - {0} be a fixed function and define [empty set] : R [right arrow] R by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Differentiating [empty set] yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (a.1)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (a.2)

If t > 0 is a critical point of [empty set], then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Thus every critical point of [empty set] in (0, [infinity]) is a strict local maximum, and hence [empty set] has at most one critical point in (0, [infinity]). By (a.2) we see that [infinity]'(t) > 0 for t > 0 small. Since we also have [infinity](0) = 0 the above comments imply that J has a local minimum of 0 at 0 [member of] H. On the other hand [infinity]'(t) [right arrow] -[infinity] as t [right arrow] [infinity] therefore [empty set] has unique zero [bar.t] [member of] (0, [infinity]) and [bar.t] [member of] S thus [infinity]'(t) = [gamma](tu) t > 0 for t < [bar.t] and similarly [infinity]'(t) = [gamma](tu) t < 0 for t > [bar.t] in particular, this shows that given u [member of] H such that [gamma](u) < 0 ([gamma](u) > 0) there exist [alpha] < 1 ([alpha] > 1) such that [gamma]([alpha]u) = 0, i.e, [alpha]u [member of] S.

Lemma 2.3. [J.sub.|S] is coercive, i.e, J(u) [right arrow] [infinity] as [parallel]u[parallel] [right arrow] [infinity] in S. also 0 [member of] S and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Proof. Suppose 0 < [lambda] < [[lambda].sub.1](g). Then it is well known that the principal eigenvalue [mu]([lambda]) of the problem -[DELTA]u - [lambda]g(x)u = [mu]([lambda])u is positive and so

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (b.1)

It is easy to deduce that there exists [delta] > 0 such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Suppose otherwise; then [for all][[delta].sub.n] = 1/n there exists [u.sub.n] [member of] H - {0} such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

from (b.1) it follows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and so

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and this is a contradiction.

We know that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4)

so [J.sub.|S] is coercive. By continuity of [gamma], S is closed and 0 [??] S, there exists [[delta].sub.1] > 0 such that if u [member of] S then [[parallel]u[parallel] [lambda] [[delta].sub.1]. This and (4) imply [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

3. Existence of Positive Solution

Now we want to prove theorem (1). We know that there exists a minimizer on S which is a critical point of J(u) and so a nontrivial solution of (1). Let {[u.sub.n]} [subset] S be a minimizing sequence i.e. lim [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. Since J is coercive {[u.sub.n]} is bounded in H. Without loss of generality we can assume [u.sub.n] [right arrow] [bar.u] in H

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Since the functional [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] is weakly continuous in H(see [3]) and [L.sup.3] imbedded in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] is also weakly continuous. It follows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

so [bar.u] [not equal to] 0. We show that [u.sub.n] [right arrow] [bar.u] in H. Suppose otherwise, [parallel][bar.u][parrallel] < [lim.bar] [parrallel][u.sub.n][parrallel] and so

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

from Lemma(2) there exists 0 < [alpha] < 1 such that [alpha]u [member of] S. Now [alpha][u.sub.n] [??] [alpha][bar.u] and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and this is a contradiction hence we conclude that [u.sub.n] [right arrow] [bar.u] in H. It follows easily that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] and so [bar.u] [member of] S. Also [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] and so [bar.u] is a minimizer on S. Since J(|u|) = J(u) then without loss of generality we may assume that [bar.u] is positive.

We proved that for any [lambda] < [[lambda].sup.+.sub.1] (g) there exists a positive solution for the problem (1), where [[lambda].sup.+.sub.1] (g) denotes the positive principal eigenvalue of corresponding linear problem. Moreover in [5] it is proved that there does not exist a positive solution of the equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

i.e. ,u [equivalent to] 0 is the only solution of the problem in [[lambda].sup.+.sub.1] (g), and hence the branch of solution bifurcates to the left at ([[lambda].sup.+.sub.1] (g); 0) and has no other intersection point with the line [lambda] = [[lambda].sup.+.sub.1] (g), and since there are no positive solutions when [lambda] = 0, the branch of solutions bifurcated from [[lambda].sup.+.sub.1] (g) must lie strictly between [lambda] = 0 and [lambda] = [[lambda].sup.+.sub.1] (g) and so must approach1in such way that [[parallel]u[parallel] ! 1in this region. The analogous result holds in the case [[lambda].sup.-.sub.1] (g). In next section we present our numerical method.

4. Numerical Algorithm

Given a nonzero element u [member of] H and a piece-wise smooth region [OMEGA] [subset] [R.sup.N], we will use the notation u to represent an array of real numbers agreeing with u on a grid [OMEGA] [subset] [bar.[OMEGA]]. We will take the grid to be regular.

At each step of the iterative process, we are required to project nonzero elements of H onto the submanifold S. By Lemma (2), we see that the projection of [nabla]J(u) on to the ray {[lambda]u : [lambda] > 0} is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Let u be a nonzero element of H, represented by u over the grid [OMEGA]. Let [s.sub.1] = 0.5 or another perhaps optimally determined small positive constant. Define [u.sub.0] = u and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

We will use notation [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], provided that limit exists, to represent unique positive multiple of u lying on S.

We use following algorithm to find the solution:

1. Initialize [u.sub.0] with appropriate initial guess.

2. Project u on to S.(ray projection in Ascent direction element u onto S, that we explain it above).

The standard [L.sup.2] gradient is not the gradient we are considering,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

3. Begin loop with k = 0

3.1. Solve linear system -[DELTA]grad = g(x)f([u.sub.k]) for grad allows one to explicitly construct the array [nabla]J([u.sub.k]) [equivalent to] [u.sub.k] - grad, representing [nabla]J(u).

3.2. Take gradient descent [u.sub.k] = [u.sub.k] - [s.sub.2] [nabla] J([u.sub.k])

3.3. Reproject [u.sub.k] on to S.

3.4. Increment k and repeat step (3.1),(3,2),(3,3) until convergence criteria are met:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

In the following sections we look for the solutions that the existence of them is mentioned in introduction. In [2] analogous result is obtained by a Finite difference method.

Numerical result:

Let [OMEGA] = [0, 1] x [0; 1] and g(x) = 1 - xy that is positive in [OMEGA]. Using initial function [u.sub.0] = 2sin([pi]x)sin([pi]y), we numerically computed the positive solution and we present a subset of grid values for [lambda] = 1 and [lambda] = 100
 approximation of positive solution, [lambda] = 100

y \ x .1 .3

 .1 0.0644 x [10.sup.-3] 0.1655 x [10.sup.-3]
 .3 0.1704 x [10.sup.-3] 0.4356 x [10.sup.-3]
 .5 0.2117 x [10.sup.-3] 0.5397 x [10.sup.-3]
 .7 0.1721 x [10.sup.-3] 0.4374 x [10.sup.-3]

y \ x .5 .7

 .1 0.2030 x [10.sup.-3] 0.1645 x [10.sup.-3]
 .3 0.5332 x [10.sup.-3] 0.5067 x [10.sup.-3]
 .5 0.6594 x [10.sup.-3] 0.5331 x [10.sup.-3]
 .7 0.5333 x [10.sup.-3] 0.4302 x [10.sup.-3]

approximation of positive solution, [lambda] = 1

y \ x .1 .3 .5 .7

 .1 3.77 9.87 12.20 9.87
 .3 9.89 25.86 31.95 25.85
 .5 12.24 31.99 39.51 31.96
 .7 9.90 25.88 31.96 25.85


For very small [lambda] we had guessed that the branch of nonzero solutions tends to [infinity] and the solutions certify that (represented in following) and for 0 < [lambda] < [[lambda].sup.+.sub.1] (g) there is positive solution. (We execute our MATLAB program for eps= 2.22 x [10.sup.-16])
[lambda] eps 10 50

[[parallel]u[parallel] 5.93 x [10.sup.8] 2.66 0.0014
.sub.[infinity]]

[lambda] 100 400

[[parallel]u[parallel] 6 x [10.sup.-4] 6 x [10.sup.-5]
.sub.[infinity]]


So according to the obtained results in the above table we guess that [[lambda].sup.+.sub.1] (g) is greater than 50.

References

[1] Adams R.A., 1975, Sobolev spaces. Pure and Applied Mathematics, Vol. 65, Academic Press, New York.

[2] Afrouzi G.A. and Khademloo S., 2005, A numerical method to find a positive solution of semilinear elliptic Dirichlet problems, Applied Mathematics and Computation, Article in Press.

[3] Drabek P. and Pohozev S.I., 1997, Positive Solution for the P-Laplacian: application of the fibrering method. Proceedings of the Royal society of Edinburgh, 127A, pp. 703-726.

[4] Fleming W.H., 1975, A selection-migration model in population genetics. J. Math. Biol., 2, pp. 219-223.

[5] Ko B. and Brown K.J., 2000, The existence of positive solution for a class of indefinite weight semilinear boundary value problem. Noninear Anal., 39, pp. 587-597.

[6] Willem M., 1996, Minimax Theorems, Birkhauser, Boston, basel, Belin.

G.A. Afrouzi, Z. Naghizadeh and S. Mahdavi

Department of Mathematics, Faculty of Basic Sciences,

Mazandaran University, Babolsar, Iran

E-mail: afrouzi@umz.ac.ir
COPYRIGHT 2006 Research India Publications
No portion of this article can be reproduced without the express written permission from the copyright holder.
Copyright 2006 Gale, Cengage Learning. All rights reserved.

Article Details
Printer friendly Cite/link Email Feedback
Author:Afrouzi, G.A.; Naghizadeh, Z.; Mahdavi, S.
Publication:Global Journal of Pure and Applied Mathematics
Geographic Code:7IRAN
Date:Aug 1, 2006
Words:2403
Previous Article:Large deviation principle for white noise Gaussian measures.
Next Article:On the existence theorem for invariant differential operators on the Heisenberg group.
Topics:


Related Articles
Old problem, new solution.
Recent Advances in Operator-Related Function Theory: Proceedings.
Multiple Dirichlet series, automorphic forms, and analytic number theory; proceedings.
A computational procedure to find positive numerical solutions of a boundary value problem with nonlinear terms.
Existence of three solutions for the Dirichlet problem involving the p-Laplacian.
On the eigenstructure of a Sturm-Liouville problem with an impedance boundary condition.
The Dirichlet problem for a class of quasilinear elliptic equation (1).
Analytic number theory; a tribute to Gauss and Dirichlet; proceedings.
Discontinuous functional [phi]-Laplacian boundary value problems on time scales.
Advances in inequalities for special functions.

Terms of use | Copyright © 2017 Farlex, Inc. | Feedback | For webmasters