# 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.]

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).

[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