2004-Fez conference on Differential Equations and Mechanics Electronic Journal of Differential Equations, Conference 11, 2004, pp. 129–134. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp)

STABILITY OF THE PRINCIPAL EIGENVALUE OF A NONLINEAR ELLIPTIC SYSTEM

ABDELOUAHED EL KHALIL, MOHAMMED OUANAN, ABDELFATTAH TOUZANI

Abstract. This paper concerns some special properties of the principal eigenvalue of nonlinear elliptic systems with Dirichlet boundary conditions. We study the stability with respect to the exponents p and q; and the dependence on the domain variations.

1. Introduction

In this work, we survey two results concerning the nonlinear elliptic system

−Δ_{p}u = λ|u|^{α}|v|^{β}v in Ω
−Δ_{q}v = λ|u|^{α}|v|^{β}u in Ω
(u, v) ∈ W_{0}^{1,p}(Ω) × W_{0}^{1,q}(Ω),

(1.1)

where Ω ⊂ ℝ^{N}, ( N ≥ 3), is a bounded domain not necessary regular; α, β, p, q are reel numbers such that p > 1, q > 1, α ≥ 0, β ≥ 0 satisfying

\frac{α + 1}{p} + \frac{β + 1}{q} = 1.  (1.2)

The principal eigenvalue λ_{1}(p, q) of (1.1) is defined as

λ_{1} = inf {A(u, v) : (u, v) ∈ W_{0}^{1,p}(Ω) × W_{0}^{1,q}(Ω), B(u, v) = 1},  (1.3)

where

A(u, v) = \frac{α + 1}{p} ∫_{Ω} |∇u|^{p} dx + \frac{β + 1}{q} ∫_{Ω} |∇v|^{q} dx,

B(u, v) = ∫_{Ω} |u|^{α}|v|^{β}uv dx.

It is known (see [2]) that this eigenvalue is simple, isolated, and can be expressed as (1.3). It is also known that the associated principal eigenfunction (u, v) can be chosen strictly positive in Ω. This eigenfunction is a regular pair of functions where the infimum of the energy functional A is attained on the C^{1}-manifold {(u, v) ∈ W_{0}^{1,p}(Ω) × W_{0}^{1,q}(Ω) : B(u, v) = 1}.

2000 Mathematics Subject Classification. 35P20, 35J70, 35J20.

Key words and phrases. p-Laplacian operator; principal eigenvalue; stability;

segment property; dependence on the domain.

\begin{matrix}c \\ ◯ \end{matrix}2004 Texas State University - San Marcos.

Published October 15, 2004.

The operator −Δ_{p} arises in problems from pure mathematics, such as the theory of quasiregular and quasiconformal mapping, as well as in problems from a variety of applications, e.g. non-Newtonian fluids, reaction diffusion problems, flow through porous media, nonlinear elasticity, glaciology, petroleum extraction, astronomy, etc.

The main purpose of this work is to prove that the mapping (p, q) ↦ λ_{1}(p, q) is continuous (stable) on the set

I_{α,β} = {(p, q) ∈ (]1, +∞[)^{2} such that (1.2) is satisfied},

for any bounded domain Ω having the segment property. Recall that an open subset Ω of ℝ^{N} is said to have the segment property if, given any x ∈ ∂Ω, there exist an open set G_{x} in ℝ^{N} with x ∈ G_{x} and y_{x} of ℝ^{N} \ \{0\} such that, if z ∈ \overline{Ω} ∩ G_{x} and t ∈]0, 1[, then z + ty_{x} ∈ Ω.

This property allows us by a translation to push the support of a function u in Ω. This class of domains for which the boundary is sufficiently regular to guarantee that

W^{1,t}(Ω) ∩_{1<r<t} W_{0}^{1,r}(Ω) = W_{0}^{1,t}(Ω),

for any t > 1. See [3] for more details about this property.

The difficulty is that as the exponents (p, q) change, the appropriate spaces L^{p}(Ω) and L^{q}(Ω) also change. In the case of equation Δ_{p}u+λ|u|^{p−2}u = 0, the same result is studied by Lindqvist [5] but in a regular bounded domain. El Khalil et al. [3] studied the equation Δ_{p}u + λg|u|^{p−2}u = 0 with g ∈ L_{loc}^{∞}(Ω) ∩ L^{r}(Ω) and Ω having the segment property. Note that the stability was partially studied in [4]. For the dependence of λ_{1} upon variations of the domain, instead of the exponents p and q, we prove that λ_{1}^{(p,q)}(Ω_{j}) → λ_{1}^{(p,q)}(Ω) as j → +∞, where Ω = ∪_{j∈ℕ^{∗}}Ω_{j}. In the case of only one equation, see [1] for contracting domain, and see [6] for variations of the domain.

2. Stability

Theorem 2.1. Let Ω be a bounded domain in ℝ^{N} having the segment property. Then, the function (p, q) ↦ λ_{1}(p, q) is continuous from I_{α,β} to ℝ^{+}. where

I_{α,β} = {(p, q) ∈ (]1, +∞[)^{2} such that (1.2) is satisfied }.

Proof. Let (t_{n})_{n≥1}, t_{n} = (p_{n}, q_{n}) be a sequence in I_{α,β} converging at t = (p, q) ∈ I_{α,β}. We will prove that

lim_{n→+∞} λ_{1}(p_{n}, q_{n}) = λ_{1}(p, q).

Indeed, let (ϕ, ψ) ∈ C_{0}^{∞}(Ω) × C_{0}^{∞}(Ω) such that B(ϕ, ψ) > 0; hence,

λ_{1}(p_{n}, q_{n}) ≤ \frac{\frac{α+1}{p_{n}}∥∇ϕ∥_{p_{n}}^{p_{n}} + \frac{β+1}{q_{n}}∥∇ψ∥_{q_{n}}^{q_{n}}}{B(ϕ, ψ)} ,

since λ_{1}(p_{n}, q_{n}) being the infimum. Letting n tend to infinity, we deduce from Lebesgue’s theorem

lim sup_{n→+∞} λ_{1}(p_{n}, q_{n}) ≤ \frac{\frac{α+1}{p}∥∇ϕ∥_{p}^{p} + \frac{β+1}{q}∥∇ψ∥_{q}^{q}}{B(ϕ, ψ)} .  (2.1)

Then,

lim sup_{n→+∞} λ_{1}(p_{n}, q_{n}) ≤ λ_{1}(p, q).  (2.2)

On the other hand, let {(p_{n_{k}}, q_{n_{k}})}_{k≥1} be a subsequence of (t_{n})_{n} such that

lim_{k→+∞} λ_{1}(p_{n_{k}}, q_{n_{k}}) = lim inf_{n→+∞} λ_{1}(p_{n}, q_{n}).

Let us fix ε_{0} > 0 small enough, so that for all ε ∈ (0, ε_{0}), we have

1 < min(p − ε, q − ε),  (2.3)
max(p + ε, q + ε) < min((p − ε)^{∗}, (q − ε)^{∗}).  (2.4)

For each k ∈ ℕ^{∗}, let (u_{(p_{n_{k}} ,q_{n_{k}})}, v_{(p_{n_{k}} ,q_{n_{k}})}) ∈ W_{0}^{1,p_{n_{k}}} (Ω) × W_{0}^{1,q_{n_{k}}} (Ω) be a principal eigenfunction of (S_{p_{n_{k}} ,q_{n_{k}}}) related with λ_{1}(p_{n_{k}}, q_{n_{k}}). Then, by Hölder’s inequality, for ε ∈ (0, ε_{0}), the following inequalities hold

∥∇u_{(p_{n_{k}} ,q_{n_{k}})}∥_{p−ε} ≤ ∥∇u_{(p_{n_{k}} ,q_{n_{k}})}∥_{p_{n_{k}}} |Ω|^{\frac{pn_{k} −p+ε}{pn_{k} (p−ε)}} ,  (2.5)
∥∇v_{(p_{n_{k}} ,q_{n_{k}})}∥_{q−ε} ≤ ∥∇v_{(p_{n_{k}} ,q_{n_{k}})}∥_{q_{n_{k}}} |Ω|^{\frac{qn_{k} −q+ε}{qn_{k} (q−ε)}} .  (2.6)

Combining these two inequalities and using the variational characterization of λ_{1}, we have

∥∇u_{(p_{n_{k}} ,q_{n_{k}})}∥_{p−ε} ≤ \{\frac{p_{n_{k}}λ_{1}(p_{n_{k}}, q_{n_{k}})}{α + 1}\}^{\frac{1}{pn_{k}}} |Ω|^{\frac{pn_{k} −p+ε}{pn_{k} (p−ε)}}  (2.7)
∥∇v_{(p_{n_{k}} ,q_{n_{k}})}∥_{q−ε} ≤ \{\frac{q_{n_{k}}λ_{1}(p_{n_{k}}, q_{n_{k}})}{β + 1}\}^{\frac{1}{qn_{k}}} |Ω|^{\frac{qn_{k} −q+ε}{qn_{k} (q−ε)}} .  (2.8)

Therefore, via (2.3) and (2.4), for a subsequence

(u_{(p_{n_{k}} ,q_{n_{k}})}, v_{(p_{n_{k}} ,q_{n_{k}})}) ⇀ (u, v) weakly in W_{0}^{1,p−ε}(Ω) × W_{0}^{1,q−ε}(Ω),
(u_{(p_{n_{k}} ,q_{n_{k}})}, v_{(p_{n_{k}} ,q_{n_{k}})}) → (u, v) strongly in L^{p+ε}(Ω) × L^{q+ε}(Ω).

Passing to the limit in (2.7) and (2.8), respectively as k → ∞ and as ϵ → 0^{+}, we have

∥∇u∥_{p}^{p} ≤ \frac{p}{α + 1} lim_{k→+∞} λ_{1}(p_{n_{k}}, q_{n_{k}}) < +∞,

∥∇v∥_{q}^{q} ≤ \frac{q}{β + 1} lim_{k→+∞} λ_{1}(p_{n_{k}}, q_{n_{k}}) < +∞.

Then,

u ∈ W_{0}^{1,p−ε}(Ω) ∩ W^{1,p}(Ω) = W_{0}^{1,p}(Ω), v ∈ W_{0}^{1,q−ε}(Ω) ∩ W^{1,q}(Ω) = W_{0}^{1,q}(Ω),

because Ω satisfies the segment property.

On the other hand, from the variational characterization of λ_{1}(p_{n_{k}}, q_{n_{k}}), (2.5), (2.6), and using weak lower semi continuity of the norm; it follows that

\frac{1}{|Ω|^{\frac{ε}{p−ε}}} \frac{α + 1}{p}∥∇u∥_{p−ε}^{p} + \frac{1}{|Ω|^{\frac{ε}{q−ε}}} \frac{β + 1}{q}∥∇v∥_{q−ε}^{q} ≤ lim_{k→+∞} λ_{1}(p_{n_{k}}, q_{n_{k}}).

Letting ε → 0^{+} in (2.10), the Fatou lemma yields,

\frac{α + 1}{p}∥∇u∥_{p}^{p} + \frac{β + 1}{q}∥∇v∥_{q}^{q} ≤ lim_{k→+∞} λ_{1}(p_{n_{k}}, q_{n_{k}}).

Since B(u_{(p_{n_{k}} ,q_{n_{k}})}, v_{(p_{n_{k}} ,q_{n_{k}})}) = 1 via compactness of B, (u, v) is admissible in the variational characterization of λ_{1}(p, q); hence

λ_{1}(p, q) ≤ lim_{k→+∞} λ_{1}(p_{n_{k}}, q_{n_{k}}) = lim inf_{n→+∞} λ_{1}(p_{n}, q_{n}).

This and (2.2) complete the proof. □

Remark 2.2. Observe that the segment property is used only for to prove

λ_{1}(p, q) ≤ lim inf_{n→+∞} λ_{1}(p_{n}, q_{n}).

3. Dependence with variations domain

Theorem 3.1. Let Ω_{1} ⊂ Ω_{2} ⊂ Ω_{3} ⊂ … be an exhaustion of Ω such that

Ω = ⋃_{j∈ℕ^{∗}} Ω_{j}.

Then

lim_{j→+∞} λ_{1}^{(p,q)}(Ω_{j}) = λ_{1}^{(p,q)}(Ω),  (3.1)
lim_{j→+∞} ∫_{Ω} |∇ũ_{j} − ∇u|^{p} dx = lim_{j→+∞} ∫_{Ω} |∇ṽ_{j} − ∇v|^{q} dx = 0,  (3.2)

where (u, v) is a positive principal eigenfunction of (1.1) related to λ_{1}^{(p,q)}(Ω); (u_{j}, v_{j}) is the principal eigenfunction related to λ_{1}^{(p,q)}(Ω_{j}) and (ũ_{j}, ṽ_{j}) is the extended by 0 in Ω\Ω_{j} of (u_{j}, v_{j}).

Proof. To prove (3.1), we note that

λ_{1}^{(p,q)}(Ω_{1}) ≥ λ_{1}^{(p,q)}(Ω_{2}) ≥ ⋯ ≥ λ_{1}^{(p,q)}(Ω).

For each ε > 0, there is a pair of function (φ_{ε}, ψ_{ε}) ∈ C_{0}^{∞}(Ω) × C_{0}^{∞}(Ω) such that

∫_{Ω} |φ_{ε}|^{α}|ψ_{ε}|^{β}φ_{ε}ψ_{ε} dx > 0

λ_{1}^{(p,q)}(Ω) > \frac{\frac{α+1}{p}∫_{Ω}|∇φ_{ε}|^{p} dx + \frac{β+1}{q}∫_{Ω}|∇ψ_{ε}|^{q} dx}{_{∫Ω}|φ_{ε}|^{α}|ψ_{ε}|^{β}φ_{ε}ψ_{ε} dx} − ε,

since λ_{1}^{(p,q)}(Ω) is the infimum.

The support of φ_{ε} and ψ_{ε} being compact sets, are covered a finite number of the sets Ω_{j}; hence, there exist j_{0} ∈ ℕ^{∗} such that

supp φ_{ε} ⊂ Ω_{j} and supp ψ_{ε} ⊂ Ω_{j} ∀j ≥ j_{0}.

Then,

λ_{1}^{(p,q)}(Ω_{j}) ≤ \frac{\frac{α+1}{p}∫_{Ω_{j}}|∇φ_{ε}|^{p} dx + \frac{β+1}{q}∫_{Ω_{j}}|∇ψ_{ε}|^{q} dx}{_{∫Ω_{j}}|φ_{ε}|^{α}|ψ_{ε}|^{β}φ_{ε}ψ_{ε} dx}

= \frac{\frac{α+1}{p}∫_{Ω}|∇φ_{ε}|^{p} dx + \frac{β+1}{q}∫_{Ω}|∇ψ_{ε}|^{q} dx}{_{∫Ω}|φ_{ε}|^{α}|ψ_{ε}|^{β}φ_{ε}ψ_{ε} dx}

≤ λ_{1}^{(p,q)}(Ω) + ϵ

for all large j. It is clear that λ_{1}^{(p,q)}(Ω) ≥ lim_{j→+∞} λ_{1}^{(p,q)}(Ω_{j}). which proves the desired result.

For the strong convergence (3.2), we proceed as follows: First, we have (ũ_{j}, ṽ_{j}) ∈ W_{0}^{1,p}(Ω) × W_{0}^{1,q}(Ω); ũ_{j} ≥ 0, ṽ_{j} ≥ 0 a.e in Ω;

∫_{Ω} |ũ_{j}|^{α}|ṽ_{j}|^{β}ũ_{j}ṽ_{j} dx = 1,

because

∫_{Ω} |ũ_{j}|^{α}|ṽ_{j}|^{β}ũ_{j}ṽ_{j} dx = ∫_{Ω_{j}} |u_{j}|^{α}|v_{j}|^{β}u_{j}v_{j} dx = 1;

and ∇ũ_{j} = ˜∇u_{j} and ∇ṽ_{j} = ˜∇v_{j} a.e. in Ω. Hence, (ũ_{j}, ṽ_{j}) is admissible function in definition of λ_{1}^{(p,q)}(Ω). Which implies

λ_{1}^{(p,q)}(Ω) ≤ \frac{α + 1}{p} ∫_{Ω} |∇ũ_{j}|^{p} dx + \frac{β + 1}{q} ∫_{Ω} |∇ṽ_{j}|^{q} dx = λ_{1}^{(p,q)}(Ω_{j}).

Therefore, via (3.1) and for a subsequence,

(ũ_{j}, ṽ_{j}) ⇀ (φ, ψ) weakly in W_{0}^{1,p}(Ω) × W_{0}^{1,q}(Ω),
(ũ_{j}, ṽ_{j}) → (u, v) strongly in L^{p}(Ω) × L^{q}(Ω),
φ ≥ 0, ψ ≥ 0 a.e in Ω

Since B is compact, we have ∫_{Ω} φ^{α+1}ψ^{β+1}dx = 1. Then (φ, ψ) is admissible function in definition of λ_{1}^{(p,q)}(Ω). We deduce that

λ_{1}^{(p,q)}(Ω) ≤ \frac{α + 1}{p} ∫_{Ω} |∇ϕ|^{p} dx + \frac{β + 1}{q} ∫_{Ω} |∇ψ|^{q} dx

≤ lim inf_{j→∞} (\frac{α + 1}{p} ∫_{Ω} |∇ũ_{j}|^{p} dx + \frac{β + 1}{q} ∫_{Ω} |∇ṽ_{j}|^{q} dx)

= lim inf_{j→∞} λ_{1}^{(p,q)}(Ω_{j}).

Then

λ_{1}^{(p,q)}(Ω) = \frac{α + 1}{p} ∫_{Ω} |∇ϕ|^{p} dx + \frac{β + 1}{q} ∫_{Ω} |∇ψ|^{q} dx

which implies (φ, ψ) is positive eigenfunction related to λ_{1}^{(p,q)}(Ω) that it is simple (see [2]) and ∫_{Ω} |φ|^{α}|ψ|^{β}φψdx = 1, hence ϕ ≡ u and ψ ≡ v . Also, (u, v) is independent of the subsequence. Then {(ũ_{j}, ṽ_{j})}_{j} converge to (u, v) least in L^{p}(Ω)×L^{q}(Ω).

For the strong convergence (2.3), we use Clarckson inequality and distinguish three possible cases for (p, q) ∈ I_{α,β}. case 1: p ≥ 2 and q ≥ 2. For all j ≥ j_{0}, we have

\frac{α + 1}{p} ∫_{Ω} |\frac{∇ũ_{j} − ∇u}{2}|^{p} dx + \frac{β + 1}{q} ∫_{Ω} |\frac{∇ṽ_{j} − ∇v}{2} |^{q} dx

≤ \frac{α + 1}{p}[ − ∫_{Ω} |\frac{∇ũ_{j} + ∇u}{2}|^{p} dx + \frac{1}{2} ∫_{Ω} |∇ũ_{j}|^{p} dx + \frac{1}{2} ∫_{Ω} |∇u|^{p} dx]

+ \frac{β + 1}{q}[ − ∫_{Ω} |\frac{∇ṽ_{j} + ∇v}{2} |^{p} dx + \frac{1}{2} ∫_{Ω} |∇ṽ_{j}|^{q} dx + \frac{1}{2} ∫_{Ω} |∇v|^{q} dx].

Also, we have

λ_{1}^{(p,q)}(Ω) ∫_{Ω} (\frac{ũ_{j} + u}{2})^{α+1}(\frac{ṽ_{j} + v}{2} )^{β+1}dx

≤ \frac{α + 1}{p} ∫_{Ω} |\frac{∇ũ_{j} + ∇u}{2}|^{p} dx + \frac{β + 1}{q} ∫_{Ω} |\frac{∇ṽ_{j} + ∇v}{2} |^{q} dx.

Combining the last inequality and using Lebegue’s Theorem; we obtain

lim sup_{j→+∞} \{\frac{α + 1}{p} ∫_{Ω} |\frac{∇ũ_{j} − ∇u}{2}|^{p} dx + \frac{β + 1}{q} ∫_{Ω} |\frac{∇ṽ_{j} − ∇v}{2} |^{q} dx\} ≤ 0,

Case 2: 1 < p < 2 and 2 < q or 1 < q < 2 and 2 < p. Using the second Clarckson inequality, we prove that

lim_{j→+∞} ∫_{Ω} |∇ũ_{j} − ∇u|^{p} dx = 0.

Using first Clarckson inequality, we prove that

lim_{j→+∞} ∫_{Ω} |∇ṽ_{j} − ∇v|^{q} dx = 0.

Which completes the proof. □

References

[1] A. Anane, Etude des valeurs propres et de la résonance pour l’opérateur p-Laplacien, thèse de Doctorat, U.L.B (1987 -1988).

[2] A. El Khalil, M. Ouanan, and A. Touzani; Bifurcation of nonlinear elliptic system from the first eigenvalue, E. J. Qualitative Theory of Diff. Equ., No. 21. (2003), pp. 1-18.

[3] A. El Khalil, P. Lindqvist, and A. Touzani, On the stability of the eigenvalue of A_{p}u + λg(x)|u|^{p−2}u = 0, with varying p, to appear in Rendiconti di Matematica, Serie VII Volume 24, Roma (2004).

[4] A. El Khalil, S. El Manouni, and M. Ouanan, Simplicity and Stability of the first eigenvalue of a Nonlinear Elleptic System, submitted in International Journal of Mathematics and Mathematical Sciences.

[5] P. Lindqvist, On Nonlinear Rayleigh Quotients, Potential Analysis 2, pp. 199-218, (1993).

[6] P. Lindqvist, On a Nonlinear eigenvalue problem, Ber. Univer. Jyväskylä Math. Inst. 68 , pp. 33-54, (1995).

Abdelouahed El Khalil Department of Mathematics and Industrial Genie, Ecole Polytechnic School Montreal, Montreal, Canada

E-mail address: lkhlil@hotmail.com

Mohammed Ouanan Department of mathematics, Faculty of sciences Dhar-Mahraz, P.O. Box 1796 Atlas, Fez 30000, Morocco

E-mail address: m ouanan@hotmail.com

Abdelfattah Touzani Department of mathematics, Faculty of sciences Dhar-Mahraz, P.O. Box 1796 Atlas, Fez 30000, Morocco

E-mail address: atouzani@iam.net.ma