2002-Fez conference on Partial Differential Equations, Electronic Journal of Differential Equations, Conference 09, 2002, pp 139–147. http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp)

Nonlinear elliptic systems with exponential nonlinearities ^{∗}

Said El Manouni & Abdelfattah Touzani

Abstract

In this paper we investigate the existence of solutions for

− div(a(|∇u|^{N})|∇u|^{N−2}u) = f(x, u, v) in Ω
− div(a(|∇v|^{N})|∇v|^{N−2}v) = g(x, u, v) in Ω
u(x) = v(x) = 0 on ∂Ω.

Where Ω is a bounded domain in ℝ^{N}, N ≥ 2, f and g are nonlinearities having an exponential growth on Ω and a is a continuous function satisfying some conditions which ensure the existence of solutions.

1 Introduction

Let Ω ⊂ ℝ^{N}, N ≥ 2 be a bounded domain with smooth boundary ∂Ω. In this paper we shall be concerned with existence of solutions for the problem

− div(a(|∇u|^{N})|∇u|^{N−2}u) = f(x, u, v) in Ω
− div(a(|∇v|^{N})|∇v|^{N−2}v) = g(x, u, v) in Ω
u(x) = v(x) = 0 on ∂Ω.

(1.1)

Where the nonlinearities f, g : Ω × ℝ^{2} → ℝ are continuous functions having an exponential growth on Ω: i.e.,

(H1) For all δ > 0

lim_{|(u,v)|→∞} \frac{|f(x, u, v)| + |g(x, u, v)|}{e^{δ(|u|^{N} +|v|^{N})^{1/(N−1)}}} = 0 Uniformly in Ω.

Let us mention that there are many results in the scalar case for problem involving exponential growth in bounded domains; see for example [4], [6]. The objective of this paper is to extend these results to a more general class of elliptic systems using variational method. Here we will make use the approach stated by Rabinowitz [8].

∗Mathematics Subject Classifications: 35J70, 35B45, 35B65.

Key words: Nonlinear elliptic system, exponential growth, Palais-Smale condition.

\begin{matrix}c \\ ◯ \end{matrix}2002 Southwest Texas State University.

Published December 28, 2002.

Note that for nonlinearities having polynomial growth, several results of such problem have been established. We can cite, among others, the articles: [9] and [10]. In order to prove the compactness condition of the functional associated to a problem (1.1) we assume the following hypothesis

(H2) u\frac{∂F}{∂u} ≥ \frac{μ}{2}F and v \frac{∂F}{∂v} ≥ \frac{μ}{2}F, where F = F (x, u, v) and such that \frac{∂F}{∂u} = f(x, u, v), \frac{∂F}{∂v} = g(x, u, v) with F (x, u, v) > 0 for u > 0 and v > 0, F (x, u, v) = 0 for u ≤ 0 or v ≤ 0 with μ > N and U = (u, v) ∈ ℝ^{2}.

We shall find weak-solution of (1.1) in the space W = W_{0}^{1,N}(Ω) × W_{0}^{1,N}(Ω) endowed with the norm

∥U∥_{W}^{N} = ∫_{Ω} |∇U|^{N} dx = ∫_{Ω}(|∇u|^{N} + |∇v|^{N}) dx

where U = (u, v) ∈ W. Motivated by the following result due to Trudinger and Moser (cf. [7].[11]), we remark that the space W is embeded in the class of Orlicz-Lebesgue space

L_{ϕ} = {U : Ω → ℝ^{2}, measurable : ∫_{Ω} ϕ(U) < ∞},

where ϕ(s, t) = exp (s^{\frac{N}{N−1}} + t^{\frac{N}{N−1}} ). Moreover,

sup_{∥(u,v)∥_{W} ≤1} ∫_{Ω} exp (γ(|u|^{\frac{N}{N−1}} + |v|^{\frac{N}{N−1}} ) dx ≤ C if γ ≤ ω_{N−1},

where C is a real number and ω_{N−1} is the dimensional surface of the unit sphere.

On this paper, we make the following assumptions on the function a.

(a1) a : ℝ^{+} → ℝ is continuous

(a2) There exist positive constants p ∈]1, N], b_{1}, b_{2}, c_{1}, c_{2} such that

c_{1} + b_{1}u^{N−p} ≤ u^{N−p}a(u^{N}) ≤ c_{2} + b_{2}u^{N−p} ∀u ∈ ℝ^{+};

(a3) The function k : ℝ → ℝ, k(u) = a(|u|^{N})|u|^{N−2}u is strictly increasing and k(u) → 0 as u → 0^{+}.

Remark Note that operator considered here has been studied by Hirano [5] and by Ubilla [11] with nonlinearities having polynomial growth.

We shall denote by λ_{1} the smallest eigenvalue [9] for the problem

−Δ_{N}u = λ|u|^{α−1}u|v|^{β+1} in Ω ⊂ ℝ^{N}
−Δ_{N}v = λ|u|^{α+1}|v|^{β−1}v in Ω ⊂ ℝ^{N}
u(x) = v(x) = 0 on ∂Ω;

i.e.,

λ_{1} = inf \{\frac{α + 1}{N} ∫_{Ω} |∇u|^{N} dx + \frac{β + 1}{N} ∫_{Ω} |∇v|^{N} dx :
(u, v) ∈ W, ∫_{Ω} |u|^{α+1}|v|^{β+1} dx = 1}

where α + β = N − 2 and α, β > −1.

Definition We say that a pair (u, v) ∈ W is a weak solution of (1.1) if for all (φ, ψ) ∈ W,

\begin{matrix}∫_{Ω} a(|∇u|^{N})|∇u|^{N−2}∇u∇φ dx = ∫_{Ω} f(x, u, v)φ dx \\ ∫_{Ω} a(|∇v|^{N})|∇v|^{N−2}∇v∇ψ dx = ∫_{Ω} g(x, u, v)ψ dx \end{matrix}  (1.2)

Now state our main results.

Theorem 1.1 Suppose that f and g are continuous functions satisfying (H1), (H2) and that a satisfies (a1), (a2) and (a3), with Nb_{2} < μb_{1}. Furthermore, assume that

lim_{|U|→0} sup \frac{pF (x, U)}{|u|^{α+1}|v|^{β+1}} < (c_{1} + b_{1}δ_{p}(N))λ_{1}  (1.3)

uniformly on x ∈ Ω, where δ_{p}(N) = 1 if N = p and δ_{p}(N) = 0 if N ≠ p. Then problem (1.1) has a nontrivial weak solution in W.

Remarks

1) Here we note that in case that (a2) holds for p = N, the condition (a2) can be rewritten as follows: (a2′) There exist c_{1}, c_{2} such that

c_{1} ≤ a(u^{N}) ≤ c_{2} for all u ∈ ℝ^{+}.

If a(t) = 1, (a2′) holds with c_{1} = c_{2} = 1 and therefore, we obtain the result given in [3].

2) If a(u) = 1 + u^{\frac{p−N}{N}} , conditions (a2) and (a3) hold, then the problem (1.1) can be formulated as follows

−Δ_{N}u − Δ_{p}u = f(x, u, v)
−Δ_{N}v − Δ_{p}v = g(x, u, v);

where Δ_{p} ≡ div(|∇u|^{p−2}∇u) is p-Laplacian operator.

2 Preliminaries

The maximal growth of f(x, u, v) and g(x, u, v) will allow us to treat variationally system (1.1) in the product Sobolev space W. This exponential growth is relatively motivated by Trudinger-Moser inequality ([4], [11]).

Note that if the functions f and g are continuous and have an exponential growth, then there exist positive constants C and γ such that

|f(x, u, v)| + |g(x, u, v)| ≤ C exp (γ(|u|^{\frac{N}{N−1}} + |v|^{\frac{N}{N−1}} )), ∀(x, u, v) ∈ Ω × ℝ^{2}.

(2.1)

Consequently the functional Ψ : W → ℝ defined as

Ψ(u, v) = ∫_{Ω} F (x, u, v) dx

is well defined, belongs to C^{1}(W, ℝ), and has

Ψ′(u, v)(φ, ψ) = ∫_{Ω} f(x, u, v)φ + g(x, u, v)ψ dx.

To prove this statements, we deduce from (2.1) that there exists C_{1} > 0 such that

|F (x, u, v)| ≤ C_{1} exp(γ(|u|^{\frac{N}{N−1}} + |v|^{\frac{N}{N−1}} )), ∀(x, u, v) ∈ Ω × ℝ^{2}.

Thus, since

exp (γ(|u|^{\frac{N}{N−1}} + |v|^{\frac{N}{N−1}} )) ∈ L^{1}(Ω), ∀(u, v) ∈ W,

we have the result.

It follows from the assumptions on the function a that for all t ∈ ℝ,

\frac{1}{N} A(|t|^{N}) ≥ \frac{b_{1}}{N} |t|^{N} + \frac{c_{1}}{p} |t|^{p}

\frac{1}{N} A(|t|^{N}) ≤ \frac{b_{2}}{N} |t|^{N} + \frac{c_{2}}{p} |t|^{p},

where A(t) = ∫_{0}^{t} a(s) ds. Furthermore the function g(t) = A(|t|^{N}) is strictly convex. Consequently, the functional Φ : W → ℝ defined as

Φ(u, v) = \frac{1}{N} ∫_{Ω} A(|∇u|^{N}) + A(|∇v|^{N}) dx

is well defined, weakly lower semicontinuous, Frechet differentiable and belongs to C^{1}(W, ℝ).

Therefore, if the function a satisfies conditions (a1), (a2) and (a3) and the nonlinearities f and g are continuous and satisfy (2.1), we conclude that the functional J : W → ℝ, given by

J(u, v) = \frac{1}{N} ∫_{Ω} A(|∇u|^{N}) + A(|∇v|^{N}) dx − ∫_{Ω} F (x, u, v) dx

is well defined and belongs to C^{1}(W, ℝ). Also for all (u, v) ∈ W,

J′(u, v)(φ, ψ) = ∫_{Ω} a(|∇u|^{N})|∇u|^{N−2}∇u∇φ + a(|∇v|^{N})|∇v|^{N−2}∇u∇ψ dx
− ∫_{Ω} f(x, u, v)φ + g(x, u, v)ψ dx .

Consequently, we are interested in using Critical Point theory to obtain weak solutions of (1.1).

Lemma 2.1 Assume that f and g are continuous and have an exponential growth. Let (u_{n}, v_{n}) be a sequence in W such that (u_{n}, v_{n}) converge weakly on (u, v) ∈ X, then

∫_{Ω} f(x, u_{n}, v_{n})(u_{n} − u) dx → 0,
∫_{Ω} g(x, u_{n}, v_{n})(v_{n} − v) dx → 0,

as n → ∞.

Proof. Let (u_{n}, v_{n}) be a sequence converging weakly to some (u, v) in W. Thus, there exist a subsequence, denoted again by (u_{n}, v_{n}) such that

u_{n} → u in L^{p}(Ω),
v_{n} → v in L^{q}(Ω),

as n → ∞ and for all p, q > 1. On the other hand, we have

∫_{Ω} |f(x, u_{n}, v_{n})|^{p} dx ≤ C ∫_{Ω} exp(pγ(|u_{n}|^{\frac{N}{N−1}} + |v_{n}|^{\frac{N}{N−1}} )) dx
≤ C(∫_{Ω} exp(spγ|u_{n}|^{\frac{N}{N−1}} ))^{\frac{1}{s}} (∫_{Ω} exp(s′pγ|v_{n}|^{\frac{N}{N−1}} ))^{\frac{1}{s′}}

≤ C( ∫_{Ω} exp(spγ∥u_{n}∥W_{0}^{1,N} (Ω)^{\frac{N}{N−1}}(\frac{|u_{n}|^{\frac{N}{N−1}}}{∥u_{n}∥_{W_{0}^{1,N} (Ω)}} )))^{1/s}

×( ∫_{Ω} exp(s′pγ∥v_{n}∥W_{0}^{1,N} (Ω)^{\frac{N}{N−1}}(\frac{|v_{n}|^{\frac{N}{N−1}}}{∥v_{n}∥_{W_{0}^{1,N} (Ω)}} )))^{1/s′}.

Since (u_{n}, v_{n}) is a bounded sequence, we may choose γ sufficiently small such that

spγ∥u_{n}∥_{W_{0}^{1,N} (Ω)}^{\frac{N}{N−1}} < α_{N} and s′pγ∥v_{n}∥_{W_{0}^{1,N} (Ω)}^{\frac{N}{N−1}} < α_{N}.

Then

∫_{Ω} |f(x, u_{n}, v_{n})|^{p} dx ≤ C_{1}

for n large and some constant C_{1} > 0. By the same argument, we have also

∫_{Ω} |g(x, u_{n}, v_{n})|^{q} dx ≤ C_{2}

for n large and some constant C_{2} > 0. Using Hölder inequality, we obtain

∫_{Ω} f(x, u_{n}, v_{n})(u_{n} − u) dx ≤ [ ∫_{Ω} |f(x, u_{n}, v_{n})|^{p′}]^{1/p′} [|u_{n} − u|^{p}]^{1/p}
≤ C [|u_{n} − u|^{p}]^{1/p}

and

∫_{Ω} g(x, u_{n}, v_{n})(v_{n} − v) dx ≤ [ ∫_{Ω} |g(x, u_{n}, v_{n})|^{q′}]^{1/q′} [|v_{n} − v|^{q}]^{1/q}
≤ C′ [|v_{n} − v|^{q}]^{1/q} .

Thus the proof is completed since u_{n} → u in L^{p}(Ω) and v_{n} → v in L^{q}(Ω). □

Lemma 2.2 Assume that f and g are continuous satisfying (H1). Then the functional J satisfies Palais-Smale condition (PS) provided that every sequence (u_{n}, v_{n}) in W is bounded.

Proof. Note that

\begin{matrix}J′(u_{n}, v_{n})(φ, ψ) = & Φ′(u_{n}, v_{n})(φ, ψ) − ∫_{Ω} f(x, u_{n}, v_{n})φ + g(x, u_{n}, v_{n})ψ dx \\ ≤ & ε_{n}∥(φ, ψ)∥_{W} , \end{matrix} (2.2)

for all (φ, ψ) ∈ W, where ε_{n} → 0 as n → ∞. Since ∥(u_{n}, v_{n})∥_{W} is bounded, we can take a subsequence, denoted again by (u_{n}, v_{n}) such that

u_{n} → u in L^{p}(Ω),
v_{n} → u in L^{q}(Ω),

as n approaches ∞ and ∀p, q > 1. Then considering in one hand φ = u_{n} − u and ψ = 0 in (2.2) and with the help of Lemma 2.1 , we obtain

Φ′(u_{n}, v_{n})(u_{n} − u, 0) → 0,

as n approaches ∞. Since u_{n} ⇀ u weakly, as n → ∞ and Φ′ ∈ (S_{+}), the result is proved. We have the same result for v_{n} by considering ψ = v_{n} − v and φ = 0 in (2.2). Finally, we conclude that (u_{n}, v_{n}) → (u, v) as n → ∞. □

Lemma 2.3 Assume that the function a satisfies (a1), (a2) and (a3) with Nb_{2} < μb_{1}, and that the nonlinearities f and g are continuous and satisfy (H1). Then the functional J satisfies the Palais-Smale condition (PS).

Proof. Using (a1), (a2) and (a3) with Nb_{2} < μb_{1}, we obtain positive constants c, d such that

\frac{μ}{N} A(t) − a(t)t ≥ ct − d ∀t ∈ ℝ^{+}.  (2.3)

Now, let (u_{n}, v_{n}) be a sequence in W satisfying condition (PS). Thus

\frac{1}{N} ∫_{Ω} A(|∇u_{n}|^{N}) + \frac{1}{N} ∫_{Ω} A(|∇v_{n}|^{N}) dx − ∫_{Ω} F (x, u_{n}, v_{n}) dx → c  (2.4)

as n goes to ∞.

\begin{matrix}| ∫_{Ω} a(|∇u_{n}|^{N})|∇u_{n}|^{N} + a(|∇v_{n}|^{N})|∇v_{n}|^{N} \\ −(∫_{Ω} f(x, u_{n}, v_{n})u_{n} + g(x, u_{n}, v_{n})v_{n}) dx| & ≤ ε_{n}∥(u_{n}, v_{n})∥_{W} , \end{matrix}  (2.5)

where ε_{n} → 0 as n → ∞. Multiplying (2.4) by μ, subtracting (2.5) from the expression obtained and using (2.3), we have

| ∫_{Ω} |∇u_{n}|^{N} +|∇v_{n}|^{N} −∫_{Ω}(μF (x, u_{n}, v_{n})−(f(x, u_{n}, v_{n})u_{n}+g(x, u_{n}, v_{n})v_{n}) dx|

≤ c + ε_{n}∥(u_{n}, v_{n})∥_{W} .

From this inequality and using hypothesis (H1), we deduce that (u_{n}, v_{n}) is bounded sequence in W. Now, with the help of Lemma 2.2, we conclude the proof. □

3 Proofs of the existence results

Lemma 3.1 Assume that the hypotheses of Theorem 1.1 hold. Then, there exist η, ρ > 0 such that J(u, v) ≥ η if ∥(u, v)∥_{X} = ρ. Moreover, J(t(u, v)) → −∞ as t → +∞ for all (u, v) ∈ W.

Proof. By (1.3) and (2.1), we can choose η_{1} < c_{1} + b_{1}δ_{p}(N) such that for r > N,

F (x, u, v) ≤ \frac{1}{p}η_{1}λ_{1}|u|^{α+1}|v|^{β+1} + C|u|^{r}e^{γ|u|^{\frac{N}{N−1}}} e^{γ|v|^{\frac{N}{N−1}}} ,

for all (x, u, v) ∈ Ω × W. For ∥u∥_{W_{0}^{1,N}} and ∥v∥_{W_{0}^{1,N}} small, from Hölder’s and Trudinger-Moser’s inequalities, we obtain

J(u, v) ≥ \frac{b_{1}}{N} ∥u∥_{W_{0}^{1,N}}^{N} + \frac{c_{1}}{p} ∥u∥_{W_{0}^{1,N}}^{p} − \frac{η_{1}}{p} ∥u∥_{W_{0}^{1,N}}^{p} − C_{1}∥u∥_{W_{0}^{1,N}}^{r}

+\frac{b_{1}}{N} ∥v∥_{W_{0}^{1,N}}^{N} + \frac{c_{1}}{p} ∥v∥_{W_{0}^{1,N}}^{p} − \frac{η_{1}}{p} ∥v∥_{W_{0}^{1,N}}^{p} − C_{1}∥v∥_{W_{0}^{1,N}}^{r} .

Since η_{1} < c_{1} + b_{1}δ_{p}(N) and p ≤ N < r, we can choose ρ > 0 such that J(u, v) ≥ η if ∥(u, v)∥_{W} = ρ for some η > 0. On the other hand, we can prove easily that

J(t(u, v)) → −∞ as t → +∞

So, by the Mountain-Pass Lemma [2], problem (1.1) has nontrivial solution (u, v) ∈ W which is a critical point of J. This completes the proof of Theorem 1.1. At the end, we give an example which illustrates conditions given on the nonlinearities f and g.

Example Let

F (x, u, v) = (1 + δ_{p}(N))\frac{λ}{p}|u|^{α+1}|v|^{β+1} + (1 − χ(u, v))exp (\frac{σ(|u|^{N} + |v|^{N})^{\frac{1}{N−1}}}{Log(|u| + |v| + 2)} )

where χ ∈ C^{1}(ℝ^{2}, [0, 1]), χ ≡ 1 on some ball B(0, r) ⊂ ℝ^{2} with r > 0 , and χ ≡ 0 on ℝ^{2}\B(0, r + 1). Thus, it follows immediately that (H_{1}), (H_{2}) and (1.3) are satisfied. Then problem (1.1) has a nontrivial weak solution provided that λ < λ_{1}.

References

[1] Adimurthi, Existence of positive solutions of the semilinear Dirichlet problem with critical growth for the n-Laplacian, Ann. Sc. Norm. Sup. Pisa 17 (1990), 393-413.

[2] A. Ambrosetti and P.H. Rabinowitz, Dual variational methods in critical point theory and applications, Funct. Anal. 14 (1973), 349-381.

[3] S. El Manouni and A. Touzani, Existence of nontrivial solutions for some elliptic systems in ℝ^{2}, Lecture Note in Pure and Applied Mathematics, Marcel Dekker, Vol 229, pp 239-248 (2002).

[4] D. G. de Figueiredo, O. H. Miyagaki and B. Ruf, Elliptic equations in ℝ^{2} with nonlinearities in the critical growth range, Calculus of Variations and PDE.

[5] N. Hirano, Multiple solutions for quasilinear elliptic equations. Nonl. Anal. Th. Meth. Appl. 15 (1990). pp.625-638.

[6] Joao Marcos B. do O, Semilinear elliptic equations with exponential nonlinearities, Appl. Nonlinear Anal. 2 (1995).

[7] J. Moser, A sharp form of an equality by N. Trudinger, Ind. Univ. Math. J. 20 (1971), 1077-1092.

[8] P.H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS, No. 65 AMS, 1986.

[9] F. De.Thelin, Première valeur propre d’un système elliptique non lineaire, C. R. Acad. Sci. Paris, t.311, Série 1 (1990), 603-606.

[10] J. Velin and F. De.Thelin , Existence and Nonexistence on Nontrivial Solutions for Some Nonlinear Elliptic Systems, Revista Matematica de la Universidad Complutense de Madrid 6, numero 1 (1993).

[11] N. S. Trudinger, On imbedding into Orlicz spaces and some applications, J. Math. Mech. 17 (1967), 473-484.

[12] P. Ubilla, Alguns resultados de multiplicidade de solucoes pam equacoes eliticas quasi-lineares. Doctoral dissertation (Unicamp). 1992.

Said El Manouni (e-mail: manouni@hotmail.com) Abdelfattah Touzani (e-mail: atouzani@iam.net.ma ) Département de Mathématiques et Informatique Faculté des Sciences Dhar-Mahraz, B. P. 1796 Atlas, Fès, Maroc.