Eigenvalue problems in anisotropic Orlicz–Sobolev spaces

Partial Differential Equations

Eigenvalue problems in anisotropic Orlicz–Sobolev spaces

Mihai Mih˘ailescu

, Gheorghe Moro¸sanu

, Vicen¸tiu R˘adulescu

aUniversity of Craiova, Department of Mathematics, Street A.I. Cuza No. 13, 200585 Craiova, Romania bDepartment of Mathematics, Central European University, 1051 Budapest, Hungary cInstitute of Mathematics “Simion Stoilow” of the Romanian Academy, 014700 Bucharest, Romania

Received 5 January 2009; accepted 6 February 2009 Available online 13 March 2009 Presented by Philippe G. Ciarlet

Abstract

We establish sufficient conditions for the existence of solutions to a class of nonlinear eigenvalue problems involving nonhomo- geneous differential operators in Orlicz–Sobolev spaces.To cite this article: M. Mih˘ailescu et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).

©2009 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.

Résumé

Problèmes de valeurs propres dans les espaces d’Orlicz–Sobolev anisotropes. On établit des conditions suffisantes pour l’existence des solutions pour une classe de problèmes non linéaires de valeurs propres avec des opérateurs différentiels non homogènes dans les espaces d’Orlicz–Sobolev.Pour citer cet article : M. Mih˘ailescu et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).

©2009 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.

Version française abrégée

SoitΩ⊂R^N (N3)un domaine borné et régulier. On considère le problème non linéaire −_N

i=1∂_i(φ_i(∂_iu))=λ|u|^q(x)−2u inΩ,

u=0 on∂Ω, (1)

oùλ >0 etq est une fonction continue telle queq(x) >1 pour toutx∈Ω. Pour chaquei∈ {1, . . . , N}on suppose qu’il existe deux constantes(p_i)₀et(p_i)⁰telles que siΦ_i(t )=_t

0φ_i(s)ds, alors 1< (p_i)0t φ_i(t )

Φ_i(t ) (p_i)⁰<∞, ∀t0.

E-mail addresses:[email protected] (M. Mih˘ailescu), [email protected] (G. Moro¸sanu), [email protected] (V. R˘adulescu).

URLs:http://www.inf.ucv.ro/~mihailescu (M. Mih˘ailescu), http://web.ceu.hu/math/People/Faculty/Morosanu/Morosanu.html (G. Moro¸sanu), http://www.inf.ucv.ro/~radulescu (V. R˘adulescu).

1631-073X/$ – see front matter ©2009 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.

doi:10.1016/j.crma.2009.02.023

Soit(P⁰)₊=max{(p₁)⁰, . . . , (p_N)⁰},(P₀)₊=max{(p₁)₀, . . . , (p_N)₀}et(P₀)₋=min{(p₁)₀, . . . , (p_N)₀}. On sup- pose que_N

i=11/(pi)₀>1 et on définit(P₀)=N/(_N

i=11/[(p_i)₀−1])etP_0,∞=max{(P₀)₊, (P₀)}.

Le résultat principal de cette Note est le suivant : Théorème 0.1. a)On suppose que

P⁰

+<min

x∈Ω

q(x)max

x∈Ω

q(x) < (P₀).

Alors, pour chaqueλ >0, le problème(1)admet une solution faible non triviale.

b)On suppose que 1<min

x∈Ω

q(x) < (P₀)₋ et max

x∈Ω

q(x) < P_0,∞.

Alors il existeλ>0tel que pour chaqueλ∈(0, λ)le problème(1)admet une solution faible non triviale.

c)On suppose que 1<min

x∈Ω

q(x)max

x∈Ω

q(x) < (P₀)₋.

Alors il existeλ>0etλ>0tels que pour chaqueλ∈(0, λ)et pour chaqueλ > λle problème(1)admet une solution faible non triviale.

1. The main result

LetΩ⊂R^N (N3) be a bounded domain with smooth boundary∂Ω. Assume that for eachi∈ {1, . . . , N},φ_i are odd, increasing homeomorphisms fromRontoR,λis a positive real andq:Ω→(1,∞)is a continuous function.

In this Note we study the following anisotropic eigenvalue problem:

−_N

i=1∂_i(φ_i(∂_iu))=λ|u|^q(x)−2u inΩ,

u=0 on∂Ω.

(2) For allt∈Randi∈ {1, . . . , N}, defineΦ_i(t )=t

0φ_i(s)ds. LetL_Φi(Ω)(i∈ {1, . . . , N}) be the corresponding Orlicz spaces (see [1,2]), which are the spaces of measurable functionsu:Ω→Rsuch that

uΦi:=inf

k >0;

Ω

Φ_i u(x)

k dx1

<∞.

The Orlicz spaceLΦ_i(Ω)endowed with the normuΦ_i is a Banach space.

Define (p_i)₀:=inf

t >0

t φi(t )

Φi(t ) and (p_i)⁰:=sup

t >0

t φi(t )

Φi(t ), i∈ {1, . . . , N}. In this Note we assume that for eachi∈ {1, . . . , N}we have

1< (pi)0t φ_i(t )

Φ_i(t ) (pi)⁰<∞, ∀t0. (3)

The above relation implies that eachΦi,i∈ {1, . . . , N}, satisfies the2-condition, that is,

Φi(2t )KΦi(t ), ∀t0, (4)

whereKis a positive constant (see [7, Proposition 2.3]).

Furthermore, in this Note we assume that for eachi∈ {1, . . . , N}the functionΦi satisfies the following condition:

the function[0,∞)t→Φ_i(√

t)is convex. (5)

Next, for eachi∈ {1, . . . , N}we build uponL_Φi(Ω)the Orlicz–Sobolev spaceW¹L_Φi(Ω)as the space of those weakly differentiable functions in Ω for which the weak derivatives belong to L_Φi(Ω). These are Banach spaces with respect to the normsu1,Φi:= uΦi+ |∇u|Φi, fori∈ {1, . . . , N}. We also define the Orlicz–Sobolev spaces

W₀¹L_Φi(Ω),i∈ {1, . . . , N}, as the closures ofC₀¹(Ω)inW¹L_Φi(Ω). OnW₀¹L_Φi(Ω),i∈ {1, . . . , N}, we may consider the equivalent normui:= |∇u|Φ_i. Moreover, the above norm is equivalent to the normui,1=_N

j=1∂_juΦ_i. Conditions (4) and (5) assure that for eachi∈ {1, . . . , N}the Orlicz spaces L_Φi(Ω)are uniformly convex spaces and thus, reflexive Banach spaces (see [7, Proposition 2.2]). That fact implies that also the Orlicz–Sobolev spaces W₀¹L_Φi(Ω),i∈ {1, . . . , N}, are reflexive Banach spaces.

Remark 1.We point out certain examples of functionsφ:R→Rwhich are odd, increasing homeomorphisms from RontoRand satisfy conditions (3) and (5):

1) φ(t)= |t|^p−2t, for allt∈R, withp >1. For this function it can be proved that(φ)₀=(φ)⁰=p.

2) φ(t)=log(1+ |t|^r)|t|^p−2t, for allt∈R, withp,r >1. In this case it can be proved that(φ)0=pand(φ)⁰= p+r.

3) φ(t)=_log(1^|t|p−_+|^2t_t|), ift=0 andφ(0)=0, withp >2. In this case we have(φ)₀=p−1 and(φ)⁰=p.

For more details the reader can consult [3, Examples 1–3, p. 243].

Finally, we introduce a natural generalization of the Orlicz–Sobolev spaces W₀¹L_Φi(Ω) that will enable us to study with sufficient accuracy problem (2). For this purpose, let us denote by Φ:Ω→R^N the vectorial function Φ=(Φ1, . . . , ΦN). We defineW₀¹L_Φ(Ω), theanisotropic Orlicz–Sobolev space, as the closure ofC₀¹(Ω)with respect to the normu_Φ=N

i=1|∂iu|Φ_i.

In the case whenΦi(t )= |t|^θi, whereθiare constants for anyi∈ {1, . . . , N}the resulting anisotropic Sobolev space is denoted byW₀^1,θ(Ω), whereθis the constant vector(θ1, . . . , θN). The theory of such spaces was developed in [4, 9,10,12,8]. It was proved thatW₀^1,θ(Ω)is a reflexive Banach space for anyθ∈R^Nwithθ_i>1 for alli∈ {1, . . . , N}. This result can be easily extended, and thus, we can show thatW₀¹L_Φ(Ω)is a reflexive Banach space.

On the other hand, in order to facilitate the manipulation of the spaceW₀¹L_Φ(Ω)we introduceP⁰,P₀∈R^Nas P⁰=

(p₁)⁰, . . . , (p_N)⁰

, P₀=

(p₁)₀, . . . , (p_N)₀ ,

and(P⁰)₊,(P0)₊,(P0)₋∈R⁺as P⁰

+=max

(p₁)⁰, . . . , (p_N)⁰

, (P₀)₊=max

(p₁)₀, . . . , (p_N)₀

, (P₀)₋=min

(p₁)₀, . . . , (p_N)₀ . Throughout this Note we assume that

N

i=1

1 (pi)0

>1, (6)

and define(P0)∈R⁺andP0,∞∈R⁺by

(P₀)= N

_N

i=11/(pi)₀−1, P_0,∞=max

(P₀)₊, (P₀) .

Next, we recall some background facts concerning the variable exponent Lebesgue spaces. For anyh∈C₊(Ω):=

{h;h∈C(Ω), h(x) >1 for allx∈Ω}we defineh⁺:=sup_x∈Ωh(x)andh⁻:=infx∈Ωh(x). For anyp∈C₊(Ω), we define the variable exponent Lebesgue spaceL^q(x)(Ω)as the set of all measurable functionsu:Ω→Rsuch that

Ω|u(x)|^q(x)dx <∞, where

|u|q(x):=inf

μ >0;

Ω

u(x) μ

^q(x)dx1

.

An important role is played by themodularof theL^q(x)(Ω)space, which is defined byρ_q(x)(u)=

Ω|u|^q(x)dx, for anyu∈L^q(x)(Ω). Ifun,u∈L^q(x)(Ω)then the following relations hold true:

|u|q(x)>1 ⇒ |u|^q_q(x)⁻

Ω

|u|^q(x)dx|u|^q_q(x)⁺ , (7)

|u|q(x)<1 ⇒ |u|^q_q(x)⁺

Ω

|u|^q(x)dx|u|^q_q(x)⁻ . (8)

In the following, for eachi∈ {1, . . . , N}we definea_i:[0,∞)→Rbya_i(t )=^φi_t^{(t )}, fort >0 anda_i(0)=0. Since φ_i are odd we deduce that actually,φ_i(t )=a_i(|t|)tfor eacht∈Rand eachi∈ {1, . . . , N}.

We say thatu∈W₀¹L_Φ(Ω)is aweak solutionof problem (2) if

Ω

_N

i=1

ai

|∂iu|

∂iu∂iw−λ|u|^q(x)−2uw

dx=0 for allw∈W₀¹L_Φ(Ω).

The main result of this Note is stated in the following theorem:

Theorem 1.1.a)Assume that the functionq(x)∈C(Ω)verifies the hypothesis P⁰

+<min

x∈Ω

q(x)max

x∈Ω

q(x) < (P₀). (9)

Then for anyλ >0problem(2)has a nontrivial solution inW₀¹L_Φ(Ω).

b)Assume that the functionq(x)∈C(Ω)verifies the hypothesis 1<min

x∈Ω

q(x) < (P₀)₋ and max

x∈Ω

q(x) < P_0,∞. (10)

Then there existsλ>0such that for anyλ∈(0, λ)problem(2)has a nontrivial solution inW₀¹L_Φ(Ω).

c)Assume that the functionq(x)∈C(Ω)verifies the hypothesis 1<min

x∈Ω

q(x)max

x∈Ω

q(x) < (P₀)₋. (11)

Then there existλ>0andλ>0such that for anyλ∈(0, λ)and anyλ > λproblem(2)has a nontrivial solution inW₀¹L_Φ(Ω).

2. Proof of Theorem 1.1

The following result extends Theorem 1 in [4]:

Proposition 2.1.AssumeΩ⊂R^N (N3)is a bounded domain with smooth boundary. Assume relation(6)is ful- filled. Assume thatq∈C(Ω)verifies1< q(x) < P0,∞, for allx∈Ω. Then the embeddingW₀¹L_Φ(Ω) →L^q(x)(Ω) is compact.

From now onEdenotes the anisotropic Orlicz–Sobolev spaceW₀¹L_Φ(Ω).

For anyλ >0 the energy functional corresponding to problem (2) is defined asT_λ:E→R, Tλ(u)=

Ω

N

i=1

Φi

|∂iu| dx−λ

Ω

1

q(x)|u|^q(x)dx.

Proposition 2.1 implies thatTλ∈C¹(E,R)and T_λ(u), v

=

Ω

N

i=1

a_i

|∂_iu|

∂_iu∂_ivdx−λ

Ω

|u|^q(x)−2uvdx,

for allu, v∈E. Thus, the weak solutions of (2) coincides with the critical points ofT_λ. The following auxiliary results show thatTλhas a mountain-pass geometry:

Lemma 2.2.Assume that the hypothesis(9)of Theorem1.1is fulfilled. Then there existη >0and α >0such that T_λ(u)α >0for anyu∈Ewithu_Φ=η.

Lemma 2.3.Assume that the hypothesis(9)of Theorem1.1is fulfilled. Then there existse∈Ewithe_Φ> η(where ηis given in Lemma2.2)such thatT_λ(e) <0.

Proof of Theorem 1.1 a). By Lemmas 2.2 and 2.3 and the mountain-pass theorem of Ambrosetti and Rabinowitz we deduce the existence of a sequence{un} ⊂Esuch that

T_λ(u_n)→c >0 and T_λ(u_n)→0(inE) asn→ ∞. (12)

We prove that{u_n}is bounded inE. Arguing by contradiction, there exists a subsequence (still denoted by{u_n}) such thatu_nΦ→ ∞. Thus, we may assume that fornlarge enough we haveu_nΦ>1.

For eachi∈ {1, . . . , N}and any positive integernwe define αi,n=

(P⁰)₊, if∂_iu_nΦ_i<1, (P₀)₋, if∂_iu_nΦ_i>1.

So, by the above considerations (combined with inequalities (C.9) and (C.10) in [3], see also [6, Lemma 1]) we deduce that fornlarge enough we have

1+c+ un_ΦTλ(un)− 1 q⁻

T_λ(un), un

N

i=1

Ω

Φi

|∂iun|

− 1 q⁻φi

|∂iun|

|∂iun| dx

1−(P⁰)₊ q⁻

N

i=1

Ω

Φi

|∂iun|

dx

1−(P⁰)₊ q⁻

N

i=1

∂iun^α_Φ^i,n_i

1−(P⁰)₊ q⁻

1

N^(P0)−−1u_n^(P_Φ⁰⁾⁻−N

. (13)

Dividing by un^(P_Φ⁰⁾⁻ in (13) and passing to the limit asn→ ∞ we obtain a contradiction. It follows that{un} is bounded inE. SinceEis reflexive, there exists a subsequence, still denoted by{un}, andu0∈E such that{un} converges weakly tou0inE. So, by Proposition 2.1,{un}converges strongly tou0inL^q(x)(Ω). The above considera- tions and relations (12) and (5) imply that actually,{u_n}converges strongly tou₀inE. Then, by relation (12) we have T_λ(u₀)=c >0 andT_λ(u₀)=0, that is,u₀is a nontrivial weak solution of equation (2). The proof of Theorem 1.1 a) is complete. 2

Proof of Theorem 1.1 b). We start with the following auxiliary result:

Lemma 2.4.Assume that the hypothesis(10)of Theorem1.1is fulfilled. Then there existsλ>0such that for any λ∈(0, λ)there areρ,a >0such thatT_λ(u)a >0for anyu∈EwithuΦ=ρ.

Letλ>0 be defined as above and fix λ∈(0, λ). By Lemma 2.4 it follows that on the boundary of the ball centered at the origin and of radiusρ inE, denoted byB_ρ(0), we have inf

∂Bρ(0)J_λ>0. Standard arguments show that there exists φ∈E,φ0, such that T_λ(t φ) <0 for allt >0 small enough. Moreover, we can show that for any u∈B_ρ(0)we have

T_λ(u)C₁· u^(P_Φ⁰⁾⁺−C₂· u^q_Φ⁻, whereC₁, C₂>0. It follows that−∞< c:= inf

Bρ(0)

T_λ<0. Fix 0< <inf∂B_ρ(0)T_λ−infB_ρ(0)T_λ. Applying Ekeland’s variational principle to the functionalT_λ:B_ρ(0)→R, we findu ∈B_ρ(0)such thatT_λ(u) < inf

B_ρ(0)

T_λ+and for allu=u,Tλ(u) < Tλ(u)+· u−u_Φ. SinceTλ(u) inf

B_ρ(0)

Tλ+ inf

Bρ(0)Tλ+ < inf

∂Bρ(0)Tλ, we deduce that u∈B_ρ(0). DefineI_λ:B_ρ(0)→RbyI_λ(u)=T_λ(u)+· u−u_Φ. Thenu is a minimum point ofI_λand thus t⁻¹[I_λ(u+t·v)−I_λ(u)]0 for smallt >0 and anyv∈B₁(0). Lettingt→0 it follows thatT_λ(u), v +· v_Φ>0, henceT_λ(u).

We deduce that there exists a sequence{w_n} ⊂B_ρ(0)such thatT_λ(w_n)→candT_λ(w_n)→0. Moreover,{w_n}is bounded inE. Thus, there existsw∈Esuch that, up to a subsequence,{w_n}converges weakly towinE. Actually, with similar arguments as those used in the end of the proof of Theorem 1.1 a) we can show that{w_n}converges strongly towinE. So,T_λ(w)=c <0 andT_λ(w)=0. We conclude thatwis a nontrivial weak solution of problem (2).

The proof of Theorem 1.1 b) is complete.

Finally, we show that Theorem 1.1 c) holds true. In order to do that we first point out that by Theorem 1.1 b) it follows that there exists λ>0 such that for anyλ∈(0, λ)problem (2) has a nontrivial weak solution. In order to show that there exists λ>0 such that for anyλ > λ problem (2) has a nontrivial weak solution we prove thatTλ possesses a nontrivial global minimum point inE. Indeed, it is not difficult to show thatTλis weakly lower semicontinuous and coercive onE. So, by Theorem 1.2 in [11], there exists a global minimizeruλ∈Eof Tλ and, thus, a weak solution of problem (2). We show thatuλis not trivial forλlarge enough. Indeed, lettingt0>1 be a fixed real andΩ1be an open subset ofΩwith|Ω1|>0 we deduce that there existsv0∈C^∞₀ (Ω)⊂Esuch thatv0(x)=t0

for anyx∈Ω₁and 0v₀(x)t₀inΩ\Ω₁. We have Tλ(v0)=

Ω

_N

i=1

Φi

|∂iv0|

− λ

q(x)|v0|^q(x)

dxL− λ q⁺

Ω₁

|v0|^q(x)dxL− λ

q⁺t₀^q−|Ω1|,

whereLis a positive constant. Thus, there existsλ>0 such thatTλ(u0) <0 for anyλ∈ [λ,∞). It follows that Tλ(uλ) <0 for anyλλand thusuλis a nontrivial weak solution of problem (2) forλlarge enough. The proof of Theorem 1.1 c) is complete. 2

We refer to [5] for complete proofs and additional results.

Acknowledgements

M. Mih˘ailescu and V. R˘adulescu have been supported by Grant CNCSIS PNII–79/2007 “Degenerate and Singular Nonlinear Processes”. V. R˘adulescu also acknowledges support through Grant CNCSIS PCCE–55/2008.

References

[1] R. Adams, Sobolev Spaces, Academic Press, New York, 1975.

[2] Ph. Clément, M. García-Huidobro, R. Manásevich, K. Schmitt, Mountain pass type solutions for quasilinear elliptic equations, Calc. Var. 11 (2000) 33–62.

[3] Ph. Clément, B. de Pagter, G. Sweers, F. de Thélin, Existence of solutions to a semilinear elliptic system through Orlicz–Sobolev spaces, Mediterr. J. Math. 1 (2004) 241–267.

[4] I. Fragalà, F. Gazzola, B. Kawohl, Existence and nonexistence results for anisotropic quasilinear equations, Ann. Inst. H. Poincaré, Analyse Non Linéaire 21 (2004) 715–734.

[5] M. Mih˘ailescu, G. Moro¸sanu, V. R˘adulescu, Eigenvalue problems for anisotropic elliptic equations: An Orlicz–Sobolev space setting, preprint.

[6] M. Mih˘ailescu, V. R˘adulescu, Eigenvalue problems associated to nonhomogeneous differential operators in Orlicz–Sobolev spaces, Analysis and Applications 6 (1) (2008) 1–16.

[7] M. Mih˘ailescu, V. R˘adulescu, Neumann problems associated to nonhomogeneous differential operators in Orlicz–Sobolev spaces, Ann. Inst.

Fourier 58 (6) (2008) 2087–2111.

[8] S.M. Nikol’skii, On imbedding, continuation and approximation theorems for differentiable functions of several variables, Russian Math.

Surv. 16 (1961) 55–104.

[9] J. Rákosník, Some remarks to anisotropic Sobolev spaces I, Beiträge zur Analysis 13 (1979) 55–68.

[10] J. Rákosník, Some remarks to anisotropic Sobolev spaces II, Beiträge zur Analysis 15 (1981) 127–140.

[11] M. Struwe, Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Springer, Heidelberg, 1996.

[12] M. Troisi, Teoremi di inclusione per spazi di Sobolev non isotropi, Ricerche Mat. 18 (1969) 3–24.