On some nonlinear elliptic systems with coercive perturbations in R N

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1 On some nonlinear ellitic systems with coercive erturbations in R Said EL MAOUI and Abdelfattah TOUZAI Déartement de Mathématiues et Informatiue Faculté des Sciences Dhar-Mahraz B.P Atlas-Fès Fès 30000, Maroc manouni@hotmail.com a.touzani@menara.ma Recibido: 14 de Octubre de 2002 Acetado: 22 de Julio de 2003 ABSTRACT A nonlinear ellitic system involving the -Lalacian is considered in the whole R. Existence of nontrivial solutions is obtained by alying critical oint theory; also a regularity result is established Mathematics Subject Classification: 35J70, 35B45, 35B65. Key words: onlinear ellitic system, coercive erturbation, Palais-Smale condition, Mountain Pass Theorem 1. Introduction In this aer, we give some existence and regularity results concerning the following class of nonlinear ellitic systems in R, 2 Δ u + a(x) u 2 u = f(x) u α 1 u v β+1 in R (S) Δ v + b(x) v 2 v = f(x) u α+1 v β 1 v in R lim x u(x) = lim x v(x) =0 where 1 <<,1 << and α, β are real constants satisfying (1.1) 0 <α 1, 0 <β 1and max(, ) < α+1 < 1. f and the erturbations a and b are measurable functions satisfying some conditions which ensure the existence and regularity of solutions. Revista Matemática Comlutense 483 ISS: htt://dx.doi.org/ /rev_rema.2003.v16.n

2 Said El Manouni and Abdelfattah Touzani On some nonlinear ellitic systems... Let us mention that many results in the scalar case have aeared for this kind of roblems involving Lalacian and -Lalacian oerators in unbounded domains. Costa in [3], among others, obtained a nontrivial solution to the roblem Δu+a(x)u = f(x, u) in R, under the assumtion that a is coercive and that the otential F (x, u) = u f(x, s) ds is nonuadratic at infinity. In [4], this result was generalized by the 0 same author to a class of ellitic systems with resect to the otential and subcritical growth. Other roblems have been considered in this direction, see e.g. P. Drabek [5] and Lao SenYuin[10]. Concerning systems of the tye (S), several studies have been devoted to investigation recently both in bounded and unbounded domains-we refer to [2, 4, 6, 7, 13, 15, 17]. The case of unbounded domains becomes more comlicate, generally the main difficulty lies in the loss of Sobolev comact imbedding. In this aer, by establishing sufficient condition, we give an extension and we comlement some results of the scalar case to a system of ellitic euations. In articular, to treat variationally this class of roblems, we assume a lower regularity condition on the function f (not necessary in L, see assumtion (H 2 ) below), so that a nontrivial solution can be obtained via Mountain Pass Theorem and local minimization of energy functional associated to our roblem resectively in connection with cases α+1 > 1and α+1 < 1. On the other hand, to overcome the lack of comactness that has arisen from the critical exonent and the unboundedness of the domain, we use a comact imbedding result essentially given by the coerciveness of the erturbations a and b (see assumtion (H 1 )). In the second section, we establish a regularity result, more recisely, we rove that such solutions (u, v) belong to L 1 L 1 for any 1 [, [ and 1 [, [. In general, this regularity cannot reach the sace L L taking into account our argument develoed here. Through this aer, we use standard notation: W 1, := W 1, (R ) is the ordinary Sobolev sace, L = L (R ) is the Lebesgue sace euied with the norm. and = is the critical Sobolev exonent, the Lebesgue integral in R will be denoted by the symbol whenever the integration is carried out over all R ; C0 := C0 (R ) is the sace of all functions with comact suort in R with continuous derivatives of arbitrary order. Let us formulate the assumtions on the erturbations a and b and on the function f = f(x). (H 1 ) a, b : R R are continuous functions satisfying a(x) a 0 > 0, b(x) b 0 > 0, i.e. in R, Revista Matemática Comlutense 484

3 Said El Manouni and Abdelfattah Touzani On some nonlinear ellitic systems... which are coercive, that is lim a(x) = lim b(x) =+, x + x + (H 2 ) the function f : R R is a nonnegative and f L ω L ω 1 δ, with ω = (α +1) (β +1). Here 0 <δ<1 is a small ositive real. We introduce the following function sace E = {(u, v) W 1, W 1, / ( u +a(x) u + v +b(x) v ) dx < } endowed with the norm (u, v) E = ( u +a(x) u dx) 1 +( v +b(x) v dx) 1 = u 1 + v 2 where u 1 =( u +a(x) u dx) 1/. dx) 1/ and v 2 =( v +b(x) v Since a(x) a 0 > 0andb(x) b 0 > 0, we clearly see that the Banach sace E is continuously embedded in W 1, W 1,. We also conclude from Sobolev s Theorem the continuous imbedding E L 1 L 1, for all 1 and 1. Definition 1.1. We say that a air (u, v) E is a weak solution of (S) if { u (SV ) 2 u ϕdx+ a(x) u 2 uϕ dx = f(x) u α 1 uϕ v β+1 dx v 2 v ψdx+ b(x) v 2 vψ dx = f(x) u α+1 v β 1 vψ dx holds for all (ϕ, ψ) E. Let us remark that the assumtions (H 1 )and(h 2 ) guarantee that integrals given in (SV ) are well defined. ow, we state the main results of this aer. Theorem 1.1. Suose 1 <,<,(1.1), (H 1 ) and (H 2 ) are satisfied. Then the system (S) has at least one nontrivial weak solution (u, v) E. Theorem 1.2. Let (u, v) be a solution of (S). Then (u, v) L σ1 L σ2, with σ 1 < and σ 2 <. Moreover u, v > 0 in R and lim x u(x) = lim x v(x) = Revista Matemática Comlutense

4 Said El Manouni and Abdelfattah Touzani On some nonlinear ellitic systems Preliminaries Let us consider the functional I : E R given by I(u, v) = α +1 u +a(x) u dx + β +1 f(x) u α+1 v β+1 dx. v +b(x) v dx By assumtion (H 2 ) and Sobolev s ineuality, we can see that the functional K defined by K(u, v) = f(x) u α+1 v β+1 dx is indeed well defined and of class C 1 on the sace E with <K (u, v); (ϕ, ψ) >= (α +1) f(x) u α 1 uϕ v β+1 dx +(β +1) f(x) u α+1 v β 1 vψ dx, for all (u, v) and(ϕ, ψ) E; where<; > denotes the duality symbol from E to E. Therefore, a weak solution of a system (S) is a critical oint (u, v) ofi, i.e I (u, v)(ϕ, ψ) =0 (ϕ, ψ) E. In order to rove our main result, we will use the following basic roerties. Lemma 2.1. There exists constant C>0 such that with m 1 ]1, [ and m 2 ]1, [. f(x) u α+1 v β+1 C(f(x) ω 1 δ + u m 1 + v m2 ) Proof. Since α +1< and β +1<, there exists 0 <δ<1 small enough such that > and >, where δ δ = α +1+ ω( 1 + 1, = β +1+ ) Then the lemma follows from Young s ineuality with m 1 = and m 2 =. ω( ). Revista Matemática Comlutense 486

5 Said El Manouni and Abdelfattah Touzani On some nonlinear ellitic systems... Lemma 2.2. Suose (H 1 ) and (H 2 ), then i) E is comactly embedded in L L. ii) K isacomactmafrome to E. Proof. i) Without loss of generality, we will show that (u n,v n ) (0, 0) strongly in E for such seuence (u n,v n ) E which converges weakly to (0, 0). Indeed, we have (u n,v n ) E C for some constant C>0. From (H 1 ), for a given ε>0andr>0 such that we have a(x) 4 C ε and b(x) 4C ε for all x R, (u n,v n ) (0, 0) weakly in W 1, (B R ) W 1, (B R ), where B R is the Ball of radius R centered at origin. By using the comact imbedding W 1, (B R ) W 1, (B R ) L (B R ) L (B R ), we get (2.1) ( u n + v n ) dx ε B R 2 n n 0 for some n 0. We also have 4 (2.2) ( u n + v n ) dx ( a(x) ε R B R R B R C u n + b(x) C v n ) dx 2 since (u n,v n ) E R B R u n dx and (u n,v n ) E R B R v n dx. Combining (2.1) and (2.2), we obtain that ( u n + v n ) dx ε, n n 0. R ii) The comactness of K follows from the estimate <K (u n,v n ) K (u 0,v 0 ); (ϕ, ψ) >= I 1 I 2, where and I 1 = I 2 = f(x)( u n α 1 u n v n β+1 u 0 α 1 u 0 v 0 β+1 )ϕdx f(x)( u n α+1 + v n β 1 v n u 0 α+1 v 0 β 1 v 0 )ψdx. The objective is to rove that I 1 0andI 2 0. On one hand, we have I 1 I 11 +I 12 where I 11 = f(x)( u n α 1 u n u 0 α 1 u 0 )( v n β+1 ϕ) dx 487 Revista Matemática Comlutense

6 Said El Manouni and Abdelfattah Touzani On some nonlinear ellitic systems... and I 12 = f(x) u 0 α 1 u 0 ( v n β+1 v 0 β+1 )ϕdx. By choosing δ sufficiently small such that x = 1 α + δ ω in view of (H 2 ) the following estimate > 1and αx <, we obtain I 11 f ω L 1 δ u n α 1 u n u 0 α 1 u 0 L x v n β+1 ϕ L L. β+1 On the other hand, since the imbedding E L L is comact, it follows from the interolation ineuality i.e. u L t u σ L u 1 σ L, u L L, where 1 t = σ + 1 σ, that the imbedding E L 1 L 1 is comact for 1 < and 1 <. Hence, we get I 11 0 (strongly) as n goes to infinity, since αx <. ow we estimate I 12 : I 12 = f(x) u 0 α 1 u 0 ( v n β+1 v 0 β+1 )ϕdx f ω L 1 δ u 0 α v L n β+1 v 0 β+1 L y ϕ L α 1 where y = β+1 > 1. A simle calculation shows that (β +1)y for δ small + δ ω enough. Conseuently, we conclude that I 12 0 (strongly) as n. Finally, using the same argument to estimate I 2, we get I 2 0 (strongly) as n. Therefore K (u n,v n ) K (u 0,v 0 ) strongly in E. as n tends to infinity. This ends the roof of Lemma 2.2. Recall that (u n,v n ) E is a Palais-Smale seuence if there exists M > 0such that, I(u n,v n ) M and I (u n,v n ) 0 strongly in E as n goes to infinity. Remark ) Let us remark that an otimal value of the generic constant δ is taken here. This allows us also to consider more lower regularity condition on the function f. 2) ote that the assumtion (H 1 ) gives a comact imbedding result which is used only to rove that the Palais Smale seuence obtained by Mountain Pass tye argument converges to a weak nontrivial solution. Lemma 2.3. Suose α+1 > 1, let (u n,v n ) be a Palais-Smale seuence. Then (u n,v n ) ossesses a subseuence which converges strongly in E. Revista Matemática Comlutense 488

7 Said El Manouni and Abdelfattah Touzani On some nonlinear ellitic systems... Proof. Let (u n,v n ) E be a Palais-Smale seuence. We have I(u n,v n ) <I (u n,v n ); ( un, vn ) >= ( 1+(α+1 since <I (u n,v n ); ( u n, v n ) > = α +1 + β +1 ( α +1 Hence, we come to the conclusion that (1 1 α+1 )( α +1 )) f(x) u n α+1 v n β+1 dx, u n +a(x) u n dx v n +b(x) v n dx + β +1 ) f(x) u n α+1 v n β+1 dx. u n +a(x) u n dx + β +1 v n +b(x) v n dx) M <I (u n,v n ); ( u n, v n ) >. From this ineuality, we easily find that (u n,v n ) is a bounded seuence in E, Since α+1 > 1. Conseuently, there exists a subseuence still denoted by (u n,v n ) such that (u n,v n ) converges weakly in E. ow, we claim that (u n,v n ) converges strongly in E. Indeed, for any air integer (i, j), we have ( u i 2 u i u j 2 u j )( u i u j ) + (a(x) u i 2 u i a(x) u j 2 u j )(u i u j ) = I (u i,v i ) <I (u j,v j ); (u i u j, 0) > + f(x)( u i α 1 u i v i β+1 u j α 1 u j v j β+1 )(u i u j ) dx. By Palais-Smale condition, it is easy to see that I (u i,v i ) <I (u j,v j ); (u i u j, 0) > 0 as i and j tend to infinity. From the Lemma 2.2 (K is comact), we have 489 Revista Matemática Comlutense

8 Said El Manouni and Abdelfattah Touzani On some nonlinear ellitic systems... f(x)( u i α 1 u i v i β+1 u j α 1 u j v j β+1 )(u i u j ) dx 0. as i and j tend to infinity. From the following algebraic relation ξ 1 ξ 2 r (( ξ 1 r 2 ξ 1 ξ 2 r 2 ξ 2 )(ξ 1 ξ 2 )) s/2 ( ξ 1 r + ξ 2 r ) 1 s/2 with s = r, for 1 <r 2ands =2for2<r,we deduce that (u n )isacauchy seuence in E, therefore it converges strongly. By the same argument, we show also that (v n ) converges strongly. This concludes the roof of Lemma Proof of the main results In this section, we give the roof of the existence results, we aly Mountain Pass Lemma and local minimization to find nontrivial solution. For that reason, we will searately distinguish two cases related to our study: α+1 α+1 > 1 and < 1. After that, we use an iterative method to rove the regularity result. Lemma 3.1. Suose (H 1 ), (H 2 ) and α+1 > 1, then 1) There exist γ, ρ, such that I(u, v) γ, for (u, v) E = ρ. 2) I(t(u, v)) as t +. Proof. 1 ) From lemma 2.1, we have I(u, v) α +1 u 1 with m 1 < and m 2 <. +β +1 v 2 C 0( u m1 m 1 + v m2 m 2 ), Denoting by θ and η resectively u 1 and v 2, we therefore obtain the following minoration of J for any (u, v) E I(u, v) θ ( α +1 C θ m1 )+η ( β +1 C η m2 ). Which imlies that there exists γ, ρ > 0 such that I(u, v) γ>0 for all (u, v) E = ρ. 2 ) From the exression I(t 1/ u, t 1/ v)= t(α+1) u 1 + t(β+1) v α+1 2 t + β+1 f(x) u α+1 v β+1 dx, it follows that I(t 1/ u, t 1/ v) as t + Revista Matemática Comlutense 490

9 Said El Manouni and Abdelfattah Touzani On some nonlinear ellitic systems... Since α+1 > 1. Hence, in view of Lemmas 2.3 and 3.1, we can aly the Mountain-Pass Theorem (c.f. [1]) which guarantees the existence of nontrivial weak solutions of (S). On the other hand, in the case α+1 < 1, we may use the local minimization of the functional I to rove the existence result. Indeed, by hyothesis (H 2 ), the functional I is weakly lower semi continuous differentiable. Moreover, I is bounded below. In fact, we have I(u, v) α +1 u 1 +β +1 v 2 C f ω u α+1 Since α+1 < 1, there exist γ 1 ]1,[,γ 2 ]1,[ such that α +1 + β +1 =1, γ 1 γ 2 which imlies that I(u, v) α +1 u 1 +β +1 v β+1. v 2 C f ω ( u γ1 + v γ2 ) since u D 1 u 1 and v D 2 v 2 for some constants D 1,D 2 > 0. Then, we have I(u, v) α +1 u 1 +β +1 v 2 C f ω ( u γ1 1 + v γ2 2 ). It follows from here that I is bounded below. Thus I has a critical oint (ū, v) I(ū, v) =inf{i(u, v) :(u, v) E}, which is solution of the system (S). We note that (u, v) must be nontrivial since I(sϕ, tψ) =s α +1 u β t v 2 sα+1 t β+1 f(x) ϕ α+1 ψ β+1 dx for some ϕ, ψ C0. Hence, since α +1<,β+1<,we get I(sϕ, tψ) < 0 for small s, t. This concludes the roof of Theorem 1.1. Proof of Theorem 1.2. In this section, we may choose u, v 0 since we can show that argument develoed here is true for u + and u. Set u k (x) =min{u(x),k},k. For any real i 1, (u i k,v) E. We have u 2 u ϕ + a(x) u 2 uϕ dx = f(x) u α 1 uϕ v β+1 dx 491 Revista Matemática Comlutense

10 Said El Manouni and Abdelfattah Touzani On some nonlinear ellitic systems... for all ϕ E. Substituting ϕ =(u k ) i in this euation, we obtain the following estimate i (u k ) i 1 u k f(x) u α+i v β+1 dx. Due to the fact (u k ) i 1 u k =( i+ 1 ) (u k ) (i+ 1) and Sobolev s ineuality, we get ( (u k ) (i+ 1) ) C f(x) u α+i v β+1 dx for some constant C>0. Setting i = i 0 =1+ δ ω,s 0 = conclude that u L s0 since n n (i 0 + 1) = 1 δ ω + α + i 0 + β +1 =1. Setting now i 1 =1+ δ ω + δ ω = i 0 + we get u L s1, where s 1 = +( 1 δ ω δ since α+i1 Iterating this rocess gives with i j =1+ δ ω + ( + δ ω ). Letting k, we δ ω, reeating the same argument, (i 1 + 1), ω )=1 and f L 1 δ(1+ ω ) u L sj where s j = (i j + 1), δ ω ( )j δ ω. Hence, it follows that for δ small enough. u L σ1 for all, σ 1 <. On the other hand, by using the same argument as above, we rove that v L σ2, σ 2 <. In order to comlete the roof of Theorem 1.2, we need the following result. Claim. fv β+1 L ε and fu α+1 L ε for some 0 <ε<1 small enough. Revista Matemática Comlutense 492

11 Said El Manouni and Abdelfattah Touzani On some nonlinear ellitic systems... Proof. Let ε ]0,min{1,δ,δ}[; using Hölder ineuality we obtain (fv β+1 ) ε dx ( f(x) ω ε ) ω ( v β+1 ε Indeed, in virtue of (1.1) (in articular, we have α+1 ω ) ω, > ), it follows that 0(since (α+1) ω > 1. Let us remark also, for reader s convenience, that + α +1 ), which imlies (3.1) (β +1) ε > ω Hence, we deduce from (3.1) and due to the fact that v L σ1 (σ 1 ), that fv β+1 L ε since ω ε [ω, ω/(1 δ)]. Similarly, we rove that fuα+1 L ε. Letting now ϕ = u as a test function in the first euation of (SV ) imlies u 0 in R. Hence, in view of revious Claim, the ositivity of solutions follows immediately from the weak Harnack tye ineuality roved in Trudinger [18, Theorem 1.2]. Finally, the decay of u and v follows directly from the result (Theorem 1) of Serrin [12]. 4. Concluding remarks Remark 4.1. When α+1 =1, one can show, by the same argument used in the case α+1 < 1 that there exists λ such that for all λ verifying 0 <λ<λ, the following eigenvalue roblem Δ u + a(x) u 2 u = λf(x) u α 1 u v β+1 in R Δ v + b(x) v 2 v = λf(x) u α+1 v β 1 v in R lim x u(x) = lim x v(x) =0 has at least one nontrivial solution in E. Remark 4.2. The existence result is also obtained for systems of the form { (S Δ u + a(x) u ) 2 u = I f i(x) u αi 1 u v βi+1 in R Δ v + b(x) v 2 v = I f i(x) u αi+1 v βi 1 v in R under the more general assumtions: f i L mi r ositive, m i [r i, i 1 δ ]where α i+1 + βi+1 r i = 1 1 ( αi+1 + βi+1 ), < 1,α i > 0,β i > 0and0<δ<1 is a small ositive real. 493 Revista Matemática Comlutense

12 Said El Manouni and Abdelfattah Touzani On some nonlinear ellitic systems... References [1] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical oint theory and alications, Funct. Anal. 14 (1973), [2] J. Chabrowski, On multile solutions for nonhomogeneous system of ellitic euations, Revista Matemática Univ. Comlutense Madrid, 9(1) (1996), [3] D. G. Costa, On nonlinear ellitic roblem in R, Center for mathematical sciences, Univ. Winconsin, CMS-Tech. Summ. Re. # 93-13, (1993). [4] D. G. Costa, On a class of ellitic systems in R, Electron. J. Diff. Ens., Vol.1994 (1994), o. 07, [5] P. Drabek, onlinear Eigenvalues for -Lalacian in R, Math.achr. 173 (1995), [6] P. Drabek, S. EL Manouni and A. Touzani, Existence and regularity of solutions for nonlinear ellitic systems in R, Atti Sem. Mat. Fis. Univ. Modena, L (2002), [7] S. EL Manouni and A. Touzani, Existence of nontrivial solutions for some ellitic systems in R 2, Lecture ote In Pure and Alied Mathematics, Marcel Dekker, Vol 229 (2002), [8] J. Fleckinger, R.F. Manasevich,. M. Stavrakakis and F. De Thélin, Princial eignvalues for some uasilinear ellitic euations on R, Advances in Differential Euations, Vol 2, n 6 (1997), [9] Ding. Yan Heng and Li. Shu Jie, Existence of entire solutions for some ellitic systems. Bull. Austral. Math. Soc. 50 (1994), no. 3, [10] Lao Sen Yu, onlinear -Lalacian roblems on unbounded domains, Proceeding of A. M. S. 115, 4 (1992), [11] Li Gongbao and You Shusen, Eigenvalue roblems for uasilinear ellitic euations on R, com. Part. Diff. Eu. 14 (1989), [12] J. Serrin, Local behaviour of solutions of uasilinear euations, Acta Math. 111 (1964), [13]. M. Stavrakakis and. B. Zograhooulos, Existence results for uasilinear ellitic systems in R, Electron. J. Diff. Ens., Vol.1999 (1999) 39, [14] F. De Thélin, J. Vélin, Existence et non-existence de solutions non triviales our des systèmes ellitiues non-linéaires, C. R. A. S. Paris 313, Serie I (1991), [15] F. De Thélin, Première valeur rore d un système ellitiue non linéaire, C. R. A. S Paris 311, Serie I (1990), [16] F. De Thélin, Résultats d existence et de non existence our la solution ositive et bornée d une EDP ellitiue non linéaire, Annales Faculté des Sciences de Toulouse VIII(3) ( ), [17] F. De Thélin, J. Vélin, Existence and nonexistence of nontrivial solutions for some nonlinear ellitic systems, Revista Matematica de la Universidad Comlutense de Madrid 6(1) (1993), [18]. S. Trudinger, On Harnack tye ineualities and their alications to uasilinear ellitic euation, Comm. Pure A. Mat. 20 (1967), Revista Matemática Comlutense 494