Multiplicity of solutions for resonant elliptic systems

J. Math. Anal. Appl. 39 006) 435 449 www.elsevier.com/locate/jmaa Multiplicity of solutions for resonant elliptic systems Marcelo F. Furtado,, Francisco O.V. de Paiva IMECC-UNICAMP, Caixa Postal 6065, 308-970 Campinas-SP, Brazil Received 6 November 004 Available online 9 July 005 Submitted by R. Manásevich Abstract We establish the existence and multiplicity of solutions for some resonant elliptic systems. The results are proved by applying minimax arguments and Morse theory. 005 Elsevier Inc. All rights reserved. Keywords: Resonant elliptic systems; Variational methods; Morse theory. Introduction Let us consider the problem { Δu = fx,u) in, u = 0 on,.) where R N is a bounded smooth domain, N 3 and f C R, R) is a subcritical nonlinearity, that is, fx,s) c + s p ) * Corresponding author. E-mail addresses: furtado@ime.unicamp.br M.F. Furtado), odair@ime.unicamp.br F.O.V. de Paiva). Supported by CAPES/Brazil. Supported by ProDoc-CAPES/Brazil. 00-47X/$ see front matter 005 Elsevier Inc. All rights reserved. doi:0.06/j.jmaa.005.06.038

436 M.F. Furtado, F.O.V. de Paiva / J. Math. Anal. Appl. 39 006) 435 449 for all x, s) R and for some <p< = N/N ). We say that the problem.) is asymptotically linear if there exists a function α such that fx,s) lim = αx). s s It is well known see [,9,3,]) that, in this case, the existence of solutions for.) is related with the interaction between the limit function αx) and the spectrum σ Δ,H0 )) of the linear problem Δu = λu in, u = 0 on..) A special class of such problems is the resonant case where αx) λ k.3) for all x and for some eigenvalue λ k σ Δ,H0 )). This kind of problem and its variants) is interesting and seems to be more difficult because the associated functional may not satisfy the classical Palais Smale condition, which is important to prove the deformation theorems that we need for applying minimax arguments. The aim of this paper is to study some classes of resonant elliptic systems. More specifically, we deal with the gradient system Δu = F u x,u,v) in, P ) Δv = F v x,u,v) in, u = v = 0 on, where R N is a bounded smooth domain and N 3. The function F C R, R) satisfies the subcritical growth condition: F ) there exist c > 0 and <p< such that Fx,z) c + z p ), x, z) R N R. Under the above hypothesis, the weak solutions of the system P ) are precisely the critical points of the C -functional I : H0 ) H 0 ) R given by Iu,v)= u + v ) dx Fx,u,v)dx. In order to obtain such critical points, we will impose some conditions in the behavior of the potential F at the infinity and at the origin. Before to presenting these assumptions, we need to introduce some notation. So, let us denote by M ) the set of all symmetric matrices of the form ) ax) bx) Ax) =, bx) cx) where the functions a,b,c C,R) satisfy

M.F. Furtado, F.O.V. de Paiva / J. Math. Anal. Appl. 39 006) 435 449 437 M ) A is cooperative, i.e., bx) 0 for all x, M ) max x max{ax),cx)} > 0. Given A M ), we can consider the weighted eigenvalue problem { Δ u ) LP) v = λax) u v) in, u = v = 0 on. In view of the conditions M ) and M ) above, we can use the spectral theory for compact operators see []) and some results contained in [7] to obtain a sequence of eigenvalues 0 <λ A) < λ A) λ k A) such that λ k A) as k see Section for more details). The problem LP) will substitute the eigenvalue problem.) in the study of our system. Since we need to control the behavior of F at infinity, we denote by z = u, v) an arbitrary vector of R and introduce the following condition: F ) there exist A M ) such that Fx,z) A x)z, z lim z z = 0, uniformly for a.e. x. Note that the natural generalization of the resonant condition.3) is to suppose A x) λ k I for some λ k σ Δ,H0 )). Here we use a more general condition, which already appears for the scalar problem in [4], by assuming that λ k A ) =, where λ k A ) is the kth positive eigenvalue of the weighted eigenvalue problem LP). In order to overcome the lack of compactness given by the resonant hypothesis, some extra conditions has been appeared in the literature see [5,6]). Here we use the nonquadraticity condition introduced by Costa and Magalhães [0]. We then suppose that F satisfies NQ) lim { } Fx,z) z Fx,z) =, uniformly for a.e. x. z Under the above hypotheses we obtain the existence of a solution for the problem P ), as stated in the result below. Theorem.. Suppose F ), F ) and NQ) holds. If λ k A ) = for some k, then the problem P ) has at least one solution. Now we observe that, if Fx,0, 0) 0, the problem P ) possesses the trivial solution u, v) = 0, 0). In this case, the key point is to assure the existence of nontrivial solutions for P ). With this aim, we need to introduce a condition that give us information about the behavior of F near the origin. More specifically, we suppose that F 0 ) there exist A 0 M ) such that Fx,z) A 0 x)z, z lim z 0 z = 0, uniformly for a.e. x.

438 M.F. Furtado, F.O.V. de Paiva / J. Math. Anal. Appl. 39 006) 435 449 Our first result concerning the multiplicity of solutions for P ) can be stated as Theorem.. Suppose Fx,0, 0) 0 and F ), F 0 ), F ) and NQ) holds. If λ k A ) = for some k and λ m A 0 )< <λ m+ A 0 ) for some m k, then the problem P ) has at least one nontrivial solution. In our next multiplicity result we suppose a nondegeneration condition and obtain the existence of two nontrivial solutions. In this case, we consider the complementary case λ A 0 )> and prove Theorem.3. Suppose Fx,0, 0) 0 and F ), F 0 ), F ) and NQ) holds. If λ k A ) = for some k 3 and λ A 0 )>, then the problem P ) has at least two nontrivial solutions provided D Fx,z) M ) for all x, z) R. In the above theorems we do not allow resonance at the first eigenvalue λ A ).However, in this case we are able to prove that the functional is coercive and we obtain the existence of two nontrivial solutions for P ). Theorem.4. Suppose Fx,0, 0) 0 and F ), F 0 ) and F ) holds. If λ m A 0 )< < λ m+ A 0 ), then the problem P ) has at least two nontrivial solutions, provided either i) λ A )>,or ii) λ A ) = and NQ) holds. For related results in the study of asymptotically linear elliptic systems we can cite [,5]. Our results are not comparable and complement it, since we deal with the weighted linear problem LP). It also complement the paper of Bartsch, Chang and Wang [4], where a Landesman Lazer condition is assumed. Finally, we would like to cite [3], where a noncooperative system is studied with no compactness condition, and the recent paper of Li and Yang [7], where the case of asymptotically linear Hamiltonian systems was studied. In the proof of our theorems we use some critical point theorems and Morse theory. As it is well known, this kind of theory is based on the existence of a linking structure and on deformation lemmas [,3,9,0]. In general, to be able to derive such deformation results, it is supposed that the functional I satisfies a compactness condition. In this article, we use the Ce) condition introduced by Cerami in [6]. We then recall that I satisfies the Cerami condition at level c R Ce) c for short), if any sequence z n ) H 0 ) H 0 ) such that Iz n) c and I z n ) + z n ) 0 possesses a convergent subsequence. The paper is organized as follows: in Section, after presenting the abstract framework, we make a detailed discussion of the weighted eigenvalue problem LP). In Section 3 we prove Theorem.. Section 4 is devoted to the proof of Theorems. and.3. Theorem.4 is proved in Section 5.

M.F. Furtado, F.O.V. de Paiva / J. Math. Anal. Appl. 39 006) 435 449 439. Abstract framework H 0 Hereafter we write u instead of ux) dx. LetH be the Hilbert space H 0 ) ) equipped with the norm z = u + v ) for all z = u, v) H. By the Sobolev theorem we know that, for any σ fixed, the embedding H L σ ) L σ ) is continuous and therefore we can find a positive constant S σ such that z σ S σ z σ..) Moreover, if σ<, the Rellich Kondrachov theorem implies that the above embedding is also compact. We proceed now with the study of the linear problem associated to P ). Let ) ax) bx) Ax) = M ) bx) cx) and consider the eigenvalue problem with weight Ax), Δu = λax)u + bx)v) in, LP) Δv = λbx)u + cx)v) in, u = v = 0 on. A simple calculation shows that λ is an eigenvalue of LP) if and only if T A u, v) = u, v), λ where T A : H H is the symmetric bounded linear operator defined by TA u, v), φ, ψ) ) ) u φ = Ax), v ψ ) ) = ax)u + bx)v φ + bx)u + cx)v ψ. Since the coefficients of A are continuous functions and the embedding H L ) L ) is compact, we can check that the operator T A is also compact. Thus, we may invoke the spectral theory for compact operators to conclude that H possesses a Hilbertian basis formed by eigenfunctions of LP). Let us denote z = u, v) and λ A) = μ A) = sup { T A z, z : z = }. Recalling that A satisfies M ) M ), we can use [7, Theorem.] see also [8, Theorem.]) to conclude that the eigenvalue μ A) is positive, simple and isolated in the

440 M.F. Furtado, F.O.V. de Paiva / J. Math. Anal. Appl. 39 006) 435 449 spectrum of T A. Moreover, if we denote by Φ A the normalized eigenfunction associated to λ A), we can suppose that the two function coordinates of Φ A are positive on. By using induction, if we suppose that μ A) > μ A) μ k A) are the k first eigenvalues of T A and {Φi A}k i= are the associated normalized eigenfunctions, we can define λ k A) = μ ka) = sup { T A z, z : z =, z span { Φ A }) },...,ΦA k. It is proved in [, Proposition.3] that, if μ k A) > 0, then it is an eigenvalue of T A with associated normalized eigenfunction Φk A. In view of the condition M ), we can argue as in the proof of [, Proposition.c)] and conclude that μ k A) > 0. Thus, we obtain a sequence of eigenvalues for LP) 0 <λ A) < λ A) λ k A) such that λ k A) as k. Moreover, if we set V k = span{φ A,...,ΦA k }, we can decompose H as H = V k Vk and the following variational inequalities hold: z λ k A) Ax)z, z, z Vk,.) and z λ k+ A) Ax)z, z, z V k..3) 3. Proof of Theorem. In this section we present the proof of Theorem.. As stated in the introduction, we will look for critical points of the C -functional I : H R given by Iz)= u + v ) Fx,z). ThefirststepistoprovethatI satisfies the Cerami condition at any level c R. In order to verify this, we first note that, given ε>0, we can use F ) to obtain R ε > 0 such that Fx,z) A x)z, z ε z, x, z R ε. This and the continuity of F provide M ε > 0 such that and Fx,z) A x)z, z ε z M ε 3.) Fx,z) A x)z, z + ε z + M ε 3.) for any x, z) R.

M.F. Furtado, F.O.V. de Paiva / J. Math. Anal. Appl. 39 006) 435 449 44 Lemma 3.. If F ) and NQ) hold, then I satisfies Ce) c for any c R. Proof. Let z n ) H be such that Iz n ) c and I z n ) H + z n ) 0. Since the nonlinearity F has subcritical growth, standard arguments [0] show that the lemma is proved if we can show that z n ) has a bounded subsequence. Suppose, by contradiction, that z n as n. Then, there exists M>0such that lim inf n Hx,z n ) = lim inf n Izn ) I z n ), z n ) M, 3.3) where Hx,z n ) = Fx,z n ) z n Fx,z n ). We obtain a contradiction by the following claim: there exists ˆ with positive measure, such that up to a subsequence, u n x) + or v n x) + as n +, for a.e. x ˆ. Assuming the claim, by Fatou s lemma and NQ), we have lim inf Hx,z n ) lim inf Hx,z n ) =, which contradicts 3.3). Now we proceed with the proof of the claim. Given ε>0, by 3.) and Iz n ) c we have, for n sufficiently large, z n c + ) + ε A x)z n,z n + z n + M ε M + A x)z n,z n + εs z n. 3.4) Defining ẑ n = û n, ˆv n ) = z n u n, v n ), we may assume that û n û, ˆv n ˆv strongly in L ) and û n x) ûx), ˆv n x) ˆvx) for a.e. x. Thus, dividing 3.4) by z n, taking n, ε 0, we conclude that A x)ẑ, ẑ, 3.5) where ẑ = û, ˆv). By denoting ) a x) b x) A x) = b x) c x) and recalling that b x) 0, we can use 3.5) and Young s inequality to obtain a x)û + b x)û ˆv + c x) ˆv ) a x) + b x) ) û + b x) + c x) ) ˆv, and therefore there exists ˆ such that ûx) 0or ˆvx) 0, a.e. x ˆ. The claim is now proved by observing that we are assuming that z n + as n +.

44 M.F. Furtado, F.O.V. de Paiva / J. Math. Anal. Appl. 39 006) 435 449 Lemma 3.. If F ) and NQ) hold, then there exists M > 0 such that Fx,z) A x)z, z M, x, z) R. Proof. Defining Gx, z) = Fx,z) A x)z, z, wehave Gx, z) z Gx, z) = Fx,z) z Fx,z). For any fixed z R with z =, it follows from NQ) that, for every M>0, there is R M > 0 such that Gx, s z) s z) Gx, s z) M, s R M. Thus, [ ] d Gx, s z) Gx, s z) s z) Gx, s z) ds s = s 3 M s 3. Integrating the above expression over the interval [t,t] [R M, + ), we obtain Gx, t z) Gx, T z) t T + M [ T ] t. Letting T +, we conclude that Gx, t z) M/, for t R M, for a.e. x. In a similar way, we have Gx, t z) M/, for t R M and a.e. x. Hence lim Gx, z) =, uniformly for a.e. x. z The lemma follows from the above equality and the continuity of G. We are now ready to prove our first theorem. Proof of Theorem.. Let k be given by the condition F ). For any j k, let Φ A j be the normalized eigenfunction associated to the jth positive eigenvalue λ j A ), as explained in Section. If we define V = span { Φ A,...,Φ A } k and W = V, we have that H = V W and dim V = k <. We claim that the functional I satisfies the geometry of the Saddle Point Theorem, that is, Claim. Iz) as z, z V. Claim. There exists γ R such that Iz) γ, for all z in W. Assuming that the above claims are true, we can use Lemma 3. and the Saddle Point Theorem [0, Theorem 4.6] see also [, Theorem.3]) to get a critical point for I.As explained before, this critical point is a solution of P ) and we have done.

M.F. Furtado, F.O.V. de Paiva / J. Math. Anal. Appl. 39 006) 435 449 443 It remains to prove the claims. Let us first consider the first one. Since k and λ A )<λ A ), we may suppose, without loss of generality, that λ k A )< λ k A ) =. Thus, the inequality.) implies that, for any z V \{0}, z λ k A ) A x)z, z < A x)z, z. Recalling that V is finite dimensional, we obtain δ>0 such that z A x)z, z δ z, z V. Thus, we can use 3.) and.) to get Iz) z A x)z, z + ε z + M ε δ + εs ) z + M ε. By taking ε = δ/s ), we conclude that Iz) as z, z V. In order the verify the second claim, we take z W and use λ k A ) =,.3) and Lemma 3., to get Iz)= z λ ka ) A x)z, z M. Hence Claim is true and the theorem is proved. Fx,z) A x)z, z ) 4. Proofs of Theorems. and.3 In this section we prove our multiplicity results concerning the resonance at higher eigenvalues. Since we will apply Morse theory, it is useful to recall some concepts. Let z 0 H be an isolated critical point of I, c = Iz 0 ) and j be a nonnegative integer. We define the jth critical group of I at z 0 as being C j I, z 0 ) = H j I c,i c \{z 0 } ), where I c ={u H : Iu) c} and H j I c,i c \{z 0 }) denotes the jth relative singular homology group with coefficients in Z, +) see [9] for more details). The critical groups will enable us to distinguish the critical points obtained by different kinds of links. Proof of Theorem.. Let w be the solution given by Theorem.. It is sufficient to show that w 0. With this aim we first note that, since w was obtained by the Saddle Point Theorem with the finite dimensional subspace having dimension k, we know by [9, Theorem.5 of Chapter ] that C k I, w) 0. 4.)

444 M.F. Furtado, F.O.V. de Paiva / J. Math. Anal. Appl. 39 006) 435 449 Recalling that F C R, R) and using some calculations, we can see that the matrix A 0 x) given by the condition F 0 ) is precisely the second derivative D Fx,0). Thus, the condition λ m A 0 )< <λ m+ A 0 ) implies that 0 is a nondegenerated critical point of I and the Morse index of I at 0, which we denote by mi, 0), is equal to m. Thus, by applying [9, Theorem 4. of Chapter ], we get { Z, if j = m, C j I, 0) = 0, otherwise. Since we are assuming that m k, the above equation and 4.) show that w 0 and the theorem is proved. Before to presenting the proof of Theorem.3, we need the following result. Lemma 4.. Suppose Fx,0, 0) 0 and F ), F 0 ) and F ) holds. If λ A 0 )> and λ k A ) = for some k 3, then the functional I has the mountain pass geometry, that is, I0) = 0 and i) there exist α, ρ > 0 such that Iz) α, for all z H such that z =ρ, ii) there exists e H such that e >ρand Ie) 0. Proof. Given ε>0, we can use F 0 ) to obtain δ ε > 0 such that Fx,z) A0 x)z, z ε z, x, z <δ ε. Moreover, condition F ) provides A ε > 0 such that Fx,z) A ε z p, x, z δ ε. It follows from the two above inequalities that Fx,z) A0 x)z, z + ε z + A ε z p 4.) for any x, z) R. Thus, for any z H, we can use the above inequality,.3) and.) to get Iz) z A0 x)z, z ε z A ε z p ) λ A 0 ) εs z A ε S p z p. Recalling that λ A 0 )>, we can take ε>0 small in such way that /λ A 0 ) εs ) = ν>0, and conclude that Iz) ν ) ν z A ε S p z p = A εs p z p z. Thus, we can check that the item i) holds for ρ = ν/4a ε S p )) /p ) and α = ρ ν/4. Let Φ A be the normalized eigenfunction associated to λ A ). In view of 3.) and.3), we have

I tφ A M.F. Furtado, F.O.V. de Paiva / J. Math. Anal. Appl. 39 006) 435 449 445 ) = tφ A t t = t F x,tφ A ) A x)φ A,Φ A t ε + λ A ) + ε ) Φ A + M ε. Φ A + M ε Since λ A )<λ k A ) =, we can choose ε>0 small in such way that /λ A ) + ε ΦA ) = κ<0, and therefore I tφ A ) t κ + M ε, as t. Thus, for sufficiently large t 0 >ρwe have It 0 Φ A ) 0. This proves item ii) and concludes the proof of the lemma. Proof of Theorem.3. Let w be the nontrivial solution given by Theorem.. In view of Lemmas 4. and 3., we can apply the Mountain Pass Theorem [0, Theorem.] to obtain a nontrivial solution w of the problem P ). In order to prove the theorem, it suffices to show that w w. Firstly, we note that C I, w) 0. Now, by the Shifting Theorem [9, Corollary 5. of Chapter ] we have that mi, w). If mi, w) = then, by the Shifting theorem, we have { Z, if j =, C j I, w) = 4.3) 0, otherwise. If mi, w) = 0 then w is a degenerated critical point and the assumption that D Fx, w) M ) implies that dim Ker I w) = see [7]). Again, we can use [9, Theorem.6 of Chapter ] to conclude that 4.3) also holds in this case. Recalling that k>, we conclude that C k I, w) = 0 and it follows from 4.) that w w. The theorem is proved. 5. Proof of Theorem.4 In this final section we consider the resonance at the first eigenvalue λ A ).Westart by proving that, in this case, the functional I is coercive on H. Lemma 5.. If F ) holds and we have either i) λ A )>,or ii) λ A ) = and NQ) holds, then the functional I is coercive on H, i.e., Iz) whenever z.

446 M.F. Furtado, F.O.V. de Paiva / J. Math. Anal. Appl. 39 006) 435 449 Proof. We first present the proof of i). Given ε>0 small we can use 3.),.3) and λ A )> to obtain Iz) z A x)z, z ε z M ε ) λ A ) εs z M ε ν z M ε 5.) for some ν>0. Thus, Iz) + as z + and the lemma is proved in this first case. For the second case ii) we note that, if Gx, z) = Fx,z) A x)z, z, then we can argue as in the proof of Lemma 3. and conclude that lim Gx, z) =, uniformly for a.e. x. 5.) z Now we suppose, by contradiction, that the lemma is false. Then there exists z n ) H such that z n as n and Iz n ) C,forsomeC R. By writing z n = u n,v n ), we define the new sequence ẑ n = z n z n = û n, ˆv n ). Passing to a subsequence if necessary, we may suppose that ẑ n ẑ = û, ˆv) weakly in H, ẑ n ẑ in L ) L ), ẑ n x) ẑx) for a.e. x. 5.3) In view of Lemma 3., we have that C Iz n ) = z n A x)z n,z n Gx, z n ) z n A x)z n,z n M. Dividing this expression by z n, letting n and using 5.3), we obtain A x)ẑ, ẑ. On the other hand, recalling the weak convergence in 5.3) and that λ A ) =, we can use.3) to get A x)ẑ, ẑ ẑ lim inf ẑ n =, n

M.F. Furtado, F.O.V. de Paiva / J. Math. Anal. Appl. 39 006) 435 449 447 from which follows that ẑ =. This and the above expression show that ẑ =±Φ A,the first eigenfunction of LP). Since in both cases the components of ẑ have the same constant sign, we conclude that u n x) and v n x) as n. This fact, λ A ) =,.3) and 5.) imply that C Iz n ) = z n A x)z n,z n Gx, z n ) Gx, z n ), as n. This is a contradiction and the lemma is proved. In order to verify that I has a local linking at the origin, we take A 0 be the function given by hypothesis F 0 ). Recalling the spectral theory of the operator T A0 discussed in Section, we set V = span { Φ A } 0,ΦA 0,...,ΦA 0 m and W = V. Thus, we have the direct decomposition H = V W. Moreover, the following holds. Lemma 5.. If F 0 ) holds and λ m A 0 )< <λ m+ A 0 ), then the functional I has a local link at the origin, i.e., i) there exists ρ > 0 such that Iz) 0, for all z V B ρ 0), ii) there exists ρ > 0 such that Iz)>0, for all nonzero z W B ρ 0). Proof. Given ε>0, we can use F 0 ) and F ) as in the proof of Theorem.3, to obtain δ ε,a ε > 0 such that Fx,z) A0 x)z, z ε z A ε z p 5.4) for any x, z) R. By taking ε>0 sufficiently small, we can use 5.4),.),.) and λ k A 0 )<to obtain Iz) z A0 x)z, z + ε z + A ε z p ) λ k A 0 ) + εs z + A ε S p z p ) z κ + A εs p z p for some κ<0 and for all z V. Hence, we have that the condition i) holds for ρ = κ/a ε S p ) /p ) > 0. In order to verify ii), we choose ε>0 small and use 4.),.3),.) and λ k+ A 0 )>, to get

448 M.F. Furtado, F.O.V. de Paiva / J. Math. Anal. Appl. 39 006) 435 449 Iz) ) λ k+ A 0 ) εs z A ε S p z p ) z μ + A εs p z p for some μ>0 and for all z W. As before, we can check that ii) holds for ρ = μ/a ε S p ) /p ) > 0. The lemma is proved. Proof of Theorem.4. In view of Lemmas 5. and 5., we can invoke the Three Critical Point Theorem [8] see also [9, Theorem.]) to obtain two nontrivial critical points for I. As before, these critical points are nontrivial solutions of P ) and the theorem is proved. References [] H. Amann, E. Zehnder, Nontrivial solutions for a class of nonresonance problems and applications to nonlinear differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 7 980) 539 603. [] A. Ambrosetti, P.H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 4 973) 349 38. [3] P. Bartolo, V. Benci, D. Fortunato, Abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity, Nonlinear Anal. 7 983) 98 0. [4] T. Bartsch, K.C. Chang, Z.Q. Wang, On the Morse index of sign change solutions of nonlinear elliptic problems, Math. Z. 33 000) 655 677. [5] T. Bartsch, S.J. Li, Critical point theory for asymptotically quadratic functionals and applications to problems with resonance, Nonlinear Anal. 8 997) 49 44. [6] G. Cerami, Un criterio de esistenza per i punti critic su varietà ilimitade, Istit. Lombardo Accad. Sci. Lett. Rend. A 978) 33 336. [7] K.C. Chang, An extension of the Hess Kato theorem to elliptic systems and its applications to multiple solutions problems, Acta Math. Sinica 5 999) 439 454. [8] K.C. Chang, Principal eigenvalue for weight in elliptic systems, Nonlinear Anal. 46 00) 49 433. [9] K.C. Chang, Infinite Dimensional Morse Theory and Multiple Solution Problems, Birkhäuser, Boston, 993. [0] D.G. Costa, C.A. Magalhães, Variational elliptic problems which are nonquadratic at infinity, Nonlinear Anal. 3 994) 40 4. [] D.G. Costa, C.A. Magalhães, A unified approach to a class of strongly indefinite functionals, J. Differential Equations 5 996) 5 547. [] D.G. defigueiredo, Positive solutions of semilinear elliptic problems, in: Differential Equations, São Paulo, 98, in: Lecture Notes in Math., vol. 957, Springer, Berlin, 98, pp. 34 87. [3] D.G. defigueiredo, Lectures on the Ekeland Variational Principle with Applications and Detours, Tata, Bombay, Springer-Verlag, Berlin, 989. [4] D.G. defigueiredo, I. Massabó, Semilinear elliptic equations with the primitive of the nonlinearity interacting with the first eigenvalue, J. Math. Anal. Appl. 56 99) 38 394. [5] M.F. Furtado, L.A. Maia, E.A.B. Silva, Solutions for a resonant elliptic system with coupling in R N,Comm. Partial Differential Equations 7 00) 55 536. [6] E.M. Landesman, A.C. Lazer, Nonlinear perturbations of linear elliptic boundary value problems at resonance, J. Math. Mech. 9 969/970) 609 63. [7] G. Li, J. Yang, Asymptotically linear elliptic systems, Comm. Partial Differential Equations 9 004) 95 954. [8] S.J. Li, J.Q. Liu, Some existence theorem on multiple critical points and their applications, Kexue Tongbao Chinese) 7 984) 05 07. [9] J. Liu, J. Su, Remarks on multiple solutions for quasi-linear resonant problems, J. Math. Anal. Appl. 58 00) 09.

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