EXISTENCE AND MULTIPLICITY OF SOLUTIONS TO EQUATIONS OF N LAPLACIAN TYPE WITH CRITICAL EXPONENTIAL GROWTH IN R N

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1 EXISTECE AD MULTIPLICITY OF SOLUTIOS TO EQUATIOS OF LAPLACIA TYPE WITH CRITICAL EXPOETIAL GROWTH I GUYE LAM AD GUOZHE LU arxiv: v1 [math.ap] 2 Jul 2011 Abstract. In this paper, we deal with the existence of solutions to the nonuniformly elliptic equation of the form 0.1 divax, u+vx u 2 u = fx,u +εhx in where 0 β <, V : R is a continuous potential satisfying Vx V 0 > 0 in and V 1 L 1 or {x : Vx M} < for every M > 0, f : R R behaveslieexp α u / 1 when u and satisfiesthe Ambrosetti- Rabinowitz condition, h W 1,, h 0 and ε is a positive parameter. In particular, in the case of Laplacian, i.e, 0.2 u+vx u 2 u = fx,u +εhx using the minimization and the Eeland variational principle, we obtain multiplicity of wea solutions of 0.2. Moreover, we prove that it is not necessary to have the small nonzero perturbation εhx to get the nontriviality of the solution to the Laplacian equation 0.3 u+vx u 2 u = fx,u Finally, we will prove the above results when our nonlinearity f doesn t satisfy the wellnown Ambrosetti-Rabinowitz condition and thus derive the existence and multiplicity of solutions for a wider class of nonlinear terms f. 1. Introduction In this paper, we consider the existence and multiplicity of nontrivial wea solution u W 1, u 0 for the nonuniformly elliptic equations of Laplacian type of the form: 1.1 divax, u+vx u 2 u = fx,u +εhx in where, in addition to some more assumptions on ax,τ and f which will be specified later in Section 2, we have ax,τ c 0 h 0 x+h 1 x τ 1 Key words and phrases. Eeland variational principle, Mountain-pass Theorem, Variational methods, Critical growth, Moser-Trudinger inequality, Laplacian, Ambrosetti-Rabinowitz condition. Corresponding Author: G. Lu at gzlu@math.wayne.edu. Research is partly supported by a US SF grant DMS

2 2 GUYE LAM AD GUOZHE LU for any τ in and a.e. x in, h 0 L / 1 and h 1 L loc R and f satisfies criticalgrowthofexponential typesuchasf : R Rbehaves lieexp α u / 1 when u and when f either satisfies or does not satisfy the Ambrosetti-Rabinowitz condition. AspecialcaseofourequationinthewholeEuclideanspacewhenax, u = u 2 u has been studied extensively, both in the case = 2 the prototype equation is the Laplacian in R 2 and in the case > 3 in for the Laplacain, see for example [11], [2], [3], [29], [19], [15, 16, 17], [5], etc. We should mention that problems involving Laplacian in bounded domains in R 2 with critical exponential growth have been studied in [4], [19], [7], [8], [12], [32], etc. and for Laplacian in bounded domains in > 2 by the authors of [2], [15], [29]. The problems of this type are important in many fields of sciences, notably the fields of electromagnetism, astronomy, and fluid dynamics, because they can be used to accurately describe the behavior of electric, gravitational, and fluid potentials. They have been extensively studied by many authors in many different cases: bounded domains and unbounded domains, different behaves of the nonlinearity, different types of boundary conditions, etc. In particular, many wors focus on the subcritical and critical growth of the nonlinearity which allows to treat the problem variationally using general critical point theory. In the case p <, by the Sobolev embedding, the subcritical and critical growth for the p Laplacian mean that the nonlinearity f cannot exceed the polynomial of degree p = p. The case p = is special, since the corresponding Sobolev space W1, p 0 Ω is a borderline case for Sobolev embeddings: one has W 1, 0 Ω L q Ω for all q 1, but W 1, 0 Ω L Ω. So, one is led to as if there is another ind of maximal growth in this situation. Indeed, this is the result of Pohozaev [30], Trudinger [34] and Moser [28], and is by now called the Moser-Trudinger inequality: it says that if Ω is a bounded domain, then 1 sup e α 1 u < Ω u W 1, 0 Ω, u L 1 where α = w and w 1 is the surface area of the unit sphere in. Moreover, the constant α is sharp in the sense that if we replace α by some β > α, the above supremum is infinite. This well-nown Moser-Trudinger inequality has been generalized in many ways. For instance, in the case of bounded domains, Adimurthi and Sandeep proved in [3] that the following inequality e α 1 u sup < α α + β u W 1, 0 Ω, u L 1 holds if and only if 1 where α > 0 and 0 β <. On the other hand, in the case of unbounded domains, B. Ruf when = 2 in [31] and Y. X. Li and B. Ruf when > 2 in [25] proved that if we replace the L -norm of u in the supremum by the standard Sobolev norm, then this supremum can still be finite Ω Ω

3 ELLIPTIC EQUATIOS WITH CRITICAL GROWTH I 3 under a certain condition for α. More precisely, they have proved the following: sup exp α u / 1 S 2 α,u u W 1, 0, u L + u L 1 where S 2 α,u = 2 =0 α u / 1.! { if α α, = + if α > α, We should mention that for α < α when = 2, the above inequality was first proved by D. Cao in [11], and proved for > 2 by Panda [29] and J.M. do O [15, 16] and Adachi and Tanaa [1]. Recently, Adimurthi and Yang generalized the above result of Li and Ruf [25] to get the following version of the singular Trudinger-Moser inequality see [5]: Lemma 1.1. For all 0 β <, 0 < α and u W 1,, there holds 1 { exp α u / 1 } S 2 α,u < Furthermore, we have for all α 1 β α and τ > 0, sup exp u 1,τ 1 1 { α u / 1 S 2 α,u } < u +τ u 1/. The inequality is sharp: for any α > where u 1,τ = 1 β α, the supremum is infinity. Motivated by this Trudinger-Moser inequality, do Ó [15, 16] and do Ó, Medeiros and Severo [17] studied the quasilinear elliptic equations when β = 0 and Adimurthi and Yang [5] studied the singular quasilinear elliptic equations for 0 β <, both with the maximal growth on the singular nonlinear term fx,u which allows them to treat the equations variationally in a subspace of W 1,. More precisely, they can find a nontrivial wea solution of mountain-pass type to the equation with the perturbation div u 2 u +Vx u 2 u = fx,u +εhx Moreover, they proved that when the positive parameter ε is small enough, the above equation has a wea solution with negative energy. However, it was not proved in [5] if those solutions are different or not. We also should stress that they need a small nonzero perturbation εhx in their equation to get the nontriviality of the solutions. In this paper, we will study further about the equation considered in the whole space [2, 15, 16, 17, 5]. More precisely, we consider the existence and multiplicity of nontrivial wea solution for the nonuniformly elliptic equations of Laplacian type of the form: 1.2 divax, u+vx u 2 u = fx,u +εhx where ax,τ c 0 h 0 x+h 1 x τ 1

4 4 GUYE LAM AD GUOZHE LU for any τ in and a.e. x in, h 0 L / 1 and h 1 L loc R. ote that the equation in [5] is a special case of our equation when ax, u = u 2 u. In fact, the elliptic equations of nonuniform type is a natural generalization of the p Laplacian equation and were studied by many authors, see [18, 20, 21, 22, 33]. As mentioned earlier, the main features of this class of equations are that they are defined in the whole and the nonuniform nonlinear operator of p-laplacian type. In spite of a possible failure of the Palais-Smale compactness condition, in this paper, we still use the Mountain-pass approach for the critical growth as in [15, 5, 16, 17] to derive a wea solution and get the nontriviality of this solution thans to the small nonzero perturbation εhx. In the case of Laplacian, i.e., and with the critical growth of the singular nonlinear term fx,u ax, u = u 2 u, our equation is exactly the equation studied in [5]: 1.3 div u 2 u +Vx u 2 u = fx,u +εhx Using the Radial lemma, Schwarz symmetrization and a modified result of Lions[27] about the singular Moser-Trudinger inequality, we will prove that two solutions derived in [5] are actually different. Thus as our second main result, we get the multiplicity of solutions to the equation 1.3. This result extends the result in [5] and also the multiplicity result in [17] β = 0 to the singular case 0 β <. Our next concern is about the existence of solution of the equation without the perturbation 1.4 div u 2 u +Vx u 2 u = fx,u. Using an approach as in [15, 16, 17], we prove that we don t even require the nonzero perturbation as in [5] to get the nontriviality of the mountain-pass type wea solution. Our main tool in this paper is critical point theory. More precisely, we will use the Mountain-pass Theorem that is proposed by Ambrosetti and Rabinowitz in the celebrated paper [6]. Critical point theory has become one of the main tools for finding solutions to elliptic equations of variational type. We stress that to use the Mountain-pass Theorem, we need to verify some types of compactness for the associated Lagrange-Euler functional, namely the Palais-Smale condition and the Cerami condition. Or at least, we must prove the boundedness of the Palais-Smale or Cerami sequence [13, 14]. In almost all of wors, we can easily establish this condition thans to the Ambrosetti-Rabinowitz AR condition, see f2 or f3 in Section 2. However, there are many interesting examples of nonlinear terms f which do not satisfy the Ambrosetti-Rabinowitz condition, but based on our theorem we can still conclude the existence and multiplicity of solutions. Thus our next result is that we will establish again the above results when the nonlinearity does not satisfy this famous AR condition. For the Laplacian equation in a bounded domain in, such a result of existence has been established by the authors in [23]. We mention in passing that the study of the existence and multiplicity results of nonuniformly elliptic equations of Laplacian type are motivated by our earlier wor on the Heisenberg group [24]. Our assumptions on the potential V are exactly those

5 ELLIPTIC EQUATIOS WITH CRITICAL GROWTH I 5 considered in [15, 16, 17, 5], namely Vx V 0 > 0 in and V 1 L 1 or {x : Vx < M} < for every M > 0. Very recently, Yang has established in [35] when ax, u = u 2 u the multiplicity of solutions when the nonlinear term f satisfies the Ambrosetti-Rabinowitz condition and the potential V is under a stronger assumption than ours. More precisely, it is assumed in [35] that V 1 L 1 1 which implies V 1 L 1 when Vx V 0 > 0 in. The stronger assumption of integrability on V 1 in [35] guarantees that the embedding E L q is compact for all 1 q <. The argument in [35], as pointed out by the author of [35], depends crucially on this compact embedding for all 1 q <. The assumption on the potential V in our paper only assures the compact embedding E L q for q. evertheless, this compact embedding for q is sufficient for us to carry out the proof of the multiplicity of solutions to equation 1.3 and existence of solutions to equation 1.4 without the perturbation term. See Proposition 5.2 and Remar 5.2 in Section 5 for more details. Moreover, our theorems hold even when f does not satisfy the Ambrosetti-Rabinowitz condition. The paper is organized as follows: In the next section, we give the main assumptions which are used throughout this paper except the last section and our main results. In Section 3, we prove some preliminary results. Section 4 is devoted to study the existence of nontrivial solutions for the nonuniformly elliptic equations of Laplacian type 1.2. The multiplicity of nontrivial solutions to Equation 1.3 is investigated in Section 5. Section 6 is about the existence of nontrivial solutions to the equation without the perturbation 1.4. Finally, in Section 7 we study the results in Sections 5 and 6 again without the well-nown Ambrosetti-Rabinowitz AR condition. 2. Assumptions and Main Results Motivated by the Trudinger-Moser inequality in Lemma 1.1, we consider here the maximal growthon the nonlinear termfx,u which allows us to treat Eq.1.2 variationallyin a subspace of W 1,. We assume that f : R R is continuous, fx,0 = 0 and f behaves lie exp α u / 1 as u. More precisely, we assume the following growth conditions on the nonlinearity fx,u as in [15, 16, 17, 5]: f1 There exist constants α 0, b 1, b 2 > 0 such that for all x,u R +, 0 < fx,u b 1 u 1 +b 2 [exp α 0 u / 1 ] S 2 α 0,u, where S 2 α 0,u = 2 =0 α 0! u / 1. f2 There exist p > such that for all x and s > 0, 0 < pfx,s = p s 0 fx,τdτ sfx,s This is the well-nown Ambrosetti-Rabinowitz condition. f3 There exist constants R 0, M 0 > 0 such that for all x and s R 0, Fx,s M 0 fx,s.

6 6 GUYE LAM AD GUOZHE LU Since we are interested in nonnegative wea solutions, it is convenient to define 2.1 fx,u = 0 for all x,u,0]. Let Abeameasurable functionon Rsuch that Ax,0 = 0andax,τ = Ax,τ is τ a Caratheodory function on R. Assume that there arepositive real numbers c 0,c 1, 1 and two nonnegative measurable functions h 0, h 1 on such that h 1 Lloc R, h 0 L / 1, h 1 x 1 for a.e. x in and the following conditions hold: A1 ax,τ c 0 h 0 x+h 1 x τ 1, τ, a.e.x A2 c 1 τ τ 1 ax,τ ax,τ 1,τ τ 1 τ,τ 1, a.e. x A3 0 ax,τ.τ Ax,τ τ, a.e.x A4 Ax,τ 0 h 1 x τ τ, a.e.x. Then A verifies the growth condition: 2.2 Ax,τ c 0 h 0 x τ +h 1 x τ τ, a.e.x ext, we introduce some notations: { E = u W 1, 0 : h 1 x u + } Vx u < R 1/ u E = h 1 x u + 1 Vx u 0, u E { } λ 1 = inf u E u x β : u E \{0} We also assume the following conditions on the potential as in [15, 16, 17, 5]: V1 V is a continuous function such that Vx V 0 > 0 for all x, we can see that E is a reflexive Banach space when endowed with the norm u E = h 1 x u Vx u 1/ and for all q <, E W 1, L q with continuous embedding. Furthermore, u E 2.3 λ 1 = inf : u E \{0} > 0 for any 0 β <. u In order to get the compactness of the embedding E L p for all p we also assume the following conditions on the potential V: V2 Vx as x ; or more generally, for every M > 0, or µ { x : Vx M } <.

7 ELLIPTIC EQUATIOS WITH CRITICAL GROWTH I 7 V3 The function [Vx] 1 belongs to L 1. ow, from f1, we obtain for all x,u R, F x,u b 3 [exp α 1 u / 1 ] S 2 α 1,u for some constants α 1, b 3 > 0. Thus, by Lemma 1.1, we have F x,u L 1 for all u W 1,. Define the functionals J, J ε : E R by J ε u = Ax, u+ 1 Vx u Fx,u Ju = 1 u + 1 Vx u ε Fx,u hu then the functionals J, J ε are well-defined by Lemma 1.1. Moreover, J, J ε are the C 1 functional on E and u,v E, DJ ε uv = ax, u v+ Vx u 2 fx,uv v ε hv R DJ uv = u 2 u v+ Vx u 2 fx,uv v. ote that in the case of Laplacian: Ax,τ = 1 τ, we choose We next state our main results. ax,τ = τ 2 τ, 0 = 1, h 1x = 1. Theorem 2.1. Suppose that V1 and V2 or V3 and f1-f2 are satisfied. Furthermore, assume that Fx,s f4 lim sup s 0+ 0 s < λ 1 uniformly in x. Then there exists ε 1 > 0 such that for each 0 < ε < ε 1, problem 1.2 has a nontrivial wea solution of mountain-pass type. Theorem 2.2. Suppose that V1 and V2 or V3 and f1-f3 are satisfied. Furthermore, assume that Fx,s f4 lim sup < λ s 0+ s 1 uniformly in x. and there exists r > 0 such that f5 limsfx,sexp α 0 s / 1 s 1 1 β > [ ] > 0 r β β eα d β/ +Cr β r β α 0 β

8 8 GUYE LAM AD GUOZHE LU uniformly on compact subsets of where d and C will be defined in section 3. Then there exists ε 2 > 0, such that for each 0 < ε < ε 2, problem 1.3 has at least two nontrivial wea solutions and one of them has a negative energy. Theorem 2.3. Under the same hypotheses in Theorem 2.2, the problem without the perturbation 1.4 has a nontrivial wea solution. As we remared earlier in the introduction, all the main theorems above remain to hold when the nonlinear term f does not satisfy the Ambrosetti-Rabinowitz condition. As a result, we then establish the existence and multiplicity of solutions in a wider class of nonlinear terms. See Section 7 for more details. 3. Preliminary Results First, we recall what we call the Radial Lemma see [10, 17] which asserts: ux ω 1 u x, x R \{0} for all u W 1, radially symmetric. Using this Radial Lemma, we can prove the following two lemmas Lemmas 3.1 and 3.2 with an easy adaptation from Lemma 2.2 and Lemma 2.3 in [17] for β = 0 and Lemma 4.2 in [5]. Lemma 3.1. For κ > 0, 0 β < and u E M with M sufficiently small and q >, we have [ exp κ u / 1 ] S 2 κ,u u q C,κ u q E. Lemma 3.2. Let κ > 0, 0 β <, u E and u E M such that M / 1 < 1 β ακ, then for some p >. ext, we have exp{α w / 1 } [ exp κ u / 1 ] S 2 κ,u u C,M,κ u p Lemma 3.3. Let {w } W 1, Ω where Ω is a bounded open set in, w L Ω 1. If w w 0 wealy and almost everywhere, w w almost everywhere, then 1/ 1. α 1 w L Ω is bounded in L 1 Ω for 0 < α < 1 β Proof. Using the Brezis-Lieb Lemma in [10], we deduce that w L Ω w w L Ω w L Ω. Thus for large enough and δ > 0 small enough: 0 < α1+δ w w / 1 L Ω < α 1 β By the singular Trudinger-Moser inequality on bounded domains [3], we get the conclusion..

9 ELLIPTIC EQUATIOS WITH CRITICAL GROWTH I 9 InthenexttwolemmaswechecthatthefunctionalJ ε satisfiesthegeometricconditions of the mountain-pass theorem. Then, we are going to use a mountain-pass theorem without a compactness condition such as theone of the PS type to prove the existence of the solution. This version of the mountain-pass theorem is a consequence of the Eeland s variational principle. Lemma 3.4. Suppose that V1, f1 and f4 hold. Then there exists ε 1 > 0 such that for 0 < ε < ε 1, there exists ρ ε > 0 such that J ε u > 0 if u E = ρ ε. Furthermore, ρ ε can be chosen such that ρ ε 0 as ε 0. Proof. From f4, there exist τ, δ > 0 such that u δ implies 3.1 Fx,u 0 λ 1 τ u for all x. Moreover, using f1 for each q >, we can find a constant C = Cq,δ such that 3.2 Fx,u C u q[ exp κ u / 1 ] S 2 κ,u for u δ and x. From 3.1 and 3.2 we have Fx,u 0 λ 1 τ u +C u q[ exp κ u / 1 ] S 2 κ,u for all x,u R. ow, by A4, Lemma 3.2, 2.3 and the continuous embedding E L, we obtain J ε u 0 u E u 0λ 1 τ C u q E ε h u E 0 1 λ 1 τ u E λ 1 C u q E ε h u E Thus [ 3.3 J ε u u E 0 1 λ 1 τ λ 1 Since τ > 0 and q >, we may choose ρ > 0 such that 0 ] u 1 E C u q 1 E ε h 1 λ 1 τ λ 1 ρ 1 Cρ q 1 > 0. Thus, if ε is sufficiently small then we can find some ρ ε > 0 such that J ε u > 0 if u = ρ ε and even ρ ε 0 as ε 0. Lemma 3.5. There exists e E with e E > ρ ε such that J ε e < inf u =ρ ε J ε u. Proof. Let u E \ {0}, u 0 with compact support Ω = suppu. By f2, we have that for p >, there exists a positive constant C > 0 such that 3.4 s 0, x Ω : F x,s cs p d. Then by 2.2, we get J ε tu Ct h 0 x u +Ct u E Ctp Ω Ω u p x β+c +εt Ω hu Since p >, we have J ε tu as t. Setting e = tu with t sufficiently large, we get the conclusion.

10 10 GUYE LAM AD GUOZHE LU ow, we define the Moser Functions which have been frequently used in the literature see, for example, [15], [17], [5]: m l x,r = 1 ω 1/ 1 logl 1/ log r x logl 1/ if x r l if r x r l 0 if x r We then immediately have m l.,r W 1,, the support of m l x,r is the ball B r, and 3.5 m l x,r = 1, and m l W 1, = ! r +o R logl l 1. Then maxvx m l E 1+ x r 1! r +o logl l 1. Consider m l x,r = m l x,r/ m l E, we can write 3.6 m / 1 l x,r = ω 1/ 1 1 logl +d l for x r/l, Using 3.5, we conclude that m l 1 as l. Consequently, as l, logl d = liminf d l l d l 2! d max x r Vxω 1/ 1 1 r. The following lemma was established in [17] when β = 0. We adapt the proof given in [17] to our case 0 β <. See also [24] for a similar result on the Heisenberg group. Lemma 3.6. Suppose that V1 and f1-f5 hold. Then there exists such that {t } max F x,tm < 1 1 β α t 0 α 0 Proof. Choose r > 0 as in the assumption f5 and β 0 > 0 such that 3.8 lim sfx,sexp α 0 s / 1 β 0 s 1 1 β > [ ], r β β eα d β/ +Cr β r β α 0 β

11 where C = lim ζ log ELLIPTIC EQUATIOS WITH CRITICAL GROWTH I 11 ζ 1 C 1 e βlogn. β 0 exp [ βlog s / 1 ζ s ] ds > 0, ζ = m, Suppose, by contradiction, that for all we get {t } max F x,tm 1 β t 0 1 α α 0 where m x = m x,r. By 3.4, for each there exists t > 0 such that t F x,t m {t } = max x F x,tm β t 0 Thus t F x,t m 1 β From Fx,u 0, we obtain 1 β 3.9 t α α 0 Since at t = t we have it follows that 3.10 t = d t dt F x,tm t m f x,t m = x r 1 α. α 0 = 0 f x,t m t m Using hypothesis f5, given τ > 0 there exists R τ > 0 such that for all u R τ and x r, we have 3.11 ufx,u β 0 τexp α 0 u / 1. From 3.10 and 3.11, for large, we obtain exp α 0 t m / 1 t β 0 τ x r = β 0 τ ω 1 r βexp α 0 t β / 1 Thus, setting L = α 0 log t / 1 α ω 1/ 1 1 log +α 0 t / 1 d +α 0 t / 1 d logt βlog

12 12 GUYE LAM AD GUOZHE LU we have 1 β 0 τ ω 1 β r β expl Consequently, the sequence t is bounded. Otherwise, up to subsequences, we would have lim L = which leads to a contradiction. Moreover, by 3.7, 3.9 and t β 0 τ ω 1 β r β exp it follows that [ α 0t / 1 β α ] log +α 0 t / 1 d 3.12 t β 1 α α 0 Setting From 3.10 and 3.11 we have A = {x B r : t m R τ } and B = B r \A 3.13 exp α 0 t m / 1 t β 0 τ + x r α 0 t m / 1 β 0 τ B exp B t m f x,t m otice that m x 0 and the characteristic functions χ B 1 for almost everywhere x in B r. Therefore the Lebesgue s dominated convergence theorem implies B t m f x,t m 0 and exp α 0 t m / 1 B ω 1 β r β Moreover, using that t 1 β α α 0

13 ELLIPTIC EQUATIOS WITH CRITICAL GROWTH I 13 we have x r = + exp α 0 t m / 1 exp α m / 1 β/ exp α m / 1 β/ exp α m / 1 β/ x r x r/ r/ x r and x r/ = exp α m / 1 β/ [ ] exp α ω 1/ 1 1 log β/ +d α β/ r β d β+α log β/ x r/ = ω 1 β = ω 1 β r β α d log β/. ow, using the change of variable x = log r s ζ log with ζ = m by straightforward computation, we have exp α m / 1 β/ r/ x r = ω 1 r β ζ log ζ 1 which converges to Cω 1 r β as where C = lim ζ log ζ exp [ βlog s / 1 ζ s ] ds exp [ βlog s / 1 ζ s ] ds > 0.

14 14 GUYE LAM AD GUOZHE LU Finally, taing in 3.13, using 3.12 and using 3.7 see [15, 17], we obtain 1 [ β α ω 1 β 0 τ α 0 β r β e αd β/ +Cω 1 r β ω ] 1 β r β which implies that β β [ ]. r β β eα d β/ +Cr β r β α 0 β This contradicts to 3.8, and the proof is complete. 4. The existence of solution for the problem 1.2 It is well nown that the failure of the PS compactness condition creates difficulties in studying this class of elliptic problems involving critical growth and unbounded domains. In next several lemmas, instead ofps sequence, we will use and analyze the compactness of Cerami sequences of J ε. Lemma 4.1. Let u E be an arbitrary Cerami sequence of J ε, i.e., J ε u c, 1+ u E DJ ε u E 0 as. Then there exists a subsequence of u still denoted by u and u E such that fx,u fx,u strongly in L 1 x β loc R u x ux almost everywhere in R ax, u ax, u wealy in L / 1 loc R u u wealy in E Furthermore u is a wea solution of 1.2. For simplicity, we will only setch the proof where includes the nonuniform terms ax, u and Ax, u. Proof. Let v E, then we have 4.1 Ax, u + 1 Vx u Fx,u and DJ ε u v = ax, u v+ Vx u 2 u v 4.2 τ v E 1+ u E ε hu c fx,u v ε hv R

15 ELLIPTIC EQUATIOS WITH CRITICAL GROWTH I 15 where τ 0 as. Choosing v = u in 4.2 and by A3, we get fx,u u +ε hu Ax, u Vx u 2 u u τ E 1+ u E 0 This together with 4.1, f2 and A4 leads to p 1 u E C1+ u E and hence u E is bounded and thus fx,u u Fx,u 4.3 C, C. Thans to the assumptions on the potential V, the embedding E L q is compact for all q, by extracting a subsequence, we can assume that Thans to Lemma 2.1 in [19], we have u u wealy in E and for almost all x. 4.4 f x,u n f x,u in L 1 loc R. ext, up to a subsequence, we can define an energy concentration set for any fixed δ > 0, { Σ δ = x : lim lim u r 0 B + u } δ rx Since u is bounded, Σ δ must be a finite set. Adapting an argument similar to [5] we omit the details here, we can prove that for any compact set K \Σ δ, f x,u u f x,uu 4.5 lim = 0 K ext we will prove that for any compact set K \Σ δ, 4.6 lim u u K = 0 It is enough to prove for any x \Σ δ, and B r x,r \Σ δ, there holds 4.7 lim u u B = 0 r/2 x

16 16 GUYE LAM AD GUOZHE LU For this purpose, we tae φ C0 B r x with 0 φ 1 and φ = 1 on B r/2 x. Obviously φu is a bounded sequence. Choose h = φu and h = φu in 4.2, we have: φax, u ax, u u u ax, u φu u B rx + B rx φax, u u u + +τ φu E +τ φu E ε B rx B rx φhu u B rx φu u f x,u ote that by Holder s inequality and the compact embedding of E L Ω, we get 4.8 lim ax, u φu u = 0 B rx Since u u and u u, there holds 4.9 lim φax, u u u = 0 and lim φhu u = 0 B rx B rx This implies that So we can conclude that lim B rx lim φu uf x,u = 0 B rx φax, u ax, u u u = 0 and hence we get 4.7 by A2. Thus we have 4.6 by a covering argument. Since Σ δ is finite, it follows that u converges to u almost everywhere. This immediately implies, up to a subsequence, ax, u ax, u wealy in L / 1 2 loc R. Using all these facts, letting tend to infinity in 4.2 and combining with 4.4, we obtain DJ ε u,v = 0 v C 0 R. This completes the proof of the Lemma. ow, we are ready to prove Theorem 2.1. The existence of the solution of 1.2 follows by a standard mountain-pass procedure The proof of Theorem 2.1. Proposition 4.1. Under the assumptions V1 and V2 or V3, and f1-f4, there exists ε 1 > 0 such that for each 0 < ε < ε 1, the problem 1.2 has a solution u M via mountain-pass theorem. Proof. For ε sufficiently small, by Lemmas 3.4 and 3.5, J ε satisfies the hypotheses of the mountain-pass theorem except possibly for the PS condition. Thus, using the mountainpass theorem without the PS condition, we can find a sequence u in E such that J ε u c M > 0 and 1+ u E DJ ε u 0 where c M is the mountain-pass level of J ε. ow, by Lemma 4.1, the sequence u converges wealy to a wea solution u M of 1.2 in E. Moreover, u M 0 since h 0.

17 ELLIPTIC EQUATIOS WITH CRITICAL GROWTH I The multiplicity results of the Problem 1.3 In this section, we deal with the problem 1.3. ote that this is the special case of the problem 1.2 with Ax,τ = τ. Some preliminary lemmas in the case β = 0 were treated in [15, 17]. We will not include details of the proof here, but refer the reader to [15, 17]. The ey ingredient of this section is the proof of Proposition 5.2 which is substantially different from those in [15, 17]. Lemma 5.1. There exists η > 0 and v E with v E = 1 such that J ε tv < 0 for all 0 < t < η. In particular, inf J ε u < 0. u E η Corollary 5.1. Under the hypotheses V1 and f1-f5, if ε is sufficiently small then {t } max J εtm = max F x,tm t εhm t 0 t 0 < 1 β α α 0 ote that we can conclude by inequality 3.3 and Lemma 5.1 that 5.1 < c 0 = inf u E ρ ε J ε u < 0. 1 ext, we will prove that this infimum is achieved and generate a solution. In order to obtain convergence results, we need to improve the estimate of Lemma 3.6. Corollary 5.2. Under the hypotheses V1 and f1-f5, there exist ε 2 0,ε 1 ] and u W 1, with compact support such that for all 0 < ε < ε 2, J ε tu < c β α α 0 1 for all t 0 ε 0 Proof. It is possible to increase the infimum c 0 by reducing ε. By Lemma 3.4, ρ ε 0. ε 0 Consequently, c 0 0. Thus there exists ε 2 > 0 such that if 0 < ε < ε 2 then, by Corollary 5.1, we have max J εtm < c β α t 0 α 0 Taing u = m W 1,, the result follows. Lemma 5.2. If u is a Cerami sequence for J ε at any level with 1/ β 5.2 lim inf u α E < α 0 then u possesses a subsequence which converges strongly to a solution u 0 of 1.3. Proof. See Lemma 5.2 in [17] for β = 0 and Lemma 4.6 in [5] for 0 β < Proof of Theorem 2.2. The proof of the existence of the second solution of 1.3 follows by a minimization argument and Eeland s variational principle. Proposition 5.1. There exists ε 2 > 0 such that for each ε with 0 < ε < ε 2, Eq. 1.3 has a minimum type solution u 0 with J ε u 0 = c 0 < 0, where c 0 is defined in 5.1.

18 18 GUYE LAM AD GUOZHE LU Proof. Let ρ ε be as in Lemma 3.4. We can choose ε 2 > 0 sufficiently small such that 1/ β α ρ ε < α 0 Since B ρε is a complete metric space with the metric given by the norm of E, convex and the functional J ε is of class C 1 and bounded below on B ρε, by the Eeland s variational principle there exists a sequence u in B ρε such that J ε u c 0 = inf u E ρ ε J ε u and DJ ε u 0 Observing that 1/ β α u E ρ ε < α 0 by Lemma 5.2 it follows that there exists a subsequence of u which converges to a solution u 0 of 1.3. Therefore, J ε u 0 = c 0 < 0. Remar 5.1. By Corollary 5.2, we can conclude that 0 < c M < c β α α 0 1 Proposition 5.2. If ε 2 > 0 is sufficiently small, then the solutions of 1.4 obtained in Propositions 4.1 and 5.1 are distinct. Remar 5.2. Before we give a proof of the proposition, we lie to mae some remars. We note the following Hardy-Littlewood inequality holds for nonegative functions f and g in : fxgx f xg x where f and g are symmetric and decreasing rearrangement of f and g respectively. However, the following inequality: fx f x does not hold for all R > 0 in general. Therefore, we will avoid using the symmetrization argument when we prove Fx,v Fx,u 0. evertheless, this can be taen care by a double truncation argument. This argument differs from those given in [15, 16, 17, 35]. Using this argument, the compact embedding E L q for q is sufficient. Proof. By Propositions 4.1 and 5.1, there exist sequences u, v in E such that and u u 0, J ε u c 0 < 0, DJ ε u u 0 v u M, J ε v c M > 0, DJ ε v v 0, v x u M x almost everywhere in

19 ELLIPTIC EQUATIOS WITH CRITICAL GROWTH I 19 ow, suppose by contradiction that u 0 = u M. As in the proof of Lemma 4.1 we obtain 5.3 Moreover, by f2, f3 fx,v fx,u 0 in L 1 B R for all R > 0 Fx,v R 0fx,v + M 0fx,v so by the Generalized Lebesgue s Dominated Convergence Theorem, We will prove that Fx,v Fx,u 0 in L 1 B R. Fx,v Fx,u 0. It s sufficient to prove that given δ > 0, there exists R > 0 such that Fx,v Fx,u 0 3δ and 3δ. To prove it, we recall the following facts from our assumptions on nonlinearity: there exists c > 0 such that for all x,s R + : 5.4 Fx,s c s +cfx,s Fx,s c s +crα 0,ss fx,v v Fx,v C, C. First, we will prove it for the case β > 0. We have that Fx,v v c x +c β v >A c R β v E +c1 A v >A fx,v fx,v v Since v E is bounded and using 5.4, we can first choose A such that c 1 fx,v v < δ for all A and then choose R such that c R β v E < δ.

20 20 GUYE LAM AD GUOZHE LU which thus v >A Fx,v 2δ. ow, note that with such A, we have for s A: So we get Fx,s c s +crα 0,ss c s α j 0 +c j! s j/ 1+1 v A s [c+c j= 1 j= 1 Cα 0,A s. Fx,v Cα 0,A R β α j 0 j! Aj/ 1+1 v A v Cα 0,A v R β E. Again, note that v E is bounded, we can choose R such that Fx,v δ. v A In conclusion, we can choose R > 0 such that Fx,v 3δ. Similarly, we can choose R > 0 such that Fx,u 0 3δ. ow, if β = 0, similarly, we have Fx,v c v +c ] fx,v v >A c A c A v +1 E v >A v +1 +c 1 A +c 1 A fx,v v fx,v v

21 ELLIPTIC EQUATIOS WITH CRITICAL GROWTH I 21 so since v E is bounded and by 5.4, we can choose A such that Fx,v 2δ. v >A ext, we have Fx,v Cα 0,A v v A v A 2 1 Cα 0,A v A v u 0 + v A u 0. ow, using the compactness of embedding E L q, q and noticing that v u 0, again we can choose R such that Fx,v δ. v A Combining all the above estimates, we have the fact that Fx,v Fx,u since δ is arbitrary and 5.3 holds. The remaining argument is similar to that in [17] when β = 0. For 0 β <, we also refer to [24]. For completeness, we will include a proof here. From the above convergence formula 5.5, we have 5.6 lim v = c M lim Vx v + Fx,u 0 +ε hu 0 ow, let v u 0 w = and w 0 = v lim v we have w = 1 for all and w w 0 in D 1,, the closure of the space C 0 R endowed withthenorm ϕ. Inparticular, w 0 1andw BR w 0 BR in W 1, B R for all R > 0. We claim that w 0 < 1. Indeed, if w 0 = 1, then we have lim v = u 0 and thus v u 0 in W 1, since v u 0 in L q, q. So we can find g W 1, for some q such that v x gx almost everywhere in. From assumption f1, we have for some α 1 > α 0 that [ fx,ss b 1 s +C exp α 1 s / 1 ] S 2 α 1,s

22 22 GUYE LAM AD GUOZHE LU for all x,s R. Thus, [ fx,v v v exp α 1 v / 1 ] S 2 α 1,v b 1 x +C β [ v exp α 1 g / 1 ] S 2 α 1,g b 1 x +C β almost everywhere in. ow, by the Lebesgue s dominated convergence theorem, fx,v v fx,u 0 u 0 = lim Similarly, since u u 0 in E, we also have lim fx,u u = fx,u 0 u 0 ow, note that DJ ε u u = u E fx,u u εhu 0 and DJ ε v v = v E fx,v v εhv 0 we conclude that lim v E = lim u E = u 0 E and thus J ε v J ε u 0 = c 0 < 0 and this is a contradiction. So w 0 < 1. Using Remar 5.1 we have c M J ε u 0 < 1 1 β α α 0 and thus α 0 < β α [ c M J ε u 0 ] 1/ 1 ow if we choose q > 1 sufficiently close to 1 and set Lw = c M 1 Vx w F x,w + +ε hw then for some δ > 0, qα 0 v / 1 β = β α v / 1 [ c M J ε u 0 ] 1/ 1 δ α Lv 1/ 1 +o 1 [ c M J ε u 0 ] 1/ 1 δ

23 ote that lim Lv 1 = c M lim and 1 c M lim c M J ε u 0 ELLIPTIC EQUATIOS WITH CRITICAL GROWTH I 23 Vx v + so for,r sufficiently large, Vx v + F x,u 0 F x,u 0 +ε hu 0 +ε hu 0 +o 1 1 Rw 0 qα 0 v / 1 β α [ ] 1/ 1 δ 1 w 0 L B R By Lemma 3.3, note that w w 0 almost everywhere since v x u M x = u 0 x almost everywhere in : 5.7 exp qα 0 v B/ 1 By f 1 and Holder s inequality, f x,v v u 0 v 1 v u 0 b 1 +b 2 v 1/ b 1 x v u 0 β 1/q v u 0 q exp +b 2 B R where q = q/q 1. By 5.7, we have w / 1 C v u 0 exp α 0 v / 1 BR 1/ qα 0 v / 1 w / 1 f x,v v u 0 C v u 0 v u 0 1 +C / 2 /q Using the Holder inequality and the compact embedding E L q, q, we get q. 1/q

24 24 GUYE LAM AD GUOZHE LU v u 0 = x <1 x <1 v u s 0 as x 1 1/s for some s > 1 sufficiently close to 1. Similarly, v u 0 q v u 0 v u 0 s x <1 0. 1/s + v u 0 Thus we can conclude that v 2 v v u 0 + Vx v 2 v v u 0 0 since DJ ε v v u 0 0. On the other hand, since v u 0 and we have u 0 2 u 0 v u 0 0 Vx u 0 2 u 0 v u 0 0 v u 0 + Vx v u 0 R C 1 v 2 v u 0 2 u 0 v u 0 +C 2 Vx where we did use the inequality v 2 v u 0 2 u 0 v u 0 x 2 x y 2 y x y 2 2 x y. So we can conclude that v u 0 in E. Thus J ε v J ε u 0 = c 0 < 0. Again, this is a contradiction. The proof is thus complete. 6. The existence result to the problem 1.4 In this section, we deal with the problem 1.4. The main result of ours shows that we don t need a nonzero small perturbation in this case to guarantee the existence of a solution.

25 ELLIPTIC EQUATIOS WITH CRITICAL GROWTH I Proof of Theorem 2.3. It s similar to the proof of Theorems 2.1 and 2.2. We can find a sequence v in E such that J v c M > 0 and 1+ v E DJ v 0 wherec M isthemountain-passlevel ofj. ow, bylemma4.1, thesequence v converges wealy to a wea solution v of 1.4 in E. ow, suppose that v = 0. Similarly as in the proof of Proposition 5.2, we have that: 6.1 So F x,v 0 lim v E = lim J v + ote that by Lemma 3.6, we have 0 < C M < 1 Thus by f1, we have Fx,v β α 1, α0 so 1/ β α lim sup v E <. α 0 = C M fx,v v v b 1 x +b Rα 0,v v β 2 ote that b 1 v x +b β 2 Rα 0,v v 0 since bylemma3.2andbythecompact embedding E L s, s, CM, v s 0. Moreover, E L and x 1 x 1 v v Rα 0,v v 0 again by the compact embedding v C v r 0 by Holder s inequality and by the compact embedding E L s, s. So we can conclude that fx,v v = 0 and it s impossible. So we get the non- which thus lim v E = lim triviality of the solution. fx,v v 0

26 26 GUYE LAM AD GUOZHE LU 7. Existence and Multiplicity Without the Ambrosetti-Rabinowitz condition The main purpose of this section is to prove that all of the results of existence and multiplicity in Sections 5 and 6 hold even when the nonlinear term f does not satisfy the Ambrosetti-Rabinowitz condition. It is not difficult to see that there are many interesting examples of such f which do not satisfy the Ambrosetti-Rabinowitz condition, but satisfy our weaer conditions listed below. In this section, instead of conditions f2 and f3, we assume that f2 Hx,t Hx,sforall0 < t < s, x where Hx,u = ufx,u Fx,u. f3 There exists c > 0 such that for all x,s R + : Fx,s c s +cfx,s. f4 Fx,u lim = uniformly on x. u u We should stress that f1+f3 will imply f3. The ey to establish the results in earlier sections is to prove that the Cerami sequence [13, 14] associated to the Lagrange-Euler functional is bounded. Once we will have proved this, the remaining should be the same as in previous sections. Therefore, we only include the proof of this essential ingredient in this section. Lemma 7.1. Let {u } be an arbitrary Cerami sequence associated to the functional Iu = 1 Fx,u u such that 1 u Fx,u C M 1+ u u 1 u v+ Vx u 1 fx,u v u v ε v ε where C M 0, 1 β α α0. Then {u } is bounded up to a subsequence. Proof. Suppose that 7.1 u Setting v = u u then v = 1. We can then suppose that v v in E up to a subsequence. We may similarly show that v + v+ in E, where w + = max{w,0}. Thans to the assumptions on the potential { V, the embedding E L q is compact for all q. So, we can v + assume that x v+ x a.e. in v + v+ in L q, q. We wish to show that v+ = 0 a.e..

27 ELLIPTIC EQUATIOS WITH CRITICAL GROWTH I 27 Indeed, if S + = { x : v + x > 0 } has a positive measure, then in S +, we have and thus by f4 : This means that lim u+ x = lim v+ x u = + F x,u + lim x u + x = + a.e. in S+ F x,u lim x n u + x and so F x,u x liminf v + x = + a.e. in S + v + x = + u + x However, since {u } is the arbitrary Cerami sequence at level C M, we see that which implies that and then 7.4 u = C M + lim inf = liminf = liminf Fx,u + x Fx,u + x + F x,u + x u + x F x,u + x u v + x Fx,u + x +o1 C M + Fx,u + x +o1 = 1 ow, note that Fx,s 0, by Fatou s lemma and 7.3 and 7.4, we get a contradiction. So v 0 a.e. which means that v + 0 in E. Letting t [0,1] such that For any given R 0, 1 β α α 0 I t u = max Itu t [0,1] 1, let ε = 1 α β α / 1 0 > 0, since f has critical growth f1 on, there exists C = CR > 0 such that 7.5 Fx,s C s 1 β + α α / 1 0 Rα 0 +ε,s, x,s R.

28 28 GUYE LAM AD GUOZHE LU Since u, we have 7.6 It u I R u u = I Rv and by 7.5, v = 1 and the fact that Fx,v 7.7 = + Fx,v, we get I Rv C C C v + x β v + x β v + x β 1 β α 1 1 β α 1 1 β α 1 α 0 α 0 α 0 R R R α0 +ε,r v + α 0 +ε 1, v + 1 β α, v Since v + 0 in E and the embedding E Lp is compact for all p, using the Holder inequality, we can show easily that v + x R1 β α, v x is bounded by a universal C. Thus using 7.6 andletting in7.7, andthenletting R we get 7.8 lim inf I t u 1 1 β 1 α > C M α 0 0. Also, by Lemma 1.1, [ 1 β α α 0 ] 1, ote that I0 = 0 and Iu C M, we can suppose that t 0,1. Thus since DIt u t u = 0, t u f x,t u t u = By f2 : I t u = t u F x,t u [f x,t u t u F x,t u ] = [f x,u u F x,u ].

29 Moreover, we have ELLIPTIC EQUATIOS WITH CRITICAL GROWTH I 29 [f x,u u F x,u ] = u +C M u +o1 = C M +o1 which is a contraction to 7.8. This proves that {u } is bounded in E. References [1] Adachi, S. and Tanaa, K Trudinger type inequalities in and their best exponents. Proc. of the Amer. Math. Soc , [2] Adimurthi Existence of positive solutions of the semilinear Dirichlet problem with critical growth for the -Laplacian. Annali della Scuola ormale Superiore di Pisa , [3] Adimurthi; Sandeep, K. A singular Moser-Trudinger embedding and its applications. odea onlinear Differential Equations Appl , no. 5-6, [4] Adimurthi and Yadava, S.L. Multiplicity results for semilinear elliptic equations in bounded domain of R 2 involving critical exponent. Ann. Scuola orm. Sup. Pisa Cl. Sci , [5] Adimurthi; Yang, Yunyan An interpolation of Hardy inequality and Trundinger-Moser inequality in and its applications. Int. Math. Res. ot. IMR 2010, no. 13, [6] Ambrosetti, Antonio; Rabinowitz, Paul H. Dual variational methods in critical point theory and applications. J. Functional Analysis , [7] Atinson, F.V. and Peletier, L.A. Elliptic equations with nearly critical growth. J. Differetnial Equations, , [8] Atinson, F.V. and Peletier, L.A. Ground states and Dirichlet problems for u = fu in R 2. Arch. Rational Mech. Anal , [9] Berestyci, H.; Lions, P.-L. onlinear scalar field equations. I. Existence of a ground state. Arch. Rational Mech. Anal , no. 4, [10] Brézis, Haïm; Lieb, Elliott A relation between pointwise convergence of functions and convergence of functionals. Proc. Amer. Math. Soc , no. 3, [11] Cao, Daomin ontrivial solution of semilinear elliptic equation with critical exponent in R 2. Comm. Partial Differential Equations , no. 3-4, [12] L. Carleson and A. Chang On the existence of an extremal function for an inequality of J. Moser. Bull. Sci. Math , [13] Cerami, Giovanna An existence criterion for the critical points on unbounded manifolds. Italian. Istit. Lombardo Accad. Sci. Lett. Rend. A, , no. 2, [14] Cerami, Giovanna On the existence of eigenvalues for a nonlinear boundary value problem. Italian. Ann. Mat. Pura Appl. 4, , [15] do Ó, João Marcos -Laplacian equations in with critical growth. Abstr. Appl. Anal , no. 3-4, [16] do Ó, João Marcos Semilinear Dirichlet problems for the -Laplacian in with nonlinearities in the critical growth range. Differential Integral Equations , no. 5, [17] do Ó, João Marcos; Medeiros, Everaldo; Severo, Uberlandio On a quasilinear nonhomogeneous elliptic equation with critical growth in R n. J. Differential Equations , no. 4, [18] Duc, Duong Minh; Thanh Vu, guyen onuniformly elliptic equations of p-laplacian type. onlinear Anal , no. 8, [19] de Figueiredo, D. G.; Miyagai, O. H.; Ruf, B. Elliptic equations in R 2 with nonlinearities in the critical growth range. Calc. Var. Partial Differential Equations , no. 2, [20] Gregori, Giovanni Generalized solutions for a class of non-uniformly elliptic equations in divergence form. Comm. Partial Differential Equations , no. 3-4,

30 30 GUYE LAM AD GUOZHE LU [21] Toan, Hoang Quoc; go, Quoc-Anh Multiplicity of wea solutions for a class of nonuniformly elliptic equations of p-laplacian type. onlinear Anal , no. 4, [22] Huang, Yisheng; Zhou, Yuying Multiple solutions for a class of nonlinear elliptic problems with a p-laplacian type operator. onlinear Anal , no. 7-8, [23] Lam, guyen; Lu, Guozhen -Laplacian equations in with subcritical and critical growth without the Ambrosetti-Rabinowitz condition. arxiv: [24] Lam, guyen; Lu, Guozhen Existence and multiplicity of solutions to the nonuniformly subelliptic equations of Q sublaplacian type with critical growth in H n. Preprint, 2010 [25] Li, Yuxiang; Ruf, Bernhard A sharp Trudinger-Moser type inequality for unbounded domains in R n. Indiana Univ. Math. J , no. 1, [26] Lieb, Elliott H.; Loss, Michael Analysis. Second edition. Graduate Studies in Mathematics, 14. American Mathematical Society, Providence, RI, xxii+346 pp. ISB: [27] Lions, P.-L. The concentration-compactness principle in the calculus of variations. The limit case. I. Rev. Mat. Iberoamericana , no. 1, [28] Moser, J. A sharp form of an inequality by. Trudinger. Indiana Univ. Math. J /71, [29] Panda, R. ontrivial solution of a quasilinear elliptic equation with critical growth in R n Proc. Indian Acad. Sci. Math. Sci , no. 4, [30] Pohožaev, S. I. On the eigenfunctions of the equation u+λfu = 0. Russian Dol. Aad. au SSSR [31] Ruf, Bernhard A sharp Trudinger-Moser type inequality for unbounded domains in R 2. J. Funct. Anal , no. 2, [32] Shaw, Mei-Chi Eigenfunctions of the nonlinear equation u+fx,u = 0 in R 2. Pacific J. Math , [33] Tersenov, A. S. On quasilinear non-uniformly elliptic equations in some non-convex domains. Comm. Partial Differential Equations , no , [34] Trudinger, eil S. On imbeddings into Orlicz spaces and some applications. J. Math. Mech [35] Yang, Yunyan Existence of positive solutions to quasilinear equations with exponential growth in the whole Euclidean space. arxiv: v1. guyen Lam and Guozhen Lu, Department of Mathematics, Wayne State University, Detroit, MI 48202, USA, s: nguyenlam@wayne.edu and gzlu@math.wayne.edu