EXISTENCE OF WEAK SOLUTIONS FOR A NONUNIFORMLY ELLIPTIC NONLINEAR SYSTEM IN R N. 1. Introduction We study the nonuniformly elliptic, nonlinear system

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1 Electronic Journal of Differential Equations, Vol ), No. 119, pp ISSN: URL: or ftp ejde.math.txstate.edu login: ftp) EXISTENCE OF WEAK SOLUTIONS FOR A NONUNIFORMLY ELLIPTIC NONLINEAR SYSTEM IN NGUYEN THANH CHUNG Abstract. We study the nonuniformly elliptic, nonlinear system divh 1 x) u) + ax)u = fx, u, v) in, divh 2 x) v) + bx)v = gx, u, v) in. Under growth and regularity conditions on the nonlinearities f and g, we obtain weak solutions in a subspace of the Sobolev space H 1, R 2 ) by applying a variant of the Mountain Pass Theorem. 1. Introduction We study the nonuniformly elliptic, nonlinear system divh 1 x) u) + ax)u = fx, u, v) in, divh 2 x) v) + bx)v = gx, u, v) in, 1.1) where N 3, h i L 1 loc RN ), h i x) 1 i = 1, 2; a, b C ). We assume that there exist a 0, b 0 > 0 such that ax) a 0, bx) b 0, x, ax), bx) as x. 1.2) System 1.1), with h 1 x) = h 2 x) = 1, has been studied by Costa [7]. There, under appropriate growth and regularity conditions on the functions fx, u, v) and gx, u, v), the weak solutions are exactly the critical points of a functional defined on a Hilbert space of functions u, v in H 1 ). In the scalar case, the problem div x α u) + bx)u = fx, u) in, with N 3 and α 0, 2), has been studied by Mihăilescu and Rădulescu [11]. In this situation, the authors overcome the lack of compactness of the problem by using the the Caffarelli-Kohn-Nirenberg inequality. In this paper, under condition 1.2), we consider 1.1) which may be a nonuniformly elliptic system. We shall reduce 1.1) to a uniformly elliptic system by using appropriate weighted Sobolev spaces. Then applying a variant of the Mountain 2000 Mathematics Subject Classification. 35J65, 35J20. Key words and phrases. Nonuniformly elliptic; nonlinear systems; mountain pass theorem; weakly continuously differentiable functional. c 2008 Texas State University - San Marcos. Submitted March 27, Published August 25,

2 2 N. T. CHUNG EJDE-2008/119 pass theorem in [8], we prove the existence of weak solutions of system 1.1) in a subspace of H 1, R 2 ). To prove our main results, we introduce the following some hypotheses: H1) There exists a function F x, w) C 1 R 2, R) such that F u = fx, w), F v = gx, w), for all x RN, w = u, v) R 2. H2) fx, w), gx, w) C 1 R 2, R), fx, 0, 0) = gx, 0, 0) = 0 for all x, there exists a positive constant τ 0 such that fx, w) + gx, w) τ 0 w p 1 for all x, w = u, v) R 2. H3) There exists a constant µ > 2 such that 0 < µf x, w) w F x, w) for all x and w R 2 \{0, 0)}. Let H 1, R 2 ) be the usual Sobolev space under the norm w 2 = u 2 + v 2 + u 2 + v 2 )dx, w = u, v) H 1, R 2 ). Consider the subspace E = {u, v) H 1, R 2 ) : u 2 + v 2 + ax) u 2 + bx) v 2 )dx < }. Then E is a Hilbert space with the norm w 2 E = u 2 + v 2 + ax) u 2 + bx) v 2 )dx. By 1.2) it is clear that w E m 0 w H 1,R 2 ), w E, m 0 > 0, and the embeddings E H 1, R 2 ) L q, R 2 ), 2 q 2 are continuous. Moreover, the embedding E L 2, R 2 ) is compact see [7]). We now introduce the space H = {u, v) E : h 1 x) u 2 + h 2 x) v 2 + ax) u 2 + bx) v 2 )dx < } endowed with the norm w 2 H = h 1 x) u 2 + h 2 x) v 2 + ax) u 2 + bx) v 2 )dx. Remark 1.1. Since h 1 x) 1, h 2 x) 1 for all x we have w E w H with w H and C 0, R 2 ) H. Proposition 1.2. The set H is a Hilbert space with the inner product w 1, w 2 = h 1 x) u 1 u 2 + h 2 x) v 1 v 2 + ax)u 1 u 2 + bx)v 1 v 2 )dx for all w 1 = u 1, v 1 ), w 2 = u 2, v 2 ) H.

3 EJDE-2008/119 EXISTENCE OF WEAK SOLUTIONS 3 Proof. It suffices to check that any Cauchy sequences {w m } in H converges to w H. Indeed, let {w m } = {u m, v m )} be a Cauchy sequence in H. Then h1 x) u m u k 2 + h 2 x) v m v k 2) dx m,k R N + ax) um u k 2 + bx) v m v k 2) dx = 0 m,k and { w m H } is bounded. Moreover, by Remark 1.1, {w m } is also a Cauchy sequence in E. sequence {w m } converges to w = u, v) E; i.e., um u 2 + v m v 2) dx R N + ax) um u 2 + bx) v m v 2) dx = 0. Hence the It follows that { w m = u m, v m )} converges to w = u, v) and {w m } converges to w in L 2, R 2 ). Therefore { w m x)} converges to { wx)} and {w m x)} converges to wx) for almost everywhere x. Applying Fatou s lemma we get h 1 x) u 2 + h 2 x) v 2 + ax) u 2 + bx) v 2 )dx inf h 1 x) u m 2 + h 2 x) v m 2 + ax) u m 2 + bx) v m 2 )dx <. Hence w = u, v) H. Applying again Fatou s lemma 0 h1 x) u m u 2 + h 2 x) v m v 2) dx R N + ax) um u 2 + bx) v m v 2) dx R [ N inf h1 x) u m u k 2 + h 2 x) v m v k 2) ] dx k R [ N + inf ax) um u k 2 + bx) v m v k 2) ] dx = 0. k We conclude that {w m } converges to w = u, v) in H. Definition 1.3. We say that w = u, v) H is a weak solution of system 1.1) if h 1 x) u ϕ + h 2 x) v ψ + ax)uϕ + bx)vψ)dx fx, u, v)ϕ + gx, u, v)ψ)dx = 0 for all Φ = ϕ, ψ) H. Our main result is stated as follows. Theorem 1.4. Assuming 1.2) and H1) H3) are satisfied, the system 1.1) has at least one non-trivial weak solution in H.

4 4 N. T. CHUNG EJDE-2008/119 This theorem will be proved by using variational techniques based on a variant of the Mountain pass theorem in [8]. Let us define the functional J : H R given by Jw) = 1 h 1 x) u 2 + h 2 x) v 2 + ax) u 2 + bx) v 2 )dx 2 R N 1.3) F x, u, v)dx = T w) P w) for w = u, v) H, where T w) = 1 2 h 1 x) u 2 + h 2 x) v 2 + ax) u 2 + bx) v 2 )dx, 1.4) P w) = F x, u, v)dx. 1.5) 2. Existence of weak solutions In general, due to hx) L 1 loc RN ), the functional J may be not belong to C 1 H) in this work, we do not completely care whether the functional J belongs to C 1 H) or not). This means that we cannot apply directly the Mountain pass theorem by Ambrosetti-Rabinowitz see [4]). In the situation, we recall the following concept of weakly continuous differentiability. Our approach is based on a weak version of the Mountain pass theorem by Duc see [8]). Definition 2.1. Let J be a functional from a Banach space Y into R. We say that J is weakly continuously differentiable on Y if and only if the following conditions are satisfied i) J is continuous on Y. ii) For any u Y, there exists a linear map DJu) from Y into R such that Ju + tv) Ju) = DJu), v, v Y. t 0 t iii) For any v Y, the map u DJu), v is continuous on Y. We denote by CwY 1 ) the set of weakly continuously differentiable functionals on Y. It is clear that C 1 Y ) CwY 1 ), where C 1 Y ) is the set of all continuously Frechet differentiable functionals on Y. The following proposition concerns the smoothness of the functional J. Proposition 2.2. Under the assumptions of Theorem 1.4, the functional Jw), w H given by 1.3) is weakly continuously differentiable on H and DJw), Φ = h 1 x) u ϕ + h 2 x) v ψ + ax)uϕ + bx)vψ)dx R N 2.1) fx, u, v)ϕ + gx, u, v)ψ)dx for all w = u, v), Φ = ϕ, ψ) H.

5 EJDE-2008/119 EXISTENCE OF WEAK SOLUTIONS 5 Proof. By conditions H1) H3) and the embedding H E is continuous, it can be shown cf. [5, Theorem A.VI]) that the functional P is well-defined and of class C 1 H). Moreover, we have DP w), Φ = fx, u, v)ϕ + gx, u, v)ψ)dx for all w = u, v), Φ = ϕ, ψ) H. Next, we prove that T is continuous on H. Let {w m } be a sequence converging to w in H, where w m = u m, v m ), m = 1, 2,..., w = u, v). Then + [h 1 x) u m u 2 + h 2 x) v m v 2 ]dx R N [ax) u m u 2 + bx) v m v 2 ]dx = 0 and { w m H } is bounded. Observe further that h 1 x) u m 2 dx h 1 x) u 2 dx = h 1 x) u m 2 u 2 )dx R N h 1 x) u m u u m + u )dx R N h 1 x) u m u u m dx + h 1 x) u m u u dx R ) N 1/2 ) 1/2 h 1 x) u m u 2 dx h 1 x) u m 2 dx R N + h 1 x) u m u 2 dx ) ) 1/2 1/2 h 1 x) u 2 dx w m H + w H ) w m w H. Similarly, we obtain h 2 x) v m 2 dx h 2 x) v 2 dx w m H + w H ) w m w H, R N ax) u m 2 dx ax) u 2 dx wm H + w H ) w m w H, R N bx) v m 2 dx bx) v 2 dx wm H + w H ) w m w H. From the above inequalities, we obtain T w m ) T w) 4 w m H + w H ) w m w H 0 as m. Thus T is continuous on H. Next we prove that for all w = u, v), Φ = ϕ, ψ) H, DJw), Φ = h 1 x) u ϕ + h 2 x) v ψ + ax)uϕ + bx)vψ)dx.

6 6 N. T. CHUNG EJDE-2008/119 Indeed, for any w = u, v), Φ = ϕ, ψ) H, any t 1, 1)\{0} and x we have h 1x) u + t ϕ 2 h 1 x) u 2 = 1 2 h 1 x) u + st ϕ) ϕds t 0 2h 1 x) u + ϕ ) ϕ h 1 x) u 2 + 3h 1 x) ϕ 2. Since h 1 x) u 2, h 1 x) ϕ 2 L 1 ), gx) = h 1 x) u 2 + 3h 1 x) ϕ 2 L 1 ). Applying Lebesgue s Dominated convergence theorem we get RN h 1 x) u + t ϕ 2 h 1 x) u 2 dx = 2 h 1 x) u ϕdx. 2.2) t 0 t Similarly, we have RN h 2 x) v + t ψ 2 h 2 x) v 2 dx = 2 h 2 x) v ψdx, 2.3) t 0 t RN ax) u + tϕ 2 ax) u 2 dx = 2 ax)uϕdx, 2.4) t 0 t RN bx) v + tψ 2 bx) v 2 dx = 2 bx)vψ dx. 2.5) t 0 t Combining 2.2)-2.5), we deduce that T w + tφ) T w) DT w), Φ = t 0 t = h 1 x) u ϕ + h 2 x) v ψ + ax)uϕ + bx)vψ) dx. Thus T is weakly differentiable on H. Let Φ = ϕ, ψ) H be fixed. We now prove that the map w DT w), Φ is continuous on H. Let {w m } be a sequence converging to w in H. We have DT wm ), Φ DT w), Φ h 1 x) u m u ϕ dx + h 2 x) v m v ψ dx R N + ax) u m u ϕ dx + bx) v m v ψ dx. It follows by applying Cauchy s inequality that DT w m ), Φ DT w), Φ 4 Φ H w m w H 0 as m. 2.6) Thus the map w DT w), Φ is continuous on H and we conclude that functional T is weakly continuously differentiable on H. Finally, J is weakly continuously differentiable on H. Remark 2.3. From Proposition 2.2 we observe that the weak solutions of system 1.1) correspond to the critical points of the functional Jw), w H given by 1.3). Thus our idea is to apply a variant of the Mountain pass theorem in [8] for obtaining non-trivial critical points of J and thus they are also the non-trivial weak solutions of system 1.1). Proposition 2.4. The functional Jw), w H given by 1.3) satisfies the Palais- Smale condition.

7 EJDE-2008/119 EXISTENCE OF WEAK SOLUTIONS 7 Proof. Let {w m = u m, v m )} be a sequence in H such that Jw m) = c, DJw m) H = 0. First, we prove that {w m } is bounded in H. We assume by contradiction that {w m } is not bounded in H. Then there exists a subsequence {w mj } of {w m } such that w mj H as j. By assumption H3) it follows that Jw mj ) 1 µ DJw m j ), w mj = ) wmj 2 H + 1 µ µ DP w m j ), w mj P w mj ) ) γ 0 w mj 2 H, where γ 0 = µ. This yields Jw mj ) γ 0 w mj 2 H + 1 µ DJw m j ), w mj γ 0 w mj 2 H 1 µ DJw m j ) H. w mj H 2.7) = w mj H γ0 w mj H 1 µ DJw ) m j ) H. Letting j, since w mj H, DJw mj ) H 0 we deduce that Jw mj ), which is a contradiction. Hence {w m } is bounded in H. Since H is a Hilbert space and {w m } is bounded in H, there exists a subsequence {w mk } of {w m } weakly converging to w in H. Moreover, since the embedding H E is continuous, {w mk } is weakly convergent to w in E. We shall prove that T w) inf k T w m k ). 2.8) Since the embedding E L 2, R 2 ) is compact, {w mk } converges strongly to w in L 2, R 2 ). Therefore, for all, {w mk } converges strongly to w in L 1, R 2 ). Besides, for any Φ = ϕ, ψ) E we have ax)u mk u)ϕ + bx)v mk v)ψ) dx max sup ) ax), sup bx) u mk u ϕ dx + Applying Cauchy inequality we obtain ax)u mk u)ϕ + bx)v mk v)ψ) dx γ 1 Φ L 2,R 2 ) w mk w L 2,R 2 ), ) v mk v ψ dx. where γ 1 = maxsup ax), sup bx)) > 0. Letting k we get ax)u mk u)ϕ + bx)v mk v)ψ) dx = ) k

8 8 N. T. CHUNG EJDE-2008/119 On the other hand, since w mk converges weakly to w in E; i.e., [ u mk u) ϕ + v mk v) ψ] dx k R N + [ax)u mk u)ϕ + bx)v mk v)ψ] dx = 0 k for all Φ = ϕ, ψ) E, by 2.9) and C0, R 2 ) H E we infer that [ u mk u) ϕ + v mk v) ψ] dx = 0, k for all. This implies that { w mk } converges weakly to w in L 1, R 2 ). Applying [15, Theorem 1.6], we obtain T w) inf k T w m k ). Thus 2.8) is proved. We now prove that DP w m k ), w mk w = F x, w mk ).w mk w)dx = ) k k Indeed, by H2), we have F x, w mk )w mk w) = fx, w mk )u mk u) + gx, w mk )v mk v) fx, θ 1 w mk ) w mk u mk u + gx, θ 2 w mk ) w mk v mk v A 1 w mk p u mk u + A 2 w mk p v mk v A 3 w mk p w mk w, 0 < θ 1, θ 2 < 1 where A i i = 1, 2, 3) are positive constants. Set 2 = 2N N 2, p 1 = 2 p 1, p 2 = p 3 = 2p1 p. We have p > 1, 2 < p 2, p 3 < 2 and 1 p p p 3 = 1. Therefore, F x, w mk ).w mk w)dx A 3 w mk p 1 w mk w w mk dx k A 3 w mk p 1 L 2 w m k w L p 2 w mk L p 3. On the other hand, using the continuous embeddings H E L q ), 2 q 2 together with the interpolation inequality where 1 p 2 = δ δ 2 ), it follows that w mk w L p 2 RN ) w mk w δ L 2 ). w m k w 1 δ L 2. Since the embedding E L 2 ) is compact we have w mk w L 2 ) 0 as k. Hence w mk w L p 2 RN ) 0 as k and 2.10) is proved. On the other hand, by 2.10) and 2.1) it follows DT w m k ), w mk w = 0. k Hence, by the convex property of the functional T we deduce that T w) k sup T w m k ) = k inf [T w) T w m k )] 2.11) k DT w m k ), w w mk = )

9 EJDE-2008/119 EXISTENCE OF WEAK SOLUTIONS 9 Relations 2.8) and 2.11) imply T w) = k T w m k ). 2.13) Finally, we prove that {w mk } converges strongly to w in H. Indeed, we assume by contradiction that {w mk } is not strongly convergent to w in H. Then there exist a constant ɛ 0 > 0 and a subsequence {w mkj } of {w mk } such that w mkj w H ɛ 0 > 0 for any j = 1, 2,.... Hence 1 2 T w m kj ) + 1 w 2 T w) T + w mkj ) = w m kj w 2 H 1 4 ɛ ) With the same arguments as in the proof of 2.8), and remark that the sequence { wm +w k j 2 } converges weakly to w in E, we have w T w) inf T + w mkj ). 2.15) j 2 Hence letting j, from 2.13) and 2.14) we infer that w T w) inf T + w mkj ) 1 j 2 4 ɛ ) Relations 2.15) and 2.16) imply ɛ2 0 > 0, which is a contradiction. Therefore, we conclude that {w mk } converges strongly to w in H and J satisfies the Palais - Smale condition on H. To apply the Mountain pass theorem we shall prove the following proposition which shows that the functional J has the Mountain pass geometry. Proposition 2.5. i) There exist α > 0 and r > 0 such that Jw) α, for all w H with w H = r. ii) There exists w 0 H such that w 0 H > r and Jw 0 ) < 0. Proof. i) From H3), it is easy to see that F x, z) min s =1 F x, s). z µ > 0 x and z 1, z R 2, 2.17) 0 < F x, z) max s =1 F x, s). z µ x and 0 < z 1, 2.18) where max s =1 F x, s) C in view of H2). It follows from 2.18) that F x, z) z 0 z 2 = 0 uniformly for x. 2.19) By using the embeddings H E L 2, R 2 ), with simple calculations we infer from 2.19) that inf w H =r Jw) = α > 0 for r > 0 small enough. This implies i). ii) By 2.17), for each compact set there exists c = c) such that F x, z) c z µ for all x, z ) Let 0 Φ = ϕ, ψ) C 1, R 2 ) having compact support, for t > 0 large enough, from 2.20) we have JtΦ) = 1 2 t2 Φ 2 H F x, tφ)dx 1 R 2 t2 Φ 2 H t µ c Φ µ dx, 2.21) N where c = c), = supp ϕ supp ψ). Then 2.21) and µ > 2 imply ii).

10 10 N. T. CHUNG EJDE-2008/119 Proof of Theorem 1.4. It is clear that J0) = 0. Furthermore, the acceptable set G = {γ C[0, 1], H) : γ0) = 0, γ1) = w 0 }, where w 0 is given in Proposition 2.5, is not empty it is easy to see that the function γt) = tω 0 G). By Proposition 1.2 and Propositions , all assumptions of the Mountain pass theorem introduced in [8] are satisfied. Therefore there exists ŵ H such that 0 < α Jŵ) = inf{max Jγ[0, 1])) : γ G} and DJŵ), Φ = 0 for all Φ H; i.e., ŵ is a weak solution of system 1.1). The solution ŵ is a non-trivial solution by Jŵ) α > 0. The proof is complete. Acknowledgments. The author would like to thank Professor Hoang Quoc Toan, Department of Mathematics, Mechanics and Informatics, College of Science, Vietnam National University, for his interest, encouragement and helpful comments. References [1] A. Abakhti-Mchachti., J. Fleckinger-Pellé; Existence of positive solutions for non-cooperative semilinear elliptic systems defined on an unbounded domain, Pitman research Notes in Maths., ), [2] R. A. Adams; Sobolev spaces, Academic Press, New York - San Francisco - London, [3] A. Ambrosetti., K.C. Chang., I. Ekeland; Nonlinear functional analysis and Applications to differential equations, Proceedings of the second school ICTP, Trieste, Italy, [4] A. Ambrosetti., P. H. Rabinowitz; Dual variational methods in critical points theory and applications, J. Funct. Anal ), [5] H. Berestycki., P. L. Lions; Nonlinear scalar field equations, I. Existence of a ground state, Arch. Rat. Mech. Anal., ), [6] H. Brezis; Analyse fonctionnelle théori et applications, Massion, [7] D. G. Costa; On a class of elliptic systems in R n, Electronic Journal of Differential Equations ) No. 07, [8] D. M. Duc; Nolinear singular elliptic equations, J. London. Math. Soc. 402) 1989), [9] L. Gasínski., N. S. Parageorgiou; Nonsmooth critical point theory and nonlinear boundary value problems, Vol. 8, A CRC Press Company, London - New York, [10] D. Gilbarg., N. C. Trudinger; Elliptic Partial Differential Equations of second order, Second Edition, Springer-Verlag, Berlin, [11] M. Mihăilescu., V. Rădulescu; Ground state solutions of nonlinear singular Schrödinger equations with lack compactness, Mathematical methods in the Applied Sciences, ), [12] P. De Nápoli., M. C. Mariani; Mountain pass solutions to equations of p-laplacian type, Nonlinear Analysis, ), [13] L. Nirenberg; Variational and topology methods in nonlinear problems, Bull. Amer. Math. Soc ), [14] P. H. Rabinowitz; Minimax methods and their applications to Partial Differential Equations, Nonlinear analysis, a collection of papers in honor of Erich Rothe, Academic Press, New York, [15] M. Struwe; Variational Methods, Second Edition, Springer-Verlag, [16] N. T. Vu., D. M. Duc; Nonuniformly elliptic equations of p-laplacian type, Nonlinear Analysis ), [17] E. Zeidler; Nonlinear Functional Analysis and its applications, Vol. III, Springer-Verlag, Nguyen Thanh Chung Department of Mathematics and Informatics, Quang Binh University, 312 Ly Thuong Kiet, Dong Hoi, Vietnam address: ntchung82@yahoo.com