NONLINEAR SCHRÖDINGER ELLIPTIC SYSTEMS INVOLVING EXPONENTIAL CRITICAL GROWTH IN R Introduction

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1 Electronic Journal of Differential Equations, Vol. 014 (014), No. 59, pp ISSN: URL: or ftp ejde.math.txstate.edu NONLINEAR SCHRÖDINGER ELLIPTIC SYSTEMS INVOLVING EXPONENTIAL CRITICAL GROWTH IN FRANCISCO SIBERIO BEZERRA ALBUQUERQUE Abstract. This article concerns the existence and multiplicity of solutions for elliptic systems with weights, and nonlinearities having exponential critical growth. Our approach is based on the Trudinger-Moser inequality and on a minimax theorem. In this article, we consider the system 1. Introduction u + V ( x )u = Q( x )f(u, v) in, v + V ( x )v = Q( x )g(u, v) in, (1.1) where the nonlinear terms f and g are allowed to have exponential critical growth. By means of the Trudinger-Moser inequality and the radial potentials V and Q may be unbounded or decaying to zero. We shall consider the variational situation in which (f(u, v), g(u, v)) = F (u, v) for some function F : R of class C 1, where F stands for the gradient of F in the variables w = (u, v). Aiming an analogy with the scalar case, we rewrite (1.1) in the matrix form w + V ( x )w = Q( x ) F (w) in, where we denote = (, ) and Q( x ) F (w) = (Q( x )f(w), Q( x )g(w)). We make the following assumptions on the potentials V ( x ) and Q( x ): (V1) V C(0, ), V (r) > 0 and there exists a > such that V (r) lim inf r + r a > 0. (Q1) Q C(0, ), Q(r) > 0 and there exist b < (a )/ and b 0 > such that Q(r) Q(r) lim sup < and lim sup r 0 r b0 r + r b <. 000 Mathematics Subject Classification. 35J0, 35J5, 35J50. Key words and phrases. Elliptic systems; exponential critical growth; Trudinger-Moser inequality. c 014 Texas State University - San Marcos. Submitted August, 013. Published February 8,

2 F. S. B. ALBUQUERQUE EJDE-014/59 This type of potentials appeared in [, 13, 14], in which the authors studied the existence and multiplicity of solutions for the scalar problem u + V ( x )u = Q( x )f(u) u(x) 0 as x, in R N where in [13, 14] the nonlinearity considered was f(u) = u p u, with < p < = N/(N ) for N 3 and < p < for N =. In [] the authors considered the critical case in the sense of Trudinger-Moser inequality [9, 15]. Let us introduce the precise assumptions under which our problem is studied. (F0) f and g have α 0 -exponential critical growth, i.e., there exists α 0 > 0 such that { f(w) g(w) 0, α > α 0, lim = lim = w + e α w w + e α w +, α < α 0 ; (F1) f(w) = o( w ) and g(w) = o( w ) as w 0; (F) there exists θ > such that 0 < θf (w) w F (w), w \{0}; (F3) there exist constants R 0, M 0 > 0 such that 0 < F (w) M 0 F (w), w R 0 ; (F4) there exist ν > and µ > 0 such that F (w) µ ν w ν, w. To establish our main results, we need to recall some notation about function spaces. In all the integrals we omit the symbol dx and we use C, C 0, C 1, C,... to denote (possibly different) positive constants. Let C 0 ( ) be the set of smooth functions with compact support and C 0,rad( ) = {u C 0 ( ) : u is radial }. Denote by D 1, rad (R ) the closure of C0,rad (R ) under the norm ( ) 1/. u = u If 1 p < we define L p ( ; Q) =. {u : R : u is mensurable, Q( x ) u p < }. Similarly we define L ( ; V ). Then we set Hrad(R 1 ; V ) =. D 1, rad (R ) L ( ; V ), which is a Hilbert space (see [14]) with the norm u H 1 rad ( ;V ). = ( u + V ( x ) u ) 1/. We will denote Hrad 1 (R ; V ) by E and its norm by E. In E E we consider the scalar product w 1, w =. [ u 1 u + V ( x )u 1 u ] + [ v 1 v + V ( x )v 1 v ],

3 EJDE-014/59 NONLINEAR SCHRÖDINGER ELLIPTIC SYSTEMS 3 where w 1 = (u 1, v 1 ) and w = (u, v ), to which corresponds the norm w = w, w 1/. Motivated by [, 13, 14] and using a minimax procedure, we obtain existence and multiplicity results for system (1.1). As in the scalar problem treated in [], there are at least two main difficulties in our problem; the possible lack of the compactness of the Sobolev embedding since the domain is unbounded and the critical growth of the nonlinearities. Denoting by S ν > 0 the best constant of the Sobolev embedding E L ν ( ; Q) (see Lemma.3 below), we have the following existence result for system (1.1). Theorem 1.1. Assume (V1) and (Q1). If (F0) (F4) are satisfied, then (1.1) has a nontrivial weak solution w 0 in E E provided µ > [ α 0 (ν ) ] (ν )/S ν/ α ν, ν where α. = min{4π, 4π(1 + b0 /)}. Our multiplicity result concerns the problem w + V ( x )w = λq( x ) F (w) in, (1.) where λ is a positive parameter. It can be stated as follows. Theorem 1.. Assume (V1) and (Q1). If F is odd and (F0) (F4) are satisfied, then for any given k N there exists Λ k > 0 such that the system (1.) has at least k pairs of nontrivial weak solutions in E E provided λ > Λ k. To close up this section, we remark that the main tool to prove Theorem 1., is the symmetric Mountain-Pass Theorem due to Ambrosetti-Rabinowitz [3]. It will be used in a more common version in comparison to the one used to prove the analogous theorem in the scalar case [, Theorem 1.5], which leads us to a more direct conclusion of the result. This article is organized as follows. Section contains some technical results. In Section 3, we set up the framework in which we study the variational problem associated with (1.1) and we prove our existence result, Theorem 1.1. Finally, in Section 4 we prove Theorem 1... Preliminaries We start by recalling a version of the radial lemma due to Strauss in [1] (see [, 13]). In the following, B r denotes the open ball in centered at the 0 with radius r and B R \ B r denotes the annulus with interior radius r and exterior radius R. Lemma.1. Assume (V1) with a. Then, there exists C > 0 such that for all u E, u(x) C u x a+ 4, x 1. Next, we recall some basic embeddings (see Su et al. [13]). Let A and define H 1 rad(a; V ) = {u A : u H 1 rad( ; V )}.

4 4 F. S. B. ALBUQUERQUE EJDE-014/59 Lemma.. Let 1 p <. For any 0 < r < R <, with R 1, (i) the embeddings H 1 rad (B R \ B r ; V ) L p (B R \ B r ; Q) are compact; (ii) the embedding H 1 rad (B R; V ) H 1 (B R ) is continuous. In particular, as a consequence of (ii) we have that H 1 rad (B R; V ) is compactly immersed in L q (B R ) for all 1 q <. If we assume that (V1) and (Q1) hold, by using Lemmas.1 and., a Hardy inequality with remainder terms (see [16]) and the same ideas from [13] we have: Lemma.3. Assume (V1) and (Q1). If a and b < a, then the embeddings E L p ( ; Q) are compact for all p <. Inspired by [1, 5, 9, 10, 15], to study system (1.1), the following version of the Trudinger-Moser inequality in the scalar case, obtained in [, Theorem 1.1], plays an important rule. Proposition.4. Assume (V1) and (Q1). Then, for any u E and α > 0, we have that Q( x )(e αu 1) L 1 ( ). Furthermore, if α < α, then there exists a constant C > 0 such that sup Q( x )(e αu 1) C. u E, u E 1 In line with Lions [8] and in order to prove our multiplicity result; Theorem 1., we establish an improvement of the Trudinger-Moser inequality on the space E E, considering our variational setting. Using Proposition.4 and following the same steps as in the proof of [7, Lemma.6] we have: Corollary.5. Assume (V1) and (Q1). Let (w n ) be in E E with w n = 1 and suppose that w n w weakly in E E with w < 1. Then, for each 0 < β < ( ) α 1 w 1, up to a subsequence, it holds sup Q( x )(e β wn 1) < +. n N 3. Variational setting The natural functional associated with (1.1) is I(w) = 1 w QF (w), w E E. Under our assumptions we have that I is well defined and it is C 1 on E E. Indeed, by (F1), for any ε > 0, there exists δ > 0 such that F (w) ε w always that w < δ. On the other hand, for α > α 0, there exist constants C 0, C 1 > 0 such that f(w) C 0 (exp(α w ) 1) and g(w) C 1 (exp(α w ) 1) for all w δ. Thus, for all w we have F (w) ε w + f(w) + g(w) ε w + C(exp(α w ) 1). Hence, using (F), (3.1) and the Hölder s inequality, we have Q F (w) (3.1)

5 EJDE-014/59 NONLINEAR SCHRÖDINGER ELLIPTIC SYSTEMS 5 ε Q w + C Q w (e α w 1) R ( ) ( ) 1/r ( ε Q u + Q v + C Q w r Q(e sα w 1/s, 1)) with r, s 1 such that 1/r + 1/s = 1. Considering Lemma.3, for r 4, we have ( ) 1/r Q w r = u + v 1/ L r/ ( ;Q) C u + v 1/ E C w <. On the other hand, by the Young s inequality and Proposition.4, 1) 1 1) + 1 1) <. (3.) Q(e sα w Q(e sαu Q(e sαv Hence, QF (w) L 1 ( ), which implies that I is well defined, for α > α 0. Using standard arguments, we can see that I C 1 (E E, R) with I (w)z = w, z Qz F (w) for all z E E. Consequently, critical points of the functional I are precisely the weak solutions of system (1.1). In the next lemma we check that the functional I satisfies the geometric conditions of the Mountain-Pass Theorem. Lemma 3.1. Assume (V1) and (Q1). If (F0) (F) hold, then: (i) there exist τ, ρ > 0 such that I(w) τ whenever w = ρ; (ii) there exists e E E, with e > ρ, such that I(e ) < 0. Proof. Just as we have obtained (3.1), we deduce that F (w) ε w + C w q 1 (e α w 1) (3.3) for all w and q 1. Thus, using (F), the Hölder s inequality and Lemma.3, we have Q F (w) ε Q w + C Q w q (e α w 1) R ( ) ( ) 1/r ( ) ε Q u + Q v + C Q w qr Q(e sα w 1/s 1) Cε w + C 0 w q( Q(e sα w 1)) 1/s, provided r and s > 1 such that 1/r+1/s = 1. Now for w M < [α /(α)] 1/, which implies that α u E αm < α and α v E αm < α, and s sufficiently close to 1, it follows from (3.) that Q F (w) Cε w + C 1 w q. Hence, I(w) ( 1 Cε ) w C 1 w q,

6 6 F. S. B. ALBUQUERQUE EJDE-014/59 which implies i), if q >. In order to verify ii), let w E E with compact support G. Thus, using (F4) we obtain I(tw) t w Ct ν Q w ν, for all t > 0, which yields I(tw) as t +, provided ν >. e = t w with t > 0 large enough, the proof is complete. G Setting To prove that a Palais-Smale sequence converges to a weak solution of system (1.1) we need to establish the following lemmas. Lemma 3.. Assume (F). Let (w n ) be a sequence in E E such that Then w n C, I(w n ) c and I (w n ) 0. QF (w n ) C, Qw n F (w n ) C. Proof. Let (w n ) be a sequence in E E such that I(w n ) c and I (w n ) 0. Thus, for any z E E, I(w n ) = 1 w n QF (w n ) = c + o n (1) (3.4) and I (w n )z = w n, z Qz F (w n ) = o n (1). (3.5) Taking z = w n in (3.5) and using (F) we have c + w n + o n (1) I(w n ) 1 θ I (w n )w n = ( 1 1 ) wn + Q[ 1 θ R θ w n F (w n ) F (w n )] ( 1 1 ) wn. θ Consequently, w n C. By (3.4) and (3.5) we obtain QF (w n ) C, Qw n F (w n ) C. We will also use the following convergence result. Lemma 3.3. Assume (F) and (F3). If (w n ) E E is a Palais-Smale sequence for I and w 0 is its weak limit then, up to a subsequence, and F (w n ) F (w 0 ) in L 1 loc(, ) QF (w n ) QF (w 0 ) in L 1 ( ). Proof. Suppose that (w n ) is a Palais-Smale sequence. According to Lemma 3., w n = (u n, v n ) w 0 = (u 0, v 0 ) weakly in E E, that is, u n u 0 and v n v 0 weakly in E. Thus, recalling that H 1 rad (B R; V ) L q (B R ) compactly for all 1 q < and R > 0 (see the consequence of ii) from Lemma.), up to a subsequence, we can assume that u n u 0 and v n v 0 in L 1 (B R ). Hence, w n w 0 in L 1 (B R, )

7 EJDE-014/59 NONLINEAR SCHRÖDINGER ELLIPTIC SYSTEMS 7 and w n (x) w 0 (x) a.e. in. Since F (w n ) L 1 (B R, ), the first convergence follows from [6, Lemma.1]. Hence, f(w n ) f(w 0 ) and g(w n ) g(w 0 ) in L 1 loc( ). Thus, there exist h 1, h L 1 (B R ) such that Q f(w n ) h 1 and Q g(w n ) h a.e. in B R. From (F3) we conclude that F (w n ) sup F (w n ) + M 0 F (w n ) [ R 0,R 0] a.e. in B R. Thus, by Lebesgue Dominated Convergence Theorem QF (w n ) QF (w 0 ) in L 1 (B R ). On the other hand, from (F) and (3.3) with q = we have QF (w n ) ε Q w n + C Q w n (e α wn 1), (3.6) for α > α 0. From Lemma.3, the Hölder s inequality, w n C and developing the exponential into a power series, we obtain ε Q w n Cε and Q w n (e α wn 1) C R ξ, for some ξ > 0. Hence, given δ > 0, there exists R > 0 sufficiently large such that Q w n < δ and Q w n (e α wn 1) < δ. Thus, from (3.6) QF (w n ) Cδ and Finally, since QF (w n ) QF (w 0 ) R QF (w n ) QF (w 0 ) + B R B R QF (w 0 ) Cδ. QF (w n ) + QF (w 0 ), BR c we obtain lim QF (w n ) QF (w 0 ) Cδ. Since δ > 0 is arbitrary, the result follows and the lemma is proved. In view of Lemma 3.1 the minimax level satisfies where c = inf max I(g(t)) τ > 0, g Γ 0 t 1 Γ = {g C([0, 1], E E) : g(0) = 0 and I(g(1)) < 0}. Hence, by the Mountain-Pass Theorem without the Palais-Smale condition (see [3]) there exists a (P S) c sequence (w n ) = ((u n, v n )) in E E, that is, I(w n ) c and I (w n ) 0. (3.7)

8 8 F. S. B. ALBUQUERQUE EJDE-014/59 Lemma 3.4. If then c < α /(4α 0 ). µ > [ α 0 (ν ) ] (ν )/S ν/ α ν, ν Proof. Since the embeddings E L p ( ; Q) are compacts for all p <, there exists a function ū E such that S ν = ū E and ū L ν ( ;Q) = 1. Thus, considering w = (ū, ū), by the definition of c and (F4), one has [ ] [ c max S ν t QF (tw) max S ν t ν/ µ t ν] t 0 t 0 ν = ν ν Now we are ready to prove our existence result. Sν ν/(ν ) α < /(ν ) µ 4α 0. Proof of Theorem 1.1. It follows from Lemmas 3. and 3.3 that the Palais-Smale sequence (w n ) is bounded and it converges weakly to a weak solution of (1.1) denoted by w 0. To prove that w 0 is nontrivial we argue by contradiction. If w 0 0, Lemma 3.3 implies that lim QF (w n ) = 0. Thus, by (3.4) lim w n = c > 0. (3.8) From this and Lemma 3.4, given ε > 0, we have that w n < α /(α 0 ) + ε for n N large. Thus, it is possible to choice s > 1 sufficiently close to 1 and α > α 0 close to α 0 such that sα w n β < α /, which implies that sα u n E β < α and sα v n E β < α. Thus, using (3.), (3.1) in combination with the Hölder s inequality and Lemma.3, up to a subsequence, we conclude that lim Qw n F (w n ) = 0. Hence, by (3.5), we obtain that lim w n = 0, which is a contradiction with (3.8). Therefore, w 0 is a nontrivial weak solution of (1.1). 4. Proof of Theorem 1. To prove our multiplicity result we shall use the following version of the Symmetric Mountain-Pass Theorem (see [3, 4, 11]). Theorem 4.1. Let X = X 1 X, where X is a real Banach space and X 1 is finite-dimensional. Suppose that J is a C 1 (X, R) functional satisfying the following conditions: (J1) J(0) = 0 and J is even; (J) there exist τ, ρ > 0 such that J(u) τ if u = ρ, u X ;

9 EJDE-014/59 NONLINEAR SCHRÖDINGER ELLIPTIC SYSTEMS 9 (J3) there exists a finite-dimensional subspace W X with dim X 1 < dim W and there exists S > 0 such that max u W J(u) S; (J4) J satisfies the (P S) c condition for all c (0, S). Then J possesses at least dim W dim X 1 pairs of nontrivial critical points. Given k N, we apply this abstract result with X = E E, X 1 = {0}, J = I λ and W = W W with W =. [ψ 1,..., ψ k ], where {ψ i } k i=1 C 0 ( ) is a collection of smooth function with disjoint supports. We see that the energy functional associated with (1.), I λ (w) =. 1 w λ QF (w), w E E, is well defined and I λ C 1 (E E, R) with derivative given by, for w, z E E, I λ(w)z = w, z λ Qz F (w). Hence, a weak solution w E E of (1.) is exactly a critical point of I λ. Furthermore, since I λ (0) = 0 and F is odd, I λ satisfies (J1) and with similar computations to prove (i) in Lemma 3.1 we conclude that I λ also verifies (J). In order to verify (J3) and (J4) we consider the following lemma. Lemma 4.. Assume (V1) and (Q1). If F satisfies (F0)-(F4), we have (i) there exists S > 0 such that max w W I λ (w) S; (ii) the functional I λ satisfies the (P S) c condition for all c (0, S), that is, any sequence (w n ) in E E such that I λ (w n ) c and I λ(w n ) 0 (4.1) admits a convergent subsequence in E E. Proof. By (F4), [ 1 ] max I λ(w) = max w W w W w λ QF (w) R [ 1 max w W u fw + 1 v fw µλ ν u ν L ν ( ;Q) µλ ] ν v ν L ν ( ;Q) [ 1 max u W f u fw µλ ] [ 1 ν u ν L ν ( ;Q) + max v W f v fw µλ ] ν v ν L ν ( ;Q). Now, once dim W <, the equivalence of the norms in this space gives a constant C > 0 such that [ 1 u fw µλ ] Cν u ν W f max u f W [ 1 + max v W f v fw µλ ] Cν v ν W f = M k (λ), where M k (λ) =. ν (C ) /(ν )λ /( ν). ν µ Since /( ν) < 0 we have that lim λ + M k (λ) = 0, which implies that there exists Λ k > 0 such that M k (λ) < α /(4α 0 ) =. S for any λ > Λ k. Therefore, i) is proved. For ii), by Lemma 3., (w n ) is bounded in E E and so, up to a subsequence, w n w weakly in E E. We claim that Qw F (w n ) Qw F (w) as n. (4.)

10 10 F. S. B. ALBUQUERQUE EJDE-014/59 Indeed, since C0,rad (R ) is dense in E, for all δ > 0, there exists v C0,rad (R, ) such that w v < δ. Observing that Qw [ F (w n ) F (w)] Q(w v) F (w n ) + v Q F (w n ) F (w) supp(v) + Q(w v) F (w) and using Cauchy-Schwarz and the fact that I λ (w n)(w v) ε n w v with ε n 0, we obtain Q(w v) F (w n ) ε n w v + w n w v C w v < Cδ, where we have used that (w n ) is bounded in E E. Similarly, since the second limit in (4.1) implies that I λ (w)(w v) = 0, we have Q(w v) F (w n ) < Cδ. From Lemma 3.3, lim Q F (w n ) F (w) = 0. supp(v) Thus, lim Qw [ F (w n ) F (w)] < Cδ. Since δ > 0 is arbitrary, the claim follows. Hence, passing to the limit when n in o n (1) = I λ(w n )w = w n, w λ Qw F (w n ) and using that w n w weakly in E E, (4.) and (F) we obtain w = λ Qw F (w) λ QF (w). Hence I λ (w) 0. (4.3) We have two cases to consider: Case 1: w = 0. This case is similar to the checking that the solution w 0 obtained in the Theorem 1.1 is nontrivial. Case : w 0. In this case, we define z n = w n w n and z = w lim w n. It follows that z n z weakly in E E, z n = 1 and z 1. If z = 1, we conclude the proof. If z < 1, it follows from Lemma 3.3 and (4.1) that 1 lim w n = c + λ QF (w). (4.4) Setting A =. ( ) c + λ QF (w) (1 z ),

11 EJDE-014/59 NONLINEAR SCHRÖDINGER ELLIPTIC SYSTEMS 11 by (4.4) and the definition of z, we obtain A = c I λ (w). Hence, coming back to (4.4) and using (4.3), we conclude that 1 lim w n A = 1 z = c I λ(w) 1 z c 1 z < α 4α 0 (1 z ). Consequently, for n N large, there are r > 1 sufficiently close to 1, α > α 0 close to α 0 and β > 0 such that rα w n β < α (1 z ) 1. Therefore, from Corollary.5, 1) r < +. (4.5) Q(e α wn Next, we claim that lim Q(w n w) F (w n ) = 0. Indeed, let r, s > 1 be such that 1/r + 1/s = 1. Invoking (3.1) and the Hölder s inequality we conclude that ( ) 1/ ( ) 1/ Q(w n w) F (w n ) ε Q w n Q w n w R ( + C 1) r) 1/r( Q w n w s) 1/s. Q(e α wn Then, from Lemma.3 and (4.5), the claim follows. This convergence together with the fact that I λ (w n)(w n w) = o n (1) imply that lim w n = w and so w n w strongly in E E. The proof is complete. Proof of Theorem 1.. Since I λ satisfies (J1) (J4), the result follows directly from Theorem 4.1. Acknowledgements. The author thanks the anonymous referee for the careful reading, valuable comments and corrections, which provided a significant improvement of the paper. This work is a part of the author s Ph.D. thesis at the UFPB Department of Mathematics and the author is greatly indebted to his thesis adviser Professor Everaldo Souto de Medeiros for many useful discussions, suggestions and comments. References [1] Adimurthi, K. Sandeep; A singular Moser-Trudinger embedding and its applications, Nonlinear Differential Equations and Applications 13 (007) [] F. S. B. Albuquerque, C. O. Alves, E. S. Medeiros; Nonlinear Schrödinger equation with unbounded or decaying radial potentials involving exponential critical growth in, J. Math. Anal. Appl. 409 (014) [3] A. Ambrosetti, P.H. Rabinowitz; Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973) [4] P. Bartolo, V. Benci, D. Fortunato; Abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity, Nonlinear Anal. TMA 7 (9) (1983)

12 1 F. S. B. ALBUQUERQUE EJDE-014/59 [5] D. M. Cao; Nontrivial solution of semilinear elliptic equation with critical exponent in, Comm. Partial Differential Equations 17 (199) [6] D. G. de Figueiredo, O. H. Miyagaki, B. Ruf; Elliptic equations in with nonlinearities in the critical growth range, Calc. Var. Partial Differential Equations 3 (1995) [7] J. M. do Ó, E. S. de Medeiros, U. Severo; A nonhomogeneous elliptic problem involving critical growth in dimension two, J. Math. Anal. Appl. 345 (008) [8] P.-L. Lions; The concentration-compactness principle in the calculus of variations. Part I, Rev. Mat. Iberoamericana 1 (1985) [9] J. Moser; A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J. 0 (1971) [10] B. Ruf; A sharp Trudinger-Moser type inequality for unbounded domains in, J. Funct. Anal. 19 () (005) [11] E. A. B. Silva; Critical point theorems and applications to differential equations, Ph.D. Thesis, University of Wisconsin-Madison, [1] W.A. Strauss; Existence of solitary waves in higher dimensions, Comm. Math. Phys. 55 (1977) [13] J. Su, Z.-Q. Wang, M. Willem; Nonlinear Schrödinger equations with unbounded and decaying radial potentials, Comm. Contemp. Math. 9 (007) [14] J. Su, Z.-Q. Wang, M. Willem; Weighted Sobolev embedding with unbounded and decaying radial potentials. J. Differential Equations 38 (007) [15] N. S. Trudinger; On the embedding into Orlicz spaces and some applications, J. Math. Mech. 17 (1967) [16] Z.-Q. Wang, M. Willem; Caffarelli-Kohn-Nirenberg inequalities with remainder terms, J. Funct. Anal. 03 (003) Francisco Siberio Bezerra Albuquerque Centro de Ciências Exatas e Sociais Aplicadas, Universidade Estadual da Paraíba , Patos, PB, Brazil address: siberio

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