EXISTENCE OF SOLUTIONS FOR CROSS CRITICAL EXPONENTIAL N-LAPLACIAN SYSTEMS

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1 Electronic Journal of Differential Equations, Vol (2014), o. 28, pp ISS: URL: or ftp ejde.math.txstate.edu EXISTECE OF SOLUTIOS FOR CROSS CRITICAL EXPOETIAL -LAPLACIA SYSTEMS XIAOZHI WAG Abstract. In this article we consider cross critical exponential -Laplacian systems. Using an energy estimate on a bounded set the Ekel variational principle, we prove the existence of a nontrivial weak solution, for a parameter large enough. 1. Introduction Let be a bounded smooth domain in R 2. Firstly we consider the problem u = au u 2 + bu u 4 2 v /2 + du( u 2 + α 0 1 u ) v exp{α 0 u 1 v = bv v 4 2 u /2 + cv v 2 + dv( v 2 + β 0 1 v ) u exp{α 0 u u = 0, v = 0 on, 1 + β0 v 1 } in, + β0 v 1 } in, (1.1) where a, b, c, d, α 0, β 0 are real constants α 0, β 0 > 0. For similar problem, to our knowledge, de Figueiredo, do O Ruf [3] firstly discussed the coupled system of exponential type in R 2 u = g(v) in, v = f(u) in, u = 0, v = 0 on, (1.2) where f(u), g(v) behave like exp{α u 2 } exp{α v 2 } respectively for some α > 0 at infinity. They obtained the existence of the positive solution by a linking theorem in Hilbert space. Recently, Lam Lu [5] extended this existence result of problem (1.2) on the condition that the nonlinear terms satisfy a weak Ambrosetti- Rabinowitz condition. Furthermore, the author [9] proved a similar result for a class of cross critical exponential system even if these critical nonlinear terms without Ambrosetti-Rabinowitz condition. For further recent researches on exponential 2000 Mathematics Subject Classification. 35J50, 35B33. Key words phrases. -Laplacian system; critical exponential growth; Ekel variational principle. c 2014 Texas State University - San Marcos. Submitted October 26, Published January 15,

2 2 X. WAG EJDE-2014/28 system, we refer to [4, 7, 8] the references therein. Our main propose of this article is to study a class nonuniform critical exponential terms similar to (1.1), which weaken the critical assumptions used in [9], further elaborate the idea of [9] that proper energy estimate guarantees the nontrivial weak solutions for some critical growth systems. In the last section, we will extend this existence result to a wider class of nonlinear terms with cross critical growth. More exactly, we study the problem u = a u 2 u + bu u /2 2 v /2 + df(x, u, v) in, v = bv v /2 2 u /2 + c v 2 v + dg(x, u, v) in, u = 0, v = 0 on, (1.3) where a, b, c, d are constants f(x, u, v), g(x, u, v) with critical growth at α 0, β 0 > 0 respectively. Here we say f(x, u, v) g(x, u, v) have critical growth at α 0, β 0 respectively, if there exist positive constants α 0, β 0 such that: For any v 0, f(x, u, v) f(x, u, v) lim = 0, α > α 0 lim = +, α < α 0 ; u exp{α u 1 } u exp{α u 1 } (1.4) for any u 0, g(x, u, v) g(x, u, v) lim = 0, β > β 0 lim = +, β < β 0. v exp{β v 1 } v exp{β v 1 } (1.5) Since the system is not variational in general, we assume that there exists the primitive F (x, u, v) such that F u (x, u, v) = f(x, u, v), F v (x, u, v) = g(x, u, v). We weaken some of the critical exponential assumptions used in [9], as follows: (F1) f(x, t, s), g(x, t, s) : R R R are Carathéodory functions satisfying f(x, t, 0) = f(x, 0, s) = g(x, t, 0) = g(x, 0, s) = 0; (F2) F (x, s, t) > 0, for t, s R + a.e. x. We note that the above assumptions have been simplified. From the exponential growth condition, the explicit exponential nonlinear term F (x, u, v) = h(x, u, v) exp{k(x, u, v)u /( 1) } exp{l(x, u, v)v /( 1) } satisfies the Ambrosetti-Rabinowitz condition, where lim u k(x, u, v) = α 0, lim v l(x, u, v) = β 0 h(x, u, v) 0. It is obvious that f(x, u, v) = h u (x, u, v) exp{k(x, u, v)u /( 1) } exp{l(x, u, v)v /( 1) } ( ) 1 + h(x, u, v) k(x, u, v)u 1 + ku (x, u, v)u 1 1 exp{k(x, u, v)u /( 1) } exp{l(x, u, v)v /( 1) }, g(x, u, v) = h v (x, u, v) exp{k(x, u, v)u /( 1) } exp{l(x, u, v)v /( 1) } ( ) 1 + h(x, u, v) k(x, u, v)v 1 + kv (x, u, v)v 1 1 exp{k(x, u, v)u /( 1) } exp{l(x, u, v)v /( 1) },

3 EJDE-2014/28 EXISTECE OF SOLUTIOS 3 Since h u (x, u, v), h v (x, u, v), k u (x, u, v), k v (x, u, v) h(x, u, v) 0, there exist constants C, M > 0 such that for all u, v C, 0 < F (x, u, v) M(f(x, u, v) + g(x, u, v)) for a.e. x ; i. e. the Ambrosetti-Rabinowitz condition is satisfied. On the other h, without the assumption lim sup t 0 = 0, we could not have mountain pass F (x,t,s) t + s geometry. A typical example is given as follows: F (x, u, v) = u v exp{α 0 e u 3 u /( 1) } exp{β 0 e v 3 v /( 1) }. Here are the main results of this article for problem (1.1). Theorem 1.1. Under the assumptions a, c < λ 1, there exists a positive constant Λ such that (1.1) has at least one solution for all d > Λ, where λ 1 as in (2.2) Λ depends on a, b, c, α 0, β 0, the dimension the domain. The following theorem extends partially the existence result of nontrivial weak solution presented in [9]. Theorem 1.2. If a, c < λ 1 the assumption (F1)-(F2) are satisfied, there exists a positive constant Θ such that (1.3) has at least one solution for all d > Θ, where λ 1 as in (2.2) Θ depends on a, b, c, α 0, β 0, the dimension the domain. This article is organized as follows. Section 2 contains the preliminaries. Section 3 shows two important estimate results. Section 4 shows the proof of Theorem 1.1. Section 5 provides a simple proof of Theorem Preliminaries Throughout this paper, we define ( ) u = u 1/ (, u = u ) 1/, I(u, v) = 1 u + 1 v a u c v 2b u /2 v /2 d u v exp{α 0 u 1 } exp{β0 v It is well known that λ 1 = u min u W 1, 0 ()\{0} u 1 }. (2.1) > 0, (2.2) The space X designates the product space W 1, 0 () W 1, 0 () equipped by the norm (u, v) X = u + v. It is well known that the maximal growth of u W 1, 0 () is of exponential type, see references [6] [9]. More precisely, we have the following uniform bound estimate (see also [2]):

4 4 X. WAG EJDE-2014/28 Trudinger-Moser inequality. Let u W 1, 0 (), then exp{ u 1 } L θ () for all 1 θ <. That is to say that for any given θ > 0, any u W 1, 0 () holds exp{θ u 1 } L 1 (). Moreover, there exists a constant C = C(, α) > 0 such that sup exp(α u 1 ) C, if 0 α α, (2.3) u 1 where is the dimension Lebesgue measure of, α = ω 1 1 ω is the 1 dimension Hausdorff measure of the unit sphere in R. Furthermore, if α > α, then C = +. Here throughout this paper, we often denote various constants by same C. The reader can recognize them easily. Thanks to Trudinger- Moser inequality, we know the functional I(u, v) is well defined. Using a stard argument, we also deduce that the functional I(u, v) is of class C 1 I (u, v), (ϕ, φ) = u 2 u ϕ + v 2 v φ a u 2 uϕ c v 2 vφ b uϕ u /2 2 v /2 b vφ v /2 2 u /2 d uϕ( u 2 + α 0 1 u ) v exp{α 0 u 1 + β0 v 1 } d vφ( v 2 + β 0 1 v ) u exp{α 0 u 1 + β0 v 1 }, (2.4) for any ϕ, φ W 1, 0 (). Obviously, the critical points of I(u, v) are precisely the weak solutions for problem (1.1). By the critical assumptions (1.4), (1.5) (F1), the functional J(u, v) = 1 u + 1 v 1 a u 1 c v 2 b u /2 v /2 d F (x, u, v), is well defined of class C 1 such that the critical points of J(u, v) are precisely the weak solutions for problem (1.3); i.e., J (u, v), (ϕ, φ) = u 2 u ϕ + v 2 v φ a u 2 uϕ c v 2 vφ b uϕ u /2 2 v /2 b vφ v /2 2 u /2 d f(x, u, v)ϕ d g(x, u, v)φ. (2.5) Lemma 3.1. If u 1 ( u 1 + α 0 < α 1 u Energy estimates v 1 < α β0, there exists q > 1 such that ) q v q exp{qα 0 u 1 + qβ0 v 1 } C

5 EJDE-2014/28 EXISTECE OF SOLUTIOS 5 ( v 1 + β 0 1 v ) q u q exp{qα 0 u 1 + qβ0 v 1 } C. Proof. By contradiction. Then for any ε 1, ε 2 > 0 any q > 1, we estimate that ( u 1 + α 0 1 u ) q v q exp{qα 0 u 1 + qβ0 v 1 } C exp{q(α 0 + ε 1 ) u 1 } exp{q(β0 + ε 2 ) v 1 } = C exp{q(α 0 + ε 1 ) u 1 ( u ) u 1 } exp{q(β0 + ε 2 ) v 1 ( v ) 1 }, v tends to infinite. Then by Trudinger-Moser inequality (2.3), we get that q(α 0 + ε 1 ) u 1 > α or q(β 0 + ε 2 ) v 1 > α. Since q > 1 ε 1, ε 2 > 0 are arbitrary, we have u 1 α or v α 0 1 α β 0, which contradicts our assumptions. Applying similar argument to ( v 1 + β 0 1 v ) q u q exp{qα 0 u 1 + qβ0 v 1 }, we deduce the conclusion. We denote the Moser functions as follows (log n) 1, x 1/n; M n (x) := ω 1/ log(1/ x ), 1/n x 1; (log n) 1/ 0, x 1; where 2 n + ω as in (2.3), i.e. 1 ω = α 1. Let r be the inner radius of x 0 such that B r (x 0 ). Then the functions M n (x) := M n ( x x 0 ) r satisfy M n = 1, M n = O(1/ log n) supp M n B r (x 0 ). We define a close convex ball as B,β 0 := {(u, v) X (u, v) 1 X min( α, α )}. α 0 β 0 ow, we give an estimate from below for the functional I(u, v) on the ball in B,β 0. Lemma 3.2. There exist a constant Λ such that for all d > Λ, inf I(u, v) = c 0 < 0, (3.1) (u,v) B,β 0 where Λ depends on a, b, c, α 0, β 0, the dimension the domain. Proof. Without loss generality, we assume that α 0 β 0. Here we take u n = M n 1 2 ( α ) 1 v n = 1 2 (α ) 1 Mn 1 α 0 2 (α ) 1 Mn. β 0

6 6 X. WAG EJDE-2014/28 Then u n = v n = 1 2 ( α ) 1 M n (x), we have a u n + c = a + 2b + c 2 ω ( α α 0 ) 1 = v n + 2b (i.e. (u n, v n ) B,β 0 ). Form the definition of u n /2 v n /2 B(x 0, r n ) (log n) 1 + a + 2b + c 2 ( α r (log ) 1 x x 0 ω α 0 B(x0,r)\B(x0, ) rn ) log n (a + 2b + c)r 2 2 n ( α α 0 ) 1 (log n) 1 + a + 2b + c 2 log n (α α 0 ) 1 = O(1/ log n), = u n v n exp{α 0 u n 1 } exp{β0 v n α 2( 1) 4 ω 2 α2( 1) 0 + α 2( 1) 4 ω 2 α2( 1) 0 exp{( 2 1 ω r 4 n 2 3 r r n α 2( 1) 0 B(x 0, r n ) (log n) 2( 1) exp{ 1 } 2 1 B(x0,r)\B(x0, rn ) (log r x x 0 )2 + β 0 2 (log r l )2 exp{( 1 ) (log (log n) 2( 1) n 2 1 (log n) 2 r x x ) 1 0 (log n) 1 1 } + β β 0 2 ) (log r l ) 1 (log n) 1 r r n (log r l ) l 1 dl log n + β log n} α 2( 1) 4 ω α 2( 1) 1 1 }l 1 dl. 0 (log n) 2 (3.2) (3.3) Obviously, for fixed n, we deduce that expression (3.2) is bounded expression (3.3) is larger than a positive constant. oticing the definitions of u n, v n, we obtain that there exists a positive constant Λ such that for all d > Λ holds I(u n, v n ) < 0, which implies that inf I(u, v) = c 0 < 0. (u,v) B,β 0 4. Proof of Theorem 1.1 Since B,β 0 is a Banach space with the norm given by the norm of X, the functional I(u, v) is of class C 1 bounded below on B,β 0. In fact, if u 1 equals to min( α, α β0 ), then v = 0. Hence that u v exp{α 0 u 1 } exp{β0 v 1 } = 0. (4.1)

7 EJDE-2014/28 EXISTECE OF SOLUTIOS 7 And same result holds for v 1 Lemma 3.1, we conclude that for u 1, v 1 bounded below on B,β 0. = min( α, α β0 ). By a similar argument of u v exp{α 0 u 1 } exp{β0 v 1 } C (4.2) < min( α, α β0 ). That is to say that the functional I(u, v) is Thanks to Ekel s variational principle [1, Corollary A.2], there exists some minimizing sequence {(u n, v n )} B,β 0 such that I(u n, v n ) inf I(u, v) = c 0 < 0, (4.3) (u,v) B,β 0 I (u n, v n ) 0 in X, as as n. (4.4) From (2.4) (4.4), taking (ϕ, φ) = (u n, 0) (ϕ, φ) = (0, v n ) respectively, we have d u n a u n b u n /2 v n /2 ( u n + α (4.5) 0 1 u n 2 1 ) vn exp{α 0 u n 1 + β0 v n 1 } 0, v n c d ( v n + β 0 v n b u n /2 v n /2 1 v n 2 (4.6) 1 ) un exp{α 0 u n 1 + β0 v n 1 } 0. Since u n, v n are uniform bounded in W 1, 0 (), by Lemma 6.1 in the Appendix, we conclude that u n u 0, v n v 0 in W 1, 0 (), (u 0, v 0 ) is a weak solution for problem (1.1). ow, we prove that this weak solution is nontrivial. Proposition 4.1. The above weak solution (u 0, v 0 ) is a nontrivial solution for problem (1.1). Proof. By the assumptions a, c < λ 1 d > 0, we have that u 0 = 0 if only if v 0 = 0. The condition d > 0 guarantees this problem is nontrivial. In fact, if u 0 = 0, then v 0 is a solution of the equation p v = c v 2 v < λ 1 v 2 v in, v = 0 on. Obviously, we have v 0 = 0. ow we suppose that u 0 = v 0 = 0. Then by u n, v n 0 weak convergence in W 1, 0 (), we have lim a u n, lim c v n 2b, lim u n /2 v n /2 = 0.

8 8 X. WAG EJDE-2014/28 These together with (4.5) (4.6), from Lemma 3.1, by Hölder inequality, we obtain u n, v n 0. i.e. (u 0, v 0 ) (0, 0) strongly in X. Obviously, u n v n exp{α 0 u n 1 } exp{β0 v n 1 } 0, as n. Hence that which contracts with (4.3). lim I(u n, v n ) = 0, Thus the proof of theorem 1.1 is complete. 5. Proof of Theorem 1.2 In this section we show the existence of nontrivial weal solution for more general quasilinear system (i.e. problem (1.3)). As the proofs are similar we will sketch from place to place. oticing the assumptions (1.4) (1.5), by similar arguments of Lemma 3.1, we would see that f(x, u, v) q, g(x, u, v) q C (5.1) for some q > 1 u 1 < α v 1 < α β0. By assumption (F2), choosing a proper constant c 0 such that (u n, v n ) = (cm n (x), cm n (x)) B,β 0, we have F (x, u n, v n ) C (5.2) for some fixed n > 1, which means that inf J(u, v) = c 0 < 0 (u,v) B,β 0 for d large enough. Form the assumption (F1), similar to equality (4.1) inequality (4.2), we have that J(u, v) is bounded below on B,β 0. This combined with B,β 0 is a Banach space with the norm given by the norm of X the functional J(u, v) is of class C 1, by Ekel s variational principle [1, Corollary A.2], there exists some minimizing sequence {(u n, v n )} B,β 0 such that J(u n, v n ) inf (u,v) B,β 0 J(u, v) = c 0 < 0, (5.3) J (u n, v n ) 0 in X, as n. (5.4) From (2.5) (5.4), taking (ϕ, φ) = (u n, 0) (ϕ, φ) = (0, v n ) respectively, we have u n a u n b u n /2 v n /2 d f(x, u n, v n )u n 0, (5.5) v n c v n b u n /2 v n /2 d g(x, u n, v n )v n 0. (5.6)

9 EJDE-2014/28 EXISTECE OF SOLUTIOS 9 Since u n, v n are uniform bounded in W 1, 0 (), by Lemma 6.1 in the appendix, we conclude that u n u 0, v n v 0 in W 1, 0 (), (u 0, v 0 ) is a weak solution for problem (1.3). ow, we will prove this weak solution is nontrivial. Proposition 5.1. The above weak solution (u 0, v 0 ) is nontrivial. Proof. By the assumptions (F1), a, c < λ 1 d > 0, using same argument for Proposition 4.1, we can get that u = 0 if only if v = 0. ow we suppose that u 0 = v 0 = 0. Then by u n, v n 0 weak convergence in W 1, 0 (), we have lim a u n, lim c v n 2b, lim u n /2 v n /2 = 0. These together with (5.1), (5.5) (5.6), by Hölder inequality, we obtain u n, v n 0. i.e. (u 0, v 0 ) (0, 0) strong convergence in X, which means F (x, u n, v n ) 0, as n. Hence lim J(u n, v n ) = 0, which contracts with (5.3). Thus the proof of Theorem 1.2 is complete. 6. Appendix Here we give a brief proof for the existence result of the weak solution for problem (1.3), see also [11], however the non-triviality of this weak solution need to be clarified. Lemma 6.1. Suppose the sequences {u n }, {v n } are bounded in W 1, 0 (), the lim J (u n, v n ) 0 in X, then there exist u 0, v 0 such that u n u 0, v n v 0 in W 1, 0 () J (u 0, v 0 ), (ϕ, φ) = 0 for all ϕ, φ W 1, 0 (). Proof. Since {u n }, {v n } are bounded in W 1, 0 (), there exist u 0, v 0 such that u n u 0 v n v 0, which implies u n u 0 v n v 0 in L 1 (). By assumptions (1.4) (1.5), using Trudinger-Moser inequality, we have f(x, u n, v n )u n C, f(x, u n, v n )v n C, g(x, u n, v n )v n C, g(x, u n, v n )u n C. Combining the above results, we find that f(x, u n, v n ) f(x, u 0, v 0 ), g(x, u n, v n ) g(x, u 0, v 0 ) in L 1 (). (6.1) ow, taking test function (τ(u n u 0 ), 0), the assumption lim J (u n, v n ) 0 becomes I 2(u n, v n ), (τ(u n u 0 ), 0)

10 10 X. WAG EJDE-2014/28 where = u n 2 u n τ(u n u 0 ) + au n τ(u n u 0 ) u n 2 + bu n τ(u n u 0 ) u n /2 2 v n /2 + f(x, u n, v n )τ(u n u 0 ) 0, τ(t) = { t, if t 1; t/ t, if t > 1. Hence by (6.1) τ(u n u 0 ) 0, we deduce ( u n p 2 u n u 0 p 2 u 0 ) τ(u n u 0 ) 0, which implies u n u 0 a.e. in ; see [10, Theorem 1.1]. Since 2, we know u n 2 u n u 0 2 u 0 in (L /( 1) ()). Using similar argument, we get the same result for sequence {v n }. By these results combined with (6.1) J (u n, v n ) 0, we obtain that J (u 0, v 0 ), (ϕ, φ) = 0 for any ϕ, φ D(). By using an argument of density, this identity holds for all ϕ, φ W 1, 0 (). Then the proof is complete. Acknowledgements. The author would like to thank the anonymous referees for the careful reading of the original manuscript for the valuable suggestions. References [1] D. G. Costa; An Invitation to Variational Methods in Differential Equations, Birkhäuser, [2] D. G. de Figueiredo, J.M. do O, B. Ruf; On an inequality by. Trudinger J. Moser related elliptic equations, Comm. Pure Appl. Math. 55 (2002), [3] D. G. de Figueiredo, J.M. do O, B. Ruf; Critical subcritical elliptic systems in dimension two, Indiana Univ. Math. J. 53 (2004), [4] D. G. de Figueiredo, J. M. do O, B. Ruf; Elliptic equations systems with critical Trudinger-Moser nonlinearites, Disc. Cont. Dyna. Syst. 30 (2011), [5]. Lam, Guozhen Lu; Elliptic equations systems with subcritical critical exponential growth without the Ambosetti-Rabinowitz condition, J. Geometry Analysis 24 (2014), [6] J. Moser; A sharp form of an inequality by.trudinger, Indiana Univ. Math. J. 20 (1971), [7] B. Ribeiro; The Ambrosetti-Prodi problem for elliptic systems with Trudinger-Moser nonlinearities, Proc. Edinburgh Math. Society 55 (2012), [8] M. D. Souza; Existence multiplicity of solutions for a singular semilinear elliptic problem in R 2, Electron. J. Diff. Equa. 2011, o. 98 (2011), [9]. S. Trudinger; On the embedding into Orlicz spaces some applications, J. Math. Mech. 17 (1967), [10] S. de Valeriola, M. Willem; On some quasilinear critical problems, Adv. onlinear Stud. 9 (2009), [11] Xiaozhi Wang; A Positive Solution for some Critical p-laplacian Systems, Preprint. Xiaozhi Wang College of Science, University of Shanghai for Science Technology, Shanghai , China address: yuanshenran@yeah.net