A PRIORI ESTIMATES AND APPLICATION TO THE SYMMETRY OF SOLUTIONS FOR CRITICAL

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1 A PRIORI ESTIMATES AND APPLICATION TO THE SYMMETRY OF SOLUTIONS FOR CRITICAL LAPLACE EQUATIONS Abstract. We establish ointwise a riori estimates for solutions in D 1, of equations of tye u = f x, u, where 1, n, := div u 2 u is the Lalace oerator, and f is a Caratheodory function with critical Sobolev growth. In the case of ositive solutions, our estimates allow us to extend revious radial symmetry results. In articular, by combining our results and a result of Damascelli Ramaswamy [6], we are able to extend a recent result of Damascelli Merchán Montoro Sciunzi [7] on the symmetry of ositive solutions in D 1, of the equation u = u 1, where := n/ n. 1. Introduction and main results In this aer, we are interested in roblems of the tye { u = f x, u in, u D 1,, 1.1 where 1, n, u := div u 2 u, D 1, is the comletion of C c with resect to the norm u D 1, := u dx 1/, and f : R R is a Caratheodory function such that f x, s Λ s 1 for all s R and a.e. x, 1.2 for some real number Λ > 0, with := n/ n. Our main result is as follows. Theorem 1.1. Let 1, n, f : R R be a Caratheodory function such that 1.2 holds true and u be a solution of 1.1. Then there exists a constant C 0 = C 0 n,, Λ, u such that u x C x 1 1 and u x C x n for all x. If moreover u 0 in and f x, u dx > 0, then we have 1 u x C x for all x, for some constant C 1 = C 1 n,, λ, Λ, u > 0, where λ is a real number such that 0 < λ < f x, u dx. Date: July 25, Revised: Aril 11, Published in Journal of Differential Equations , no. 1,

2 2 The deendence on u of the constants C 0 and C 1 will be made more recise in Remarks 4.1 and 4.3. In the case of the Lalace oerator = 2, the uer bound estimates 1.3 have been established by Jannelli Solimini [15] for nonlinearities of the form f x, u = N i=1 a i x u q i 2 u, where qi := 2 1 1/q i, q i n/2, ], a i x = O x n/q i for large x, and a i belongs to the Marcinkiewicz sace M q i for all i = 1,..., N. The case of unbounded domains Ω is also treated in [15]. Since the ioneer work of Gidas Ni Nirenberg [12] and later extensions by Li [18] in case = 2 and Damascelli Ramaswamy [6] in case 1 < < 2, decay estimates are known to be useful to derive radial symmetry results for C 1 solutions of roblems of the tye { u = f u, u > 0 in, 1.5 u x 0 as x 0. Here, we consider the following result of Damascelli Ramaswamy [6] and Li [18]: if 1 < 2, f is a locally Lischitz continuous function in 0, such that f v f u Λ max u α, v α u, v such that 0 < u < v < s v u for some real numbers Λ, s 0 > 0, and α > 2, and u is a C 1 solution of 1.5 such that u x = O x m and u x = O x m and u x C x m for large x when α < for some real numbers C > 0 and m > / α + 2, then u is radially symmetric and strictly radially decreasing about some oint x 0, i.e. there exists v C 1 0, such that v r < 0 for all r > 0 and u x = v x x 0 for all x. We also mention that other symmetry results for roblems of tye 1.5 have been established without any decay assumtion in the case where f is nonincreasing near 0 see Gidas Ni Nirenberg [12], Li [18], and Li Ni [19] in case = 2, Damascelli Pacella Ramaswamy [5], Damascelli Ramaswamy [6], and Serrin Zou [26] in case 2. In case α = 2, the conditions follow from with m = n / 1 which is greater than / α + 2 = n /. Consequently, by combining Theorem 1.1, the results of Damascelli Ramaswamy [6] and Li [18], and the regularity results that are referred to in Lemma 2.1 below, we obtain the following corollary. Corollary 1.2. Assume that 1 < 2. Let f be a locally Lischitz continuous function in 0, such that 1.2 and 1.6 hold true with α = 2. Then any nonnegative solution of 1.1 is radially symmetric and strictly radially decreasing about some oint x 0.

3 A PRIORI ESTIMATES FOR CRITICAL LAPLACE EQUATIONS 3 Let us now comment on the ositive solutions of the equation with ure ower nonlinearity, namely u = u 1, u > 0 in. 1.9 Guedda Véron [14] roved that the only ositive, radially symmetric solutions of 1.9 are of the form u µ,x0 x = nµ 2 1 n µ 2 + x x 0 n for all x, for some real number µ > 0 and oint x 0. In case = 2, Caffarelli Gidas Sruck [2] see also Chen Li [3] roved that the functions 1.10 are the only ositive solutions of 1.9. In a recent aer, Damascelli Merchán Montoro Sciunzi [7] roved that any solution in D 1, of 1.9 is radially symmetric rovided that 2n/ n + 2 < 2. The condition 2n/ n + 2 corresonds to the values of for which the function s s 1 is Lischitz continuous near 0. With the above Corollary 1.2, we extend the result of Damascelli Merchán Montoro Sciunzi [7] to the whole interval 1 < < 2. By combining the result of Guedda Véron [14] and Corollary 1.2, we obtain the following corollary. Corollary 1.3. Assume that 1 < < 2. Then the functions 1.10 are the only ositive solutions in D 1, of 1.9. As a motivation to our results, it is well known that the rofile of solutions of the equation u = u 2 u in 1.11 lays a central role in the blow-u theories of critical equations. Possible references in book form on this subject and its alications in case = 2 are Druet Hebey Robert [9], Ghoussoub [11], and Struwe [28]. In case 2, global comactness results in energy saces in the sirit of Struwe [27] have been established in different contexts by Alves [1] for equations osed in the whole, Saintier [23] in the case of a smooth, comact manifold, and Mercuri Willem [20] and Yan [34] in the case of a smooth, bounded domain. In view of these results, it is likely that the new information rovided by Theorem 1.1 and Corollary 1.2 on the solutions of 1.11 will lead to new existence and multilicity results as it is the case for = 2. The aer is organized as follows. Section 2 is concerned with global boundedness results. The key result in this section is a global bound in weak Lebesgue saces which we obtain by arguments of measure theory. In Section 3, we establish a reliminary decay estimate which is not shar but which turns out to be a crucial ingredient in what follows. To rove this estimate, we exloit the scaling law of the

4 4 equation, and we aly a doubling roerty from Poláčik Quittner Soulet [22]. In Section 4, we conclude the roof of Theorem 1.1. The roof of the uer bound estimates 1.3 follows from the results of Sections 2 and 3 together with Harnack-tye inequalities of Serrin [25] and Trudinger [30]. The roof of the lower bound estimate 1.4 relies on a Harnack inequality on annuli, which is insired from similar results used in Friedman Véron [10] and Véron [33] for the study of singular solutions of Lalace equations in ointed domains. Note. Since this aer was written, the result of Corollary 1.3 has been extended to all 1, n by Sciunzi [24]. The roof in [24] is based on the moving lane method. It uses the estimates of Theorem 1.1 together with a shar lower bound estimate for the norm of the gradient of the solutions. Acknowledgments. The author wishes to exress his gratitude to Emmanuel Hebey for helful comments on the manuscrit. 2. Global boundedness results The first result of this section refers to some known regularity results for critical equations. Lemma 2.1. Let f : R R be a Caratheodory function such that 1.2 holds true. Then any solution of 1.1 belongs to W 1, C 1,θ loc Rn for some θ 0, 1. Proof of Lemma 2.1. A straightforward adatation of Peral [21, Theorem E.0.20] which in turn is adated from Trudinger [31, Theorem 3] yields that for any solution u of 1.1, there exist constants C, R > 0 and β > 1 such that u L β Bx,R C for all x Rn, where B x, R is the Euclidean ball of center x and radius R. We then obtain a global L bound by alying Serrin [25, Theorem 1]. Once we have the L boundedness of the solutions, the results of DiBenedetto [8] and Tolksdorf [29] rovide global L bounds and local Hölder regularity for the derivatives. The next result is concerned with the boundedness of solutions of 1.1 in weak Lebesgue saces. For any s 0, and any domain Ω, we define L s, Ω as the set of all measurable functions u : Ω R such that u L s, Ω := su h meas { u > h} 1/s <, h>0 where meas { u > h} is the measure of the set {x Ω : u x > h}. The ma L s, Ω defines a quasi-norm on Ls, Ω see for instance Grafakos [13]. Our result is as follows.

5 A PRIORI ESTIMATES FOR CRITICAL LAPLACE EQUATIONS 5 Lemma 2.2. Let f : R R be a Caratheodory function such that 1.2 holds true. Then any solution of 1.1 belongs to L 1,, where := n 1 / n. Hence, by interolation see for instance Grafakos [13, Proosition ], since by Lemma 2.1 any solution of 1.1 belongs to L, we obtain that the solutions belong to L s for all s 1, ]. Proof of Lemma 2.2. We let u be a nontrivial solution of 1.1. For any h > 0, by testing 1.1 with T h u := sgn u min u, h, where sgn u denotes the sign of u, we obtain u dx = f x, u u dx + h f x, u sgn u dx. u >h 2.1 It follows from 1.2 and 2.1that u dx Λ u dx + h u 1 dx. 2.2 u >h We then write u dx = T h u dx h meas { u > h} 2.3 and u 1 dx = 1 s 2 meas { u > max s, h} ds u >h 0 = h 1 meas { u > h} + 1 h s 2 meas { u > s} ds. 2.4 It follows from that u dx Λ T h u dx + 1 h s 2 meas { u > s} ds. 2.5 Sobolev inequality gives T h u dx K h u dx n 2.6 for some constant K = K n,. By 2.3, 2.5, 2.6, and since T h u dx = o 1 as h 0, we obtain h meas { u > h} T h u dx R n n C h s 2 meas { u > s} ds 2.7 h

6 6 for small h, for some constant C = C n,, Λ. We then define G h := g s ds, where g s := s 2 meas { u > s}. h Since the function t t / is locally Lischitz in 0, and g s ds > 0 for all h < u h L, we get that G is locally absolutely continuous in 0, u L with derivative G h = n g s ds g h 2.8 n h for a.e. h 0, u L see for instance Leoni [17, Theorem 3.68]. By 2.7 and 2.8, we obtain G h C n h 2 n 2.9 for small h. Integrating 2.9 gives G h G 0 C h 2.10 for small h, where G 0 := lim h 0 G h. On the other hand, by 2.4 and dominated convergence, we have 1 hg h n h u 1 dx = o u >h as h 0. It follows from 2.10 and 2.11 that G 0 > 0, i.e. g s ds <. By 2.7 and since n = 0 1 and G is nonincreasing, we then get h 1 meas { u > h} C G h n/ C G 0 n/ for small h, and hence we obtain u L 1, <. By 1.2 and a weak version of Kato s inequality [16] see Cuesta Leon [4, Proosition 3.2], we obtain u f x, u Λ u 1 where the inequality is in the sense that u 2 u ϕ dx Λ u 1 ϕ dx in, 2.12 for all nonnegative, smooth functions ϕ with comact suort in. Our last result in this section is as follows. Lemma 2.3. For any real number Λ > 0 and any nonnegative, nontrivial solution v D 1, of the inequality v Λv 1 in, we have v L κ 0 for some constant κ 0 = κ 0 n,, Λ > 0.

7 A PRIORI ESTIMATES FOR CRITICAL LAPLACE EQUATIONS 7 Proof. By alying Sobolev inequality and testing v Λv 1 with the function v, we obtain n n v dx K v dx K Λ v dx 2.13 for some constant K = K n,. The result then follows immediately from A reliminary decay estimate The following result rovides a decay estimate which is not shar but which will serve as a reliminary ste in the roof of Theorem 1.1. Lemma 3.1. Let κ 0 be as in Lemma 2.3, f : R R be a Caratheodory function such that 1.2 holds true, and u be a solution of 1.1. For any κ > 0, we define r κ u := inf { r > 0 : u L \B0,r < κ}, where B 0, r is the Euclidean ball of center 0 and radius r. Then for any κ 0, κ 0 and r > r κ u, there exists a constant K 0 = K 0 n,, Λ, κ, r, rκ u, u L such that u x K 0 x n for all x \B 0, r. 3.1 The roof of Lemma 3.1 relies on scaling arguments and the following doubling roerty from Poláčik Quittner Soulet [22]. Lemma 3.2. Let X, dist be a comlete metric sace, D and Σ be two subsets of X such that D, D Σ, and Σ is closed. Let M be a nonnegative function on D which is bounded on comact subsets of D. Then for any oint x 0 in D and any ositive real number α 0 such that dist x 0, Σ\D Mx 0 > 2α 0, there exists a oint y 0 in D such that and dist y 0, Σ\D M y 0 > 2α 0, M x 0 M y 0, 3.2 M y 2M y 0 for all y D B X y 0, α 0 /M y 0, 3.3 where B X y 0, α 0 /M y 0 is the ball of center y 0 and radius α 0 /M y 0 with resect to the distance dist. In the case where X =, dist is the Euclidean distance, D is oen, and Σ = D, it follows from the first inequality in 3.2 that B X y 0, α 0 /M y 0 D, and hence 3.3 holds true for all y B X y 0, α 0 /M y 0. We refer to [22] for the roof of Lemma 3.2. Now we rove Lemma 3.1.

8 8 Proof of Lemma 3.1. We fix Λ > 0, κ 0, κ 0, κ > κ 0, r > 0, and r 0, r. As is easily seen, in order to rove Lemma 3.1, it is sufficient to rove that there exists a constant K 1 = K 1 n,, Λ, κ, κ, r, r such that for any solution u of 1.1 such that r κ u r and u L κ, we have dist x, B 0, r u x K1 for all x \B 0, r, 3.4 where r := r + r /2 and dist is the Euclidean distance function. We rove 3.4 by contradiction. Suose that for any α N, there exists a Caratheodory function f α : R R such that 1.2 holds true, a solution u α of 1.1 with f = f α such that r κ u α r and u α L κ, and a oint x α \B 0, r such that dist x α, B 0, r u α x α > 2α. 3.5 By 3.5 and Lemma 3.2, and since B 0, r B 0, r, we get that there exists a oint y α \B 0, r such that and dist y α, B 0, r u α y α > 2α, uα x α u α y α, 3.6 u α y 2 u α y α for all y B y α, α u α y α For any α and y, we define. 3.7 ũ α y := µ α u α µ α y + y α, 3.8 where µ α := u α y α 1. By 1.1, we obtain ũ α = µ 1 α f α µ α y + y α, µ 1 α ũ α It follows from 1.2 that µ 1 α f α µ α y + y α, µ 1 α ũ α Λ ũα 1 Moreover, by 3.7 and 3.8, we obtain in. 3.9 in ũ α 0 = 1 and ũ α y 2 for all y B 0, α By DiBenedetto [8] and Tolksdorf [29], it follows from 3.10 and 3.11 that there exists a constant C > 0 and a real number θ 0, 1 such that for oint x, we have ũ α C 1,θ Bx,1 C 3.12 for large α. By comactness of C 1,θ B x, 1 C 1 B x, 1, it follows from 3.12 that ũ α α converges u to a subsequence in Cloc 1 Rn to some function ũ. By 3.11, we obtain ũ 0 = 1. Moreover, by alying the inequality 2.12, we obtain ũ α dx = u α dx Λ u α dx Λ κ,

9 A PRIORI ESTIMATES FOR CRITICAL LAPLACE EQUATIONS 9 and hence ũ D 1,. By observing that the inequality 2.12 is invariant by the change of scale 3.8, we then get that ũ is a weak solution of ũ Λ ũ 1 in On the other hand, for any R > 0, we have ũ α L B0,R = u α L By α,rµ α By 3.6 and since r κ u α < r, we get B y α, Rµ B 0, rκ u α = 3.15 α for large α. By 3.14, 3.15, and by definition of r κ u α, we obtain ũ α L B0,R κ 3.16 for large α. Passing to the limit into 3.16 as α and then as R yields ũ L κ Since κ < κ 0, by Lemma 2.3, 3.13, and 3.17, we get that ũ 0, which is in contradiction with ũ 0 = 1. This ends the roof of Lemma Proof of Theorem 1.1 We can now rove Theorem 1.1 by alying Lemmas 2.2, 3.1, and Harnack-tye inequalities of Serrin [25] and Trudinger [30]. Proof of 1.3. We let u be a solution of 1.1. We let κ and r be as in Lemma 3.1. For any R > 0 and y, we define By 1.1, we obtain u R y := R 1 u R y. 4.1 u R = f R y, R n 1 ur It follows from 1.2 that f R y, R n 1 ur Λ R 1 ur 1 in. 4.2 in. 4.3 Moreover, similarly to 2.12, it follows from 4.2 and 4.3 that u R is a weak solution of u R Λ R 1 ur 1 in. 4.4 By writing u R 1 = u R u R 1 and alying Lemma 3.1, we obtain R 1 ur 1 K 0 u R 1 in \B 0, rovided that R r. It follows from 4.4, 4.5, and Trudinger [30, Theorem 1.3] that for any ε > 0, we have u R L B0,2\B0,4 c ε u R L 1+ε B0,5\B0,1. 4.6

10 10 for some constant c ε = c n,, Λ, K 0, ε. We fix ε 0 = ε 0 n, such that 0 < ε 0 <, where is as in Lemma 2.2. By a generalized version of Hölder s inequality see for instance Grafakos [13, Exercise ], we obtain that there exists a constant c 0 = c 0 n, such that u R L 1+ε 0 B0,5\B0,1 c 0 u R L 1, B0,5\B0, By observing that the quasi-norm L 1, is left invariant by the change of scale 4.1, we deduce from 4.6, 4.7, and Lemma 2.2 that u R L B0,2\B0,4 c for some constant c 1 = c 1 n,, Λ, K0, u L 1,. By , 4.8, and the estimates of DiBenedetto [8] and Tolksdorf [29], we get u R L B0,5/2\B0,7/2 c for some constant c 2 = c 2 n,, Λ, K0, u L 1,. Finally, for any x \B 0, 3r, by alying 4.8 and 4.9 with R = x /3, we obtain u x c 3 x n 1 and u x c 3 x 1 n for some constant c 3 = c 3 n,, Λ, K0, u L 1,. Since on the other hand u and u are uniformly bounded in B 0, 3r, we can deduce 1.3 from Remark 4.1. As one can see from the above roof, the constant C 0 in 1.3 deends on n,, Λ, κ, r, r κ u, u L 1,, u L, and u W 1, B0,3r. In order to rove the lower bound estimate 1.4, we need the following Harnack inequality on annuli. This result is insired from Friedman Véron [10] and Véron [33] where similar results are used for the study of singular solutions of Lalace equations in ointed domains. Lemma 4.2. Let f : R R be a Caratheodory function such that 1.2 holds true, u be a nonnegative solution of 1.1, κ and r be as in Lemma 3.1, and K 0 be the constant given by Lemma 3.1. Then there exists a constant c 4 = c 4 n,, Λ, K 0 such that for all R r. su u x c 4 inf u x R< x <5R 2R< x <5R Proof of Lemma 4.2. For any R > 0, we define u R as in 4.1. By 4.2, 4.3, 4.5, and Serrin [25, Theorem 5], we obtain that there exists a constant c = c n,, Λ, K 0 such that su u R z c inf u R z 4.12 z By,1/3 z By,1/3

11 A PRIORI ESTIMATES FOR CRITICAL LAPLACE EQUATIONS 11 for all oints y in the annulus A := B 0, 5 \B 0, 2. Moreover, we can join every two oints in A by 17 connected balls of radius 1/3 and centers in A. Hence 4.11 follows from 4.12 with c 4 := c 17. We can now rove 1.4 by alying Lemma 4.2. Proof of 1.4. We let u be a nonnegative solution of 1.1 such that f x, u dx > 0. In articular, in view of 1.2, we have u 0, and hence u > 0 in by the strong maximum rincile of Vázquez [32]. By Lemma 4.2, we then get that in order to rove 1.4, it is sufficient to obtain a lower bound estimate of u L B0,5R\B0,2R for large R. By 4.2, 4.3, 4.5, and Serrin [25, Theorem 1], we obtain u R L B0,4\B0,3 c 5 u R L B0,5\B0,2 c 5 u R L B0,5\B0, for some constants c 5 and c 5 deending only on n,, Λ, and K 0, where u R is as in 4.1. By changing the scale of 4.13, we then get u L B0,4R\B0,3R c 5R u L B0,5R\B0,2R Next, we claim that if f x, u dx > λ for some real number λ > 0, then we have u L B0,4R\B0,3R c 6R n for large R, for some constant c 6 = c 6 n,, λ > 0. For any x and R > 0, we define χ R x := χ x /R, where χ C 1 0, is a cutoff function such that χ 1 on [0, 3], χ 0 on [4,, 0 η 1 and η 2 on 3, 4. By testing 1.1 with χ R and alying Hölder s inequality, we obtain f x, u χ R dx = u 2 u χ R dx u 1 L su χ R χ R L su χ R, 4.16 where su χ R denotes the suort of χ R. It follows from 4.16 and the definition of χ R that f x, u χ R dx CR u 1 L B0,4R\B0,3R 4.17 for some constant C = C n, > 0. Then 4.15 follows from 4.17 with c 6 := λ/c 1 1 Finally, we deduce 1.4 from 4.11, 4.14, and Remark 4.3. As one can see from the above roof, the constant C 1 in 1.4 deends on n,, λ, Λ, κ, r, r κ u, u L, and a lower bound for u on the ball B 0, 2 max r, R λ,f u, where R λ,f u := inf { R > 0 : f x, u χ R dx > λ, R > R }.

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