GROWTH ESTIMATES THROUGH SCALING FOR QUASILINEAR PARTIAL DIFFERENTIAL EQUATIONS

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1 Annales Academiæ Scientiaum Fennicæ Mathematica Volumen 32, 2007, GROWTH ESTIMATES THROUGH SCALING FOR QUASILINEAR PARTIAL DIFFERENTIAL EQUATIONS Teo Kilpeläinen, Henik Shahgholian and Xiao Zhong Univesity of Jyväskylä, Depatment of Mathematics and Statistics P.O. Box 35 (MaD), FI Univesity of Jyväskylä, Finland; Royal Institute of Technology, Depatment of Mathematics SE Stockholm, Sweden; Univesity of Jyväskylä, Depatment of Mathematics and Statistics P.O. Box 35 (MaD), FI Univesity of Jyväskylä, Finland; Abstact. In this note we use a scaling o blow up agument to obtain estimates to solutions of equations of p-laplacian type. Weak solutions of equation 1. Intoduction div( u p 2 u) = 0, 1 < p <, ae called p-hamonic. It is known that p-hamonic functions ae in C 1,α fo some α > 0, whee fo p 2 one cannot have α 1 in geneal; see [3] fo shap egulaity in the plana case. In this note we pesent a blow up agument and show that if 0 < α 1 is such that the class of p-hamonic functions ae continuously embedded into C 1,α, then the only entie p-hamonic functions that gow at infinity slowe than x 1+α ae the linea ones. We fomulate the poof and the gowth ate esult only in the p-laplacian setting, but the agument is moe geneal. The only ingedients equied ae the following: thee is a class F of functions so that F contains cetainly escaled vesions of functions and F can be embedded into C 1,α. Then nonlinea functions in F gow at least as fast as x 1+α. As an application of the gowth ate esult we show that a nonnegative p- hamonic function in a half space is actually linea if it vanishes on the bounday of the half space. This gives an affimative answe to a quey of Bonk, who also found independently a diffeent poof fo this fact Mathematics Subject Classification: Pimay 35J60; Seconday 35J15. Key wods: p-laplace, scaling. T. K. (patially) and X. Z. suppoted by the Academy of Finland, H. S. patially suppoted by Swedish Reseach Council. The pesent wok is pat of the ESF pogamme GLOBAL.

2 596 Teo Kilpeläinen, Henik Shahgholian and Xiao Zhong 2. Gowth of entie solutions We pove the following two theoems: 2.1. Theoem. Let u be p-hamonic in R n. Thee is a numbe β > 0 depending only on p and n so that if then u is (affine) linea. u(x) = o( x 1+β ) as x, The second is an immediate consequence of the fist one Theoem. Let u be p-hamonic in R n. If then u is constant. u(x) = o( x ) as x, 2.3. Remak. It is known that thee ae no entie hamonic functions (i.e. p = 2) with nonintege gowth ate. That is, if u is hamonic (i.e. 2-hamonic) in R n with lim sup x log u(x) log x = γ ]0, [, then γ is an intege. If p 2, the situation is diffeent. Let ( ) γ = p p (p 1) 2 Obseve that γ ] 4, 2[ fo p > 2; fo 1 < p < 2 the constant γ > 2 but it 3 has nonintege value fo most of p s. Thee ae entie p-hamonic functions whose gowth ate is = γ. Constuctions fo such solutions ae done by Kol [4], Tolksdoff [9], Aonsson [2], and Iwaniec and Manfedi [3]. Basically all these examples ae quasiadial functions in the plane (highe dimensional examples ae obtained by adding dummy vaiables.) Hence Theoem 2.1 is optimal as stated. In the plane case one can choose β 4 3 fo all p (see Theoem 2.5 below), but in the highe dimensions we do not know if β needs to be close to 0. Poof of Theoem 2.1. Choose a sequence R j and wite Then the scaled functions S j = sup u. B(0,R j ) u j (x) = u(r jx) S j ae p-hamonic and u j 1 in B(0, 1). By a well known egulaity estimate (see e.g. Lewis [5]), thee is a constant β = β(n, p) > 0 so that the C 1,β (B(0, 1)) noms of u j ae bounded, independently

3 Gowth estimates though scaling fo quasilinea patial diffeential equations 597 of j. Hence the quantities C j (x) = Du j(x) Du j (0) x β = R1+β j S j Du(R j x) Du(0) R j x β ae unifomly bounded in B(0, 1). Since the gowth condition u(x) = 2 o( x 1+β ) implies R 1+β j lim =, j S j we conclude that sup y B(0, R j 2 ) But this implies that and Theoem 2.1 follows. Du(y) Du(0) y β = sup x B(0, 1 2 ) Du(R j x) Du(0) R j x β 0 as j. Du(y) = Du(0) fo all y R n, 2.4. Remak. Anothe way to pove Theoem 2.1 fo the p-laplacian goes via the estimate ( α osc u C sup u B(x 0,) B(x 0,R) R) that can be found e.g. in [7, Theoem 3.44]. Fo moe geneal opeatos the oscillation estimate might not be available but one can pove the embedding into C 1,α by othe means. We would like to emphasize hee that ou method woks also in those cases whee one can establish bounded embedding to C 1,α even though thee is no oscillation estimate fo the gadient. Appealing to the shap egulaity esult in [3] ou method immediately yields the following esult in the plana case: If 2.5. Theoem. Let u be p-hamonic in R 2 so that γ = then u is (affine) linea. u(x) = o( x γ ) as x. { 2 ( if 1 < p 2, ) if p > 2, 6 p 1 p 1 (p 1) 2 3. Nonnegative functions in the half space As an application of Theoem 2.1 we pove the following esult Theoem. If u is a nonnegative p-hamonic function on a half space H, continuous up to the bounday with u = 0 on H, then u is (affine) linea.

4 598 Teo Kilpeläinen, Henik Shahgholian and Xiao Zhong Theoem 3.1 follows by combining the following lemma with Theoem 2.1, when we obseve that u in Theoem 3.1 can be eflected though the hypeplane H and the esulting function is p-hamonic in the whole space R n (this can be easily veified by a diect computation, see [8]) Lemma. Let u be a nonnegative p-hamonic function on a half space H, continuous up to the bounday. If u = 0 on H, then u(x) = O( x ) as x. Poof. We assume, as we clealy may, that the half space H is the uppe half space H = R n + = {(x 1, x 2,..., x n ) : x n > 0}. We fist show that thee is a constant c = c(n, p) > 0 so that (3.3) u(re n ) c R u(e n ) fo all R > 2; hee e n = (0, 0,..., 0, 1) is the nth unit vecto in R n. Fo this, we wite x 0 = Re n = 2e n and obseve that by Hanack s inequality u(x) c u(x 0 ) fo all x B(x 0, ), whee c = c(n, p) > 0. Now, let v be the p-capacitay potential in B(x 0, 2) \ B(x 0, ), i.e. v(x) = x x 0 Then since v is p-hamonic in B(x 0, 2) \ B(x 0, ), we have by compaison pinciple that u(x) cu(x 0 )v(x) fo all x B(x 0, 2) \ B(x 0, ), whee c = c(n, p) > 0. x = e n, fo 1 v(e n ) =. The claim (3.3) follows fom this estimate evaluated at = ( 1)(1 n)/(p 1) (2) (1 n)/(p 1) 2 (1 n)/(p 1) c2 = cr, whee c = c(n, p). The estimate (3.3) is poved. To complete the poof the lemma, we employ the bounday Hanack pinciple (see [1] o [6]) which states that thee is a constant c depending on n and p only so

5 Gowth estimates though scaling fo quasilinea patial diffeential equations 599 that u(x) c u(re n) fo all x B(0, 2R) R n + and R > 0; x n R hee x n is the nth coodinate of x. Next we combine this with (3.3) and have u(x) c u(re n) R x n c u(e n )x n c x u(e n ) fo x B(0, 2R) and R > 2. The lemma follows. Refeences [1] Aikawa, H., T. Kilpeläinen, N. Shanmugalingam, and X. Zhong: Bounday Hanack pinciple fo p-hamonic functions in smooth Euclidean domains. - Potential Anal. 26, 2007, [2] Aonsson, G.: On cetain p-hamonic functions in the plane. - Manuscipta Math. 61:1, 1988, [3] Iwaniec, T., and J. J. Manfedi: Regulaity of p-hamonic functions on the plane. - Rev. Mat. Ibeoameicana 5:1-2, 1989, [4] Kol, I. N.: The behavio of the solutions of a cetain quasilinea equation nea zeo cusps of the bounday. - Bounday value poblems of mathematical physics, 8, Tudy Mat. Inst. Steklov. 125, 1973, , 233 (in Russian). [5] Lewis, J. L.: Regulaity of the deivatives of solutions to cetain degeneate elliptic equations. - Indiana Univ. Math. J. 32:6, 1983, [6] Lewis, J. L., and K. Nystöm: On a bounday Hanack inequality fo p-hamonic functions. - Pepint, [7] Malý, J., and W. P. Zieme: Fine egulaity of solutions of elliptic patial diffeential equations. - Math. Suveys Monog. 51, Ameican Mathematical Society, Povidence, RI, [8] Matio, O.: Reflection pinciple fo solutions of elliptic patial diffeential equations and quasiegula mappings. - Ann. Acad. Sci. Fenn. Se. A I Math. 6:1, 1981, [9] Tolksdof, P.: Regulaity fo a moe geneal class of quasilinea elliptic equations. - J. Diffeential Equations 51:1, 1984, Received 20 Mach 2007

Growth estimates through scaling for quasilinear partial differential equations

Growth estimates through scaling for quasilinear partial differential equations Tero Kilpeläinen, Henrik Shahgholian, and Xiao Zhong March 5, 2007 Abstract In this note we use a scaling or blow up argument