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A Liouville type theorem for poly-harmonic Dirichlet problems in a half space. (English) Zbl 1250.35051

The authors consider the following Dirichlet problem for poly-harmonic operators on a half space \({\mathbb R}^n_{+}\): \[ \left\{ \begin{aligned} &(-\Delta)^m u = u^p \qquad \qquad \qquad \qquad \qquad \qquad \text{in} \, {\mathbb R}^n_{+},\\ &u = \frac{\partial u}{\partial x_n} = \frac{\partial^2 u}{\partial x_n^2} = \cdots = \frac{\partial^{m-1} u}{\partial x_n^m-1} = 0 \quad \,\, \text{on}\, {\partial {\mathbb R}^n_{+}}, \end{aligned}\right.\tag{1} \] where \(m\) is any positive integer, \(2m < n\), and \(1 < p \leq \frac{n+2m}{n-2m}\).
This problem has been considered in [W. Reichelt and T. Weth, J. Differ. Equations 248, No. 7, 1866–1878 (2010; Zbl 1185.35066)]. They proved that there are no bounded classical solutions. In this paper, the authors removes their boundedness assumptions on \(u\) and all its derivatives, and under very mild growth conditions, the authors show that problem (1) is equivalent to the integral equation \[ u(x) = \int_{{\mathbb R}^n_+} G(x, y) u^p\, dy,\tag{2} \] where \(G(x, y)\) is the Green’s function on the half space. Using this equivalence and showing that there is no positive solution for integral equation (2), the authors prove that there is no positive solution for the equation (1) in both subcritical and critical cases.

MSC:

35B53 Liouville theorems and Phragmén-Lindelöf theorems in context of PDEs
35J40 Boundary value problems for higher-order elliptic equations
35B45 A priori estimates in context of PDEs
35A08 Fundamental solutions to PDEs

Citations:

Zbl 1185.35066
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References:

[1] Berestycki, H.; Nirenberg, L., On the method of moving planes and sliding method, Bol. Soc. Brasil. Mat. (N.S.), 22, 1, 1-37 (1991) · Zbl 0784.35025
[2] Bianchi, G., Non-existence of positive solutions to semilinear elliptic equations in \(R^N\) and \(R_+^N\) through the method of moving plane, Comm. Partial Differential Equations, 22, 1671-1690 (1997) · Zbl 0910.35048
[3] Boggio, T., Sulle funzioni di Green dʼordine \(m\), Rend. Circ. Mat. Palermo, 20, 97-135 (1905) · JFM 36.0827.01
[4] Chang, A.; Yang, P., On uniqueness of an \(n\)-th order differential equation in conformal geometry, Math. Res. Lett., 4, 1-12 (1997)
[5] Chen, W.; Li, C., Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63, 615-622 (1991) · Zbl 0768.35025
[6] Chen, W.; Li, C., A priori estimates for prescribing scalar curvature equations, Ann. of Math., 145, 547-564 (1997) · Zbl 0877.35036
[7] Chen, W.; Li, C., Moving planes, moving spheres, and a priori estimates, J. Differential Equations, 195, 1-13 (2003) · Zbl 1134.35331
[8] Chen, W.; Li, C., Classification of positive solutions for nonlinear differential and integral systems with critical exponents, Acta Math. Sci. Ser. B Engl. Ed., 29, 4, 949-960 (2009) · Zbl 1212.35103
[9] Chen, W.; Li, C., An integral system and the Lane-Emden conjecture, Discrete Contin. Dyn. Syst., 4, 1167-1184 (2009) · Zbl 1176.35067
[10] Chen, W.; Li, C., A \(\sup + \inf\) inequality near \(R = 0\), Adv. Math., 220, 219-245 (2009) · Zbl 1157.35364
[11] Chen, W.; Li, C., Methods on Nonlinear Elliptic Equations, AIMS Ser. Differ. Equ. Dyn. Syst., vol. 4 (2010)
[12] W. Chen, C. Li, Super polyharmonic property of solutions for PDE systems and its applications, Calc. Var. Partial Differential Equations (2011), submitted for publication.; W. Chen, C. Li, Super polyharmonic property of solutions for PDE systems and its applications, Calc. Var. Partial Differential Equations (2011), submitted for publication.
[13] Chen, W.; Zhu, J., Radial symmetry and regularity of solutions for poly-harmonic Dirichlet problems, J. Math. Anal. Appl., 377, 744-753 (2011) · Zbl 1211.35101
[14] Chen, W.; Jin, C.; Li, C.; Lim, C., Weighted Hardy-Littlewood-Sobolev inequalities and system of integral equations, Discrete Contin. Dyn. Syst. Ser. S, 164-172 (2005) · Zbl 1147.45301
[15] Chen, W.; Li, C.; Ou, B., Qualitative problems of solutions for an integral equation, Discrete Contin. Dyn. Syst., 12, 347-354 (2005) · Zbl 1081.45003
[16] Chen, W.; Li, C.; Ou, B., Classification of solutions for a system of integral equations, Comm. Partial Differential Equations, 30, 59-65 (2005) · Zbl 1073.45005
[17] Chen, W.; Li, C.; Ou, B., Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59, 330-343 (2006) · Zbl 1093.45001
[18] Y. Fang, J. Zhang, Nonexistence of positive solution for an integral equation on a half-space \(R_+^n\); Y. Fang, J. Zhang, Nonexistence of positive solution for an integral equation on a half-space \(R_+^n\)
[19] Gidas, B.; Spruck, J., A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations, 6, 883-901 (1981) · Zbl 0462.35041
[20] Gidas, B.; Spruck, J., Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 34, 525-598 (1981) · Zbl 0465.35003
[21] Gidas, B.; Ni, W.; Nirenberg, L., Symmetry of positive solutions of nonlinear elliptic equations in \(R^n\), (Mathematical Analysis and Applications. Mathematical Analysis and Applications, Adv. Math. Suppl. Stud., vol. 7A (1981), Academic Press: Academic Press New York) · Zbl 0469.35052
[22] Grunau, H.; Sweers, G., Positivity for equations involving polyharmonic operators with Dirichlet boundary conditions, Math. Ann., 307, 589-626 (1997) · Zbl 0892.35031
[23] Guo, Y.; Liu, J., Liouville-type theorems for polyharmonic equations in \(R^N\) and in \(R_+^N\), Proc. Roy. Soc. Edinburgh Sect. A, 138, 339-359 (2008) · Zbl 1153.35028
[24] Hang, F., On the integral systems related to Hardy-Littlewood-Sobolev inequality, Math. Res. Lett., 14, 373-383 (2007) · Zbl 1144.26031
[25] Jin, C.; Li, C., Symmetry of solutions to some systems of integral equations, Proc. Amer. Math. Soc., 134, 1661-1670 (2006) · Zbl 1156.45300
[26] Li, C., Local asymptotic symmetry of singular solutions to nonlinear elliptic equations, Invent. Math., 123, 221-231 (1996) · Zbl 0849.35009
[27] Li, Y. Y., Remarks on some conformally invariant integral equations: the method of moving spheres, J. Eur. Math. Soc. (JEMS), 6, 153-180 (2004) · Zbl 1075.45006
[28] Li, C.; Ma, L., Uniqueness of positive bound states to Shrödinger systems with critical exponents, SIAM J. Appl. Anal., 40, 1049-1057 (2008) · Zbl 1167.35347
[29] Li, Y.; Zhu, M., Uniqueness theorems through the method of moving spheres, Duke Math. J., 80, 38-417 (1995) · Zbl 0846.35050
[30] Li, D.; Zhuo, R., An integral equation on Half space, Proc. Amer. Math. Soc., 138, 2779-2791 (2010) · Zbl 1200.45001
[31] Li, D.; Strohmer, G.; Wang, L., Symmetry of integral equations on bounded domains, Proc. Amer. Math. Soc., 137, 3695-3702 (2009) · Zbl 1188.45001
[32] Lin, C. S., A classification of solutions of a conformally invariant fourth order equation in \(R^n\), Comment. Math. Helv., 73, 206-231 (1998) · Zbl 0933.35057
[33] Liu, S., Regularity, symmetry, and uniqueness of some integral type quasilinear equations, Nonlinear Anal., 71, 1796-1806 (2009) · Zbl 1171.45300
[34] Liu, C.; Qiao, S., Symmetry and monotonicity for a system of integral equations, Commun. Pure Appl. Anal., 6, 1925-1932 (2009) · Zbl 1185.45011
[35] G. Lu, J. Zhu, The axial symmetry and regularity of solutions to an integral equation in a half space, Pacific J. Math. (2010), in press.; G. Lu, J. Zhu, The axial symmetry and regularity of solutions to an integral equation in a half space, Pacific J. Math. (2010), in press.
[36] Ma, L.; Chen, D., A Liouville type theorem for an integral system, Commun. Pure Appl. Anal., 5, 855-859 (2006) · Zbl 1134.45007
[37] Ma, L.; Liu, B., Symmetry results for decay solutions of elliptic systems in the whole space, Adv. Math., 225, 3052-3063 (2010) · Zbl 1202.35080
[38] Ma, L.; Zhao, L., Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Ration. Mech. Anal., 195, 455-467 (2010) · Zbl 1185.35260
[39] Ma, C.; Chen, W.; Li, C., Regularity of solutions for an integral system of Wolff type, Adv. Math., 226, 2676-2699 (2011) · Zbl 1209.45006
[40] Reichel, W.; Weth, T., A prior bounds and a Liouville theorem on a half-space for higher-order elliptic Dirichlet problems, Math. Z., 261, 805-827 (2009) · Zbl 1167.35014
[41] Reichel, W.; Weth, T., Existence of solutions to nonlinear, subcritical higher order elliptic Dirichlet problems, J. Differential Equations, 248, 1866-1878 (2010) · Zbl 1185.35066
[42] Serrin, J., A symmetry problem in potential theory, Arch. Ration. Mech. Anal., 43, 304-318 (1971) · Zbl 0222.31007
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