Liouville type theorems for poly-harmonic Dirichlet problems of Hénon-Hardy type equations on a half space or a ball

• Dai, Wei ^[1]
  1. [1] Beihang University

     Beihang University

     China

     
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 74, Fasc. 3, 2023, págs. 729-751
• Idioma: inglés
• DOI: 10.1007/s13348-022-00371-8
• Texto completo no disponible (Saber más ...)
• Resumen

  • In this paper, we are concerned with the poly-harmonic Dirichlet problems for Hénon-Hardy type equations \begin{aligned} (-\Delta )^{m}u(x)=f(x,u(x)) \,\,\,\,\,\,\,\,\,\,\,\, \text {in} \,\,\, {\mathbb {R}}^{n}_{+} \,\,\, \text {or} \,\,\, B_{R}(0) \end{aligned} with n\ge 2, m\ge 1 and R>0. We prove Liouville theorems for nonnegative solutions to the above poly-harmonic Dirichlet problems and equivalent integral equations in {\mathbb {R}}^{n}_{+} and B_{R}(0) under general assumptions on f. A typical case is the Hénon-Hardy type nonlinearity f(x,u)=|x|^{a}u^{p} with a\in (-2m,+\infty ) and p>0. Our results extend the Liouville results on poly-harmonic Dirichlet problems in Reichel and Weth (Math. Z. 261:805–827, 2009), Fang and Chen (Adv. Math. 229:2835–2867, 2012), Pucci and Serrin (Indiana Univ. Math. J. 35:681–703, 1986, J. Math. Pures Appl. 69:55–83, 1990) from f=u^{p} to general f(x, u).

• Referencias bibliográficas
  • Boggio, T.: Sulle funzioni di Green d‘ordine m. Rend. Circ. Mat. Palermo 20, 97–135 (1905)
  • Bidaut-Véron, M.F., Giacomini, H.: A new dynamical approach of Emden-Fowler equations and systems. Adv. Differ. Equ. 15(11–12), 1033–1082...
  • Berchio, E., Gazzola, F., Weth, T.: Radial symmetry of positive solutions to nonlinear poly-harmonic Dirichlet problems. J. Reine Angew. Math....
  • Bidaut-Véron, M.F., Pohozaev, S.: Nonexistence results and estimates for some nonlinear elliptic problems. J. Anal. Math. 84, 1–49 (2001)
  • Cao, D., Dai, W.: Classification of nonnegative solutions to a bi-harmonic equation with Hartree type nonlinearity. Proc. Royal Soc. Edinburgh...
  • Cao, D., Dai, W., Qin, G.: Super poly-harmonic properties, Liouville theorems and classification of nonnegative solutions to equations involving...
  • Chen, W., Fang, Y., Yang, R.: Liouville theorems involving the fractional Laplacian on a half space. Adv. Math. 274, 167–198 (2015)
  • Caffarelli, L., Gidas, B., Spruck, J.: Asymptotic symmetry and local behavior of semilinear elliptic equation with critical Sobolev growth....
  • Chen, W., Li, C.: Classification of solutions of some nonlinear elliptic equations. Duke Math. J. 63(3), 615–622 (1991)
  • Chen, W., Li, C.: On Nirenberg and related problems—a necessary and sufficient condition. Comm. Pure Appl. Math. 48, 657–667 (1995)
  • Chen, W., Li, C.: A priori estimates for prescribing scalar curvature equations. Annals Math. 145(3), 547–564 (1997)
  • Chen, W., Li, C., Li, Y.: A direct method of moving planes for the fractional Laplacian. Adv. Math. 308, 404–437 (2017)
  • Chen, W., Li, C., Ou, B.: Classification of solutions for an integral equation. Comm. Pure Appl. Math. 59, 330–343 (2006)
  • Chen, W., Li, Y., Zhang, R.: A direct method of moving spheres on fractional order equations. J. Funct. Anal. 272(10), 4131–4157 (2017)
  • Chang, S.-Y.A., Yang, P.C.: On uniqueness of solutions of n-th order differential equations in conformal geometry. Math. Res. Lett. 4, 91–102...
  • Dai, W.: Nonexistence of positive solutions to n-th order equations in {\mathbb{R}}^n, Bulletin des Sciences Mathématiques, 174, Paper No....
  • Dai, W., Duyckaerts, T.: Uniform a priori estimates for positive solutions of higher order Lane-Emden equations in {\mathbb{R}}^n. Publicacions...
  • Dai, W., Liu, Z.: Classification of nonnegative solutions to static Schrödinger-Hartree and Schrödinger-Maxwell equations with combined nonlinearities,...
  • Dai, W., Liu, Z., Lu, G.: Liouville type theorems for PDE and IE systems involving fractional Laplacian on a half space. Potential Anal. 46,...
  • Dai, W., Liu, Z., Qin, G.: Classification of nonnegative solutions to static Schrödinger-Hartree-Maxwell type equations. SIAM J. Math. Anal....
  • Dai, W., Qin, G.: Classification of nonnegative classical solutions to third-order equations. Adv. Math. 328, 822–857 (2018)
  • Dai, W., Qin, G.: Liouville type theorems for fractional and higher order Hénon-Hardy type equations via the method of scaling spheres, Int....
  • Dai, W., Qin, G.: Liouville type theorem for critical order Hénon-Lane-Emden type equations on a half space and its applications, J. Funct....
  • Dai, W., Qin, G.: Liouville type theorems for elliptic equations with Dirichlet conditions in exterior domains. J. Differ. Equ. 269, 7231–7252...
  • Dai, W., Qin, G., Zhang, Y.: Liouville type theorem for higher order Hénon equations on a half space, Nonlinear. Analysis 183, 284–302 (2019)
  • Edmunds, D.E., Fortunato, D., Jannelli, E.: Critical exponents, critical dimensions and the biharmonic operator. Arch. Rational Mech. Anal....
  • Fazly, M.: Liouville theorems for the poly-harmonic Hénon-Lane-Emden system. Methods Appl. Anal. 21(2), 265–281 (2014)
  • Fang, Y., Chen, W.: A Liouville type theorem for poly-harmonic Dirichlet problems in a half space. Adv. Math. 229, 2835–2867 (2012)
  • Fazly, M., Ghoussoub, N.: On the Hénon-Lane-Emden conjecture. Discrete Contin. Dyn. Syst.-A 34(6), 2513–2533 (2014)
  • Fall, M.M., Weth, T.: Nonexistence results for a class of fractional elliptic boundary value problems. J. Funct. Anal. 263, 2205–2227 (2012)
  • Fall, M.M., Weth, T.: Monotonicity and nonexistence results for some fractional elliptic problems in the half space. Comm. Contemp. Math....
  • Gazzola, F., Grunau, H.C., Sweers, G.: Polyharmonic Boundary Value Problems, Lecture Notes in Mathematics, 1991 (2010): xviii+423
  • Guo, Y., Liu, J.: Liouville-type theorems for poly-harmonic equations in {\mathbb{R}}^n and in {\mathbb{R} }^n_+. Proc. Roy. Soc. Edinburgh...
  • Gidas, B., Ni, W., Nirenberg, L.: Symmetry and related properties via maximum principle. Comm. Math. Phys. 68, 209–243 (1979)
  • Gidas, B., Spruck, J.: A priori bounds for positive solutions of nonlinear elliptic equations. Comm. PDE 6(8), 883–901 (1981)
  • Grunau, H., Sweers, G.: Positivity for equations involving poly-harmonic operators with Dirichlet boundary conditions. Math. Ann. 307, 589–626...
  • Jin, Q., Li, Y.Y., Xu, H.: Symmetry and asymmetry: The method of moving spheres. Adv. Differ. Equ. 13(7), 601–640 (2007)
  • Lin, C.: A classification of solutions of a conformally invariant fourth order equation in {\mathbb{R} }^n. Comment. Math. Helv. 73, 206–231...
  • Li, Y.Y.: Remark on some conformally invariant integral equations: The method of moving spheres. J. Eur. Math. Soc. 6, 153–180 (2004)
  • Li, Y.Y., Zhu, M.: Uniqueness theorems through the method of moving spheres. Duke Math. J. 80, 383–417 (1995)
  • Li, Y.Y., Zhang, L.: Liouville type theorems and Harnack type inequalities for semilinear elliptic equations. J. Anal. Math. 90, 27–87 (2003)
  • Luo, S., Zou, W.: Liouville theorems for integral systems related to fractional Lane-Emden systems in {\mathbb{R} }^n_+. Diff. Integral...
  • Mitidieri, E.: Nonexistence of positive solutions of semilinear elliptic systems in {\mathbb{R} }^n. Differ. Integral Equ. 9, 465–479 (1996)
  • Mitidieri, E., Pohozaev, S.I.: A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities,...
  • Padilla, P.: On some nonlinear elliptic equations. Thesis, Courant Institute (1994)
  • Phan, Q.: Liouville-type theorems for poly-harmonic Hénon-Lane-Emden system. Adv. Nonlinear Stud. 15(2), 415–432 (2015)
  • Poho\check{a}aev, S. I.: Eigenfunctions of the equation \Delta u +\lambda f(u)= 0 (Russian), Dokl. Akad. Nauk SSSR, 165 (1965), no....
  • Pol\acute{a}\check{c}ik, P., Quittner, P., Souplet, P.: Singularity and decay estimates in superlinear problems via Liouville type theorems....
  • Phan, Q., Souplet, P.: Liouville-type theorems and bounds of solutions of Hardy-Hon equations. J. Diff. Equ. 252, 2544–2562 (2012)
  • Pucci, P., Serrin, J.: A general variational identity. Indiana Univ. Math. J. 35, 681–703 (1986)
  • Pucci, P., Serrin, J.: Critical exponents and critical dimensions for poly-harmonic operators. J. Math. Pures Appl. 69, 55–83 (1990)
  • Reichel, W., Weth, T.: A priori bounds and a Liouville theorem on a half-space for higher-order elliptic Dirichlet problems. Math. Z. 261,...
  • Reichel, W., Weth, T.: Existence of solutions to nonlinear, subcritical higher order elliptic Dirichlet problems. J. Differ. Equ. 248(7),...
  • Reichel, W., Zou, H.: Non-existence results for semilinear cooperative elliptic systems via moving spheres. J. Diff. Equ. 161(1), 219–243...
  • Serrin, J.: A symmetry problem in potential theory. Arch. Rational Mech. Anal. 43, 304–318 (1971)
  • Souplet, P.: The proof of the Lane-Emden conjecture in four space dimensions. Adv. Math. 221(5), 1409–1427 (2009)
  • Souto, M.A.S.: A priori estimates and existence of positive solutions of non-linear cooperative elliptic systems. Differ. Integral Equ. 8,...
  • Serrin, J., Zou, H.: Non-existence of positive solutions of Lane-Emden systems. Diff. Integral Equ. 9(4), 635–653 (1996)
  • Wei, J., Xu, X.: Classification of solutions of higher order conformally invariant equations. Math. Ann. 313(2), 207–228 (1999)