American Institute of Mathematical Sciences

• Previous Article
Asymptotic symmetry for a class of nonlinear fractional reaction-diffusion equations
• DCDS Home
• This Issue
• Next Article
The role of the scalar curvature in some singularly perturbed coupled elliptic systems on Riemannian manifolds
June  2014, 34(6): 2561-2580. doi: 10.3934/dcds.2014.34.2561

Liouville type results and regularity of the extremal solutions of biharmonic equation with negative exponents

 1 Department of Mathematics, Henan Normal University, Xinxiang, 453007, China 2 Department of Mathematics, Chinese University of Hong Kong, Shatin, New Territories, Hong Kong

Received  June 2013 Revised  June 2013 Published  December 2013

We first obtain Liouville type results for stable entire solutions of the biharmonic equation $-\Delta^2 u=u^{-p}$ in $\mathbb{R}^N$ for $p>1$ and $3 \leq N \leq 12$. Then we consider the Navier boundary value problem for the corresponding equation and improve the known results on the regularity of the extremal solution for $3 \leq N \leq 12$. As a consequence, in the case of $p=2$, we show that the extremal solution $u^{*}$ is regular when $N =7$. This improves earlier results of Guo-Wei [21] ($N \leq 4$), Cowan-Esposito-Ghoussoub [2] ($N=5$), Cowan-Ghoussoub [4] ($N=6$).
Citation: Zongming Guo, Juncheng Wei. Liouville type results and regularity of the extremal solutions of biharmonic equation with negative exponents. Discrete and Continuous Dynamical Systems, 2014, 34 (6) : 2561-2580. doi: 10.3934/dcds.2014.34.2561
References:
 [1] C. Cowan, Liouville theorems for stable Lane-Emden systems and biharmonic problems, arXiv:1207.1081v1, 2012. doi: 10.1088/0951-7715/26/8/2357. [2] C. Cowan, P. Esposito and N. Ghoussoub, Regularity of extremal solutions in fourth order nonlinear eigevalue problems on general domains, Discrete Contin. Dyn. Syst., 28 (2010), 1033-1050. doi: 10.3934/dcds.2010.28.1033. [3] C. Cowan, P. Esposito, N. Ghoussoub and A. Moradifam, The critical dimension for a fourth order elliptic problem with singular nonlinearity, Arch. Ration. Mech. Anal., 198 (2010), 763-787. doi: 10.1007/s00205-010-0367-x. [4] C. Cowan and N. Ghoussoub, Regularity of semi-stable solutions to fourth order nonlinear eigevalue problems on general domains, Calculus of Variations and Partial Differential Equations, 2012. doi: 10.1007/s00526-012-0582-4. [5] A. Capella, J. Dávila, L. Dupaigne and Y. Sire, Regularity of radial extremal solutions for some non-local semilinear equations, Comm. PDEs, 36 (2011), 1353-1384. doi: 10.1080/03605302.2011.562954. [6] M. Crandall and P. H. Rabinowitz, Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems, Arch. Rational Mech. Anal., 58 (1975), 207-218. doi: 10.1007/BF00280741. [7] E. N. Dancer, Y. Du and Z. M. Guo, Finite Morse index solutions of an elliptic equation with supercritical exponent, J. Differential Equations, 250 (2011), 3281-3310. doi: 10.1016/j.jde.2011.02.005. [8] J. Dávila, I. Flores and I. Guerra, Multiplicity of solutions for a fourth order equation with power-type nonlinearity, Math. Ann., 348 (2010), 143-193. doi: 10.1007/s00208-009-0476-8. [9] J. Dávila, L. Dupaigne, I. Guerra and M. Montenegro, Stable solutions for the bilaplacian with exponential nonlinearity, SIAM J. Math. Anal., 39 (2007), 565-592. doi: 10.1137/060665579. [10] J. Dávila, L. Dupaigne and A. Farina, Partial regularity of finite Morse index solutions to the Lane-Emden equation, J. Funct. Anal., 261 (2011), 218-232. doi: 10.1016/j.jfa.2010.12.028. [11] J. Davila, L. Dupaigne, K. Wang and J. Wei, Monotonicity formula and Liouville theorems for fourth order supercritical problems, preprint, 2013. [12] L. Dupaigne, M. Ghergu, O. Goubet and G. Warnault, Entire large solutions for semilinear elliptic equations, J. Differential Equations, 253 (2012), 2224-2251. doi: 10.1016/j.jde.2012.05.024. [13] , L. Dupaigne and P. Esposito, private communications. [14] L. Dupaigne and A. Farina, Stable solutions of $-\Delta u=f(u)$ in $\mathbb{R}^N2$, J. Eur. Math. Soc., 12 (2010), 855-882. doi: 10.4171/JEMS/217. [15] Y. Du and Z. Guo, Positive solutions of an elliptic equation with negative exponent: Stability and critical power, J. Differential Equations, 246 (2009), 2387-2414. doi: 10.1016/j.jde.2008.08.008. [16] P. Esposito, Compactness of a nonlinear eigenvalue problem with a singular nonlinearity, Comm. Contemp. Math., 10 (2008), 17-45. doi: 10.1142/S0219199708002697. [17] P. Esposito, N. Ghoussoub and Y. Guo, Compactness along the branch of semi-stable and unstable solutions for an elliptic problem with a singular nonlinearity, Comm. Pure Appl. Math., 60 (2007), 1731-1768. doi: 10.1002/cpa.20189. [18] P. Esposito, N. Ghoussoub and Y. Guo, Mathematical Analysis of Partial Differential Equations Modeling Electrostatic MEMS, Courant Lecture Notes in Mathematics, 20, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2010. [19] A. Farina, On the classification of solutions of the Lane-Emden equation on unbounded domains of $\mathbb{R}^N2$, J. Math. Pures Appl. (9), 87 (2007), 537-561. doi: 10.1016/j.matpur.2007.03.001. [20] F. Gazzola and H.-Ch. Grunau, Radial entire solutions for supercritical biharmonic equations, Math. Ann., 334 (2006), 905-936. doi: 10.1007/s00208-005-0748-x. [21] Z. Guo and J. Wei, On a fourth order elliptic problem with negative exponent, SIAM J. Math. Anal., 40 (2008/09), 2034-2054. doi: 10.1137/070703375. [22] Z. Guo and J. Wei, Qualitative properties of entire radial solutions for a biharmonic equation with supcritical nonlinearity, Proc. American Math. Soc., 138 (2010), 3957-3964. doi: 10.1090/S0002-9939-10-10374-8. [23] H. Hajlaoui, A. A. Harrabi and D. Ye, On stable solutions of biharmonic problem with polynomial growth, arXiv:1211.2223v2, 2012. [24] P. Karageorgis, Stability and intersection properties of solutions to the nonlinear biharmonic equation, Nonlinearity, 22 (2009), 1653-1661. doi: 10.1088/0951-7715/22/7/009. [25] P. J. McKenna and W. Reichel, Radial solutions of singular nonlinear biharmonic equations and applications to conformal geometry, Electron J. Differential Equations, 2003, 13 pp. [26] L. Ma and J. Wei, Properties of positive solutions to an elliptic equation with negative exponent, Journal of Functional Analysis, 254 (2008), 1058-1087. doi: 10.1016/j.jfa.2007.09.017. [27] A. Moradiam, On the critical dimension of a fourth order elliptic problem with negative exponent, J. Differential Equations, 248 (2010), 594-616. doi: 10.1016/j.jde.2009.09.011. [28] F. Rellich, Perturbation theory of eigenvalue problems, Gordon and Breach Science Publisher, New York-London-Paris, 1969. [29] P. Souplet, The proof of the Lane-Emden conjecture in four space dimensions, Adv. Math., 221 (2009), 1409-1427. doi: 10.1016/j.aim.2009.02.014. [30] J. Wei and D. Ye, Liouville theorems for stable solutions of biharmonic problem, Math. Ann., 356 (2013), 1599-1612. doi: 10.1007/s00208-012-0894-x.

show all references

References:
 [1] C. Cowan, Liouville theorems for stable Lane-Emden systems and biharmonic problems, arXiv:1207.1081v1, 2012. doi: 10.1088/0951-7715/26/8/2357. [2] C. Cowan, P. Esposito and N. Ghoussoub, Regularity of extremal solutions in fourth order nonlinear eigevalue problems on general domains, Discrete Contin. Dyn. Syst., 28 (2010), 1033-1050. doi: 10.3934/dcds.2010.28.1033. [3] C. Cowan, P. Esposito, N. Ghoussoub and A. Moradifam, The critical dimension for a fourth order elliptic problem with singular nonlinearity, Arch. Ration. Mech. Anal., 198 (2010), 763-787. doi: 10.1007/s00205-010-0367-x. [4] C. Cowan and N. Ghoussoub, Regularity of semi-stable solutions to fourth order nonlinear eigevalue problems on general domains, Calculus of Variations and Partial Differential Equations, 2012. doi: 10.1007/s00526-012-0582-4. [5] A. Capella, J. Dávila, L. Dupaigne and Y. Sire, Regularity of radial extremal solutions for some non-local semilinear equations, Comm. PDEs, 36 (2011), 1353-1384. doi: 10.1080/03605302.2011.562954. [6] M. Crandall and P. H. Rabinowitz, Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems, Arch. Rational Mech. Anal., 58 (1975), 207-218. doi: 10.1007/BF00280741. [7] E. N. Dancer, Y. Du and Z. M. Guo, Finite Morse index solutions of an elliptic equation with supercritical exponent, J. Differential Equations, 250 (2011), 3281-3310. doi: 10.1016/j.jde.2011.02.005. [8] J. Dávila, I. Flores and I. Guerra, Multiplicity of solutions for a fourth order equation with power-type nonlinearity, Math. Ann., 348 (2010), 143-193. doi: 10.1007/s00208-009-0476-8. [9] J. Dávila, L. Dupaigne, I. Guerra and M. Montenegro, Stable solutions for the bilaplacian with exponential nonlinearity, SIAM J. Math. Anal., 39 (2007), 565-592. doi: 10.1137/060665579. [10] J. Dávila, L. Dupaigne and A. Farina, Partial regularity of finite Morse index solutions to the Lane-Emden equation, J. Funct. Anal., 261 (2011), 218-232. doi: 10.1016/j.jfa.2010.12.028. [11] J. Davila, L. Dupaigne, K. Wang and J. Wei, Monotonicity formula and Liouville theorems for fourth order supercritical problems, preprint, 2013. [12] L. Dupaigne, M. Ghergu, O. Goubet and G. Warnault, Entire large solutions for semilinear elliptic equations, J. Differential Equations, 253 (2012), 2224-2251. doi: 10.1016/j.jde.2012.05.024. [13] , L. Dupaigne and P. Esposito, private communications. [14] L. Dupaigne and A. Farina, Stable solutions of $-\Delta u=f(u)$ in $\mathbb{R}^N2$, J. Eur. Math. Soc., 12 (2010), 855-882. doi: 10.4171/JEMS/217. [15] Y. Du and Z. Guo, Positive solutions of an elliptic equation with negative exponent: Stability and critical power, J. Differential Equations, 246 (2009), 2387-2414. doi: 10.1016/j.jde.2008.08.008. [16] P. Esposito, Compactness of a nonlinear eigenvalue problem with a singular nonlinearity, Comm. Contemp. Math., 10 (2008), 17-45. doi: 10.1142/S0219199708002697. [17] P. Esposito, N. Ghoussoub and Y. Guo, Compactness along the branch of semi-stable and unstable solutions for an elliptic problem with a singular nonlinearity, Comm. Pure Appl. Math., 60 (2007), 1731-1768. doi: 10.1002/cpa.20189. [18] P. Esposito, N. Ghoussoub and Y. Guo, Mathematical Analysis of Partial Differential Equations Modeling Electrostatic MEMS, Courant Lecture Notes in Mathematics, 20, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2010. [19] A. Farina, On the classification of solutions of the Lane-Emden equation on unbounded domains of $\mathbb{R}^N2$, J. Math. Pures Appl. (9), 87 (2007), 537-561. doi: 10.1016/j.matpur.2007.03.001. [20] F. Gazzola and H.-Ch. Grunau, Radial entire solutions for supercritical biharmonic equations, Math. Ann., 334 (2006), 905-936. doi: 10.1007/s00208-005-0748-x. [21] Z. Guo and J. Wei, On a fourth order elliptic problem with negative exponent, SIAM J. Math. Anal., 40 (2008/09), 2034-2054. doi: 10.1137/070703375. [22] Z. Guo and J. Wei, Qualitative properties of entire radial solutions for a biharmonic equation with supcritical nonlinearity, Proc. American Math. Soc., 138 (2010), 3957-3964. doi: 10.1090/S0002-9939-10-10374-8. [23] H. Hajlaoui, A. A. Harrabi and D. Ye, On stable solutions of biharmonic problem with polynomial growth, arXiv:1211.2223v2, 2012. [24] P. Karageorgis, Stability and intersection properties of solutions to the nonlinear biharmonic equation, Nonlinearity, 22 (2009), 1653-1661. doi: 10.1088/0951-7715/22/7/009. [25] P. J. McKenna and W. Reichel, Radial solutions of singular nonlinear biharmonic equations and applications to conformal geometry, Electron J. Differential Equations, 2003, 13 pp. [26] L. Ma and J. Wei, Properties of positive solutions to an elliptic equation with negative exponent, Journal of Functional Analysis, 254 (2008), 1058-1087. doi: 10.1016/j.jfa.2007.09.017. [27] A. Moradiam, On the critical dimension of a fourth order elliptic problem with negative exponent, J. Differential Equations, 248 (2010), 594-616. doi: 10.1016/j.jde.2009.09.011. [28] F. Rellich, Perturbation theory of eigenvalue problems, Gordon and Breach Science Publisher, New York-London-Paris, 1969. [29] P. Souplet, The proof of the Lane-Emden conjecture in four space dimensions, Adv. Math., 221 (2009), 1409-1427. doi: 10.1016/j.aim.2009.02.014. [30] J. Wei and D. Ye, Liouville theorems for stable solutions of biharmonic problem, Math. Ann., 356 (2013), 1599-1612. doi: 10.1007/s00208-012-0894-x.
 [1] Xavier Cabré, Manel Sanchón. Semi-stable and extremal solutions of reaction equations involving the $p$-Laplacian. Communications on Pure and Applied Analysis, 2007, 6 (1) : 43-67. doi: 10.3934/cpaa.2007.6.43 [2] Olivier Goubet. Regularity of extremal solutions of a Liouville system. Discrete and Continuous Dynamical Systems - S, 2019, 12 (2) : 339-345. doi: 10.3934/dcdss.2019023 [3] Mostafa Fazly. Regularity of extremal solutions of nonlocal elliptic systems. Discrete and Continuous Dynamical Systems, 2020, 40 (1) : 107-131. doi: 10.3934/dcds.2020005 [4] Tomás Sanz-Perela. Regularity of radial stable solutions to semilinear elliptic equations for the fractional Laplacian. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2547-2575. doi: 10.3934/cpaa.2018121 [5] Mostafa Fazly, Yuan Li. Partial regularity and Liouville theorems for stable solutions of anisotropic elliptic equations. Discrete and Continuous Dynamical Systems, 2021, 41 (9) : 4185-4206. doi: 10.3934/dcds.2021033 [6] Guillaume Warnault. Regularity of the extremal solution for a biharmonic problem with general nonlinearity. Communications on Pure and Applied Analysis, 2009, 8 (5) : 1709-1723. doi: 10.3934/cpaa.2009.8.1709 [7] Yu-Juan Sun, Li Zhang, Wan-Tong Li, Zhi-Cheng Wang. Entire solutions in nonlocal monostable equations: Asymmetric case. Communications on Pure and Applied Analysis, 2019, 18 (3) : 1049-1072. doi: 10.3934/cpaa.2019051 [8] Craig Cowan, Pierpaolo Esposito, Nassif Ghoussoub. Regularity of extremal solutions in fourth order nonlinear eigenvalue problems on general domains. Discrete and Continuous Dynamical Systems, 2010, 28 (3) : 1033-1050. doi: 10.3934/dcds.2010.28.1033 [9] Congming Li, Jisun Lim. The singularity analysis of solutions to some integral equations. Communications on Pure and Applied Analysis, 2007, 6 (2) : 453-464. doi: 10.3934/cpaa.2007.6.453 [10] Jong-Shenq Guo, Yoshihisa Morita. Entire solutions of reaction-diffusion equations and an application to discrete diffusive equations. Discrete and Continuous Dynamical Systems, 2005, 12 (2) : 193-212. doi: 10.3934/dcds.2005.12.193 [11] Limei Dai. Entire solutions with asymptotic behavior of fully nonlinear uniformly elliptic equations. Communications on Pure and Applied Analysis, 2011, 10 (6) : 1707-1714. doi: 10.3934/cpaa.2011.10.1707 [12] Soohyun Bae. Positive entire solutions of inhomogeneous semilinear elliptic equations with supercritical exponent. Conference Publications, 2005, 2005 (Special) : 50-59. doi: 10.3934/proc.2005.2005.50 [13] Alan V. Lair, Ahmed Mohammed. Entire large solutions of semilinear elliptic equations of mixed type. Communications on Pure and Applied Analysis, 2009, 8 (5) : 1607-1618. doi: 10.3934/cpaa.2009.8.1607 [14] Peter Poláčik. On uniqueness of positive entire solutions and other properties of linear parabolic equations. Discrete and Continuous Dynamical Systems, 2005, 12 (1) : 13-26. doi: 10.3934/dcds.2005.12.13 [15] Peter Poláčik, Darío A. Valdebenito. Existence of quasiperiodic solutions of elliptic equations on the entire space with a quadratic nonlinearity. Discrete and Continuous Dynamical Systems - S, 2020, 13 (4) : 1369-1393. doi: 10.3934/dcdss.2020077 [16] Filippo Gazzola. On the moments of solutions to linear parabolic equations involving the biharmonic operator. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3583-3597. doi: 10.3934/dcds.2013.33.3583 [17] Ziqing Yuana, Jianshe Yu. Existence and multiplicity of nontrivial solutions of biharmonic equations via differential inclusion. Communications on Pure and Applied Analysis, 2020, 19 (1) : 391-405. doi: 10.3934/cpaa.2020020 [18] Veronica Felli, Elsa M. Marchini, Susanna Terracini. On the behavior of solutions to Schrödinger equations with dipole type potentials near the singularity. Discrete and Continuous Dynamical Systems, 2008, 21 (1) : 91-119. doi: 10.3934/dcds.2008.21.91 [19] Gawtum Namah, Mohammed Sbihi. A notion of extremal solutions for time periodic Hamilton-Jacobi equations. Discrete and Continuous Dynamical Systems - B, 2010, 13 (3) : 647-664. doi: 10.3934/dcdsb.2010.13.647 [20] Ruyun Ma, Yanqiong Lu. Disconjugacy and extremal solutions of nonlinear third-order equations. Communications on Pure and Applied Analysis, 2014, 13 (3) : 1223-1236. doi: 10.3934/cpaa.2014.13.1223

2020 Impact Factor: 1.392