Existence and uniqueness of positive radial solutions of some weighted fourth order elliptic Navier and Dirichlet problems in the unit ball $ B $ are studied. The weights can be singular at $ x = 0 \in B $. Existence of positive radial solutions of the problems is obtained via variational methods in the weighted Sobolev spaces. To obtain the uniqueness results, we need to know exactly the asymptotic behavior of the solutions at the singular point $ x = 0 $.
Citation: Zongming Guo, Fangshu Wan. Weighted fourth order elliptic problems in the unit ball[J]. Electronic Research Archive, 2021, 29(6): 3775-3803. doi: 10.3934/era.2021061
Existence and uniqueness of positive radial solutions of some weighted fourth order elliptic Navier and Dirichlet problems in the unit ball $ B $ are studied. The weights can be singular at $ x = 0 \in B $. Existence of positive radial solutions of the problems is obtained via variational methods in the weighted Sobolev spaces. To obtain the uniqueness results, we need to know exactly the asymptotic behavior of the solutions at the singular point $ x = 0 $.
[1] | First order interpolation inequalities with weights. Compositio Math. (1984) 53: 259-275. |
[2] | On Caffarelli-Kohn-Nirenberg type inequalities for the weighted biharmonic operator in cones. Milan J. Math. (2011) 79: 657-687. |
[3] | On the Caffarelli-Kohn-Nirenberg inequalities: Sharp constants, existence (and nonexistence), and symmetry of extremal functions. Comm. Pure Appl. Math. (2001) 54: 229-258. |
[4] | On the best constant for a weighted Sobolev-Hardy inequality. J. London Math. Soc. (1993) 48: 137-151. |
[5] | Finite Morse index solutions of an elliptic equation with supercritical exponent. J. Differential Equations (2011) 250: 3281-3310. |
[6] | Finite Morse index solutions of weighted elliptic equations and the critical exponents. Calc. Var. PDEs. (2015) 54: 3161-3181. |
[7] | Monotonicity formula and $\epsilon$-regularity of stable solutions to supercritical problems and applications to finite Morse index solutions. Calc. Var. PDEs. (2014) 50: 615-638. |
[8] | Sobolev spaces of symmetric functions and applications. J. Funct. Anal. (2011) 261: 3735-3770. |
[9] | Existence and regularity of positive solutions of a degenerate elliptic problem. Math. Nachr. (2019) 292: 56-78. |
[10] | Z. Guo, X. Huang and D. Ye, Existence and nonexistence results for a weighted elliptic equation in exterior domains, Z. Angew. Math. Phy., 71 (2020), Paper No. 116, 9 pp. doi: 10.1007/s00033-020-01338-0 |
[11] | Asymptotic behavior at the isolated singularities of solutions of some equations on singular manifolds with conical metrics. Comm. Partial Differential Equations (2020) 45: 1647-1681. |
[12] | Further study of a weighted elliptic equation. Sci. China Math. (2017) 60: 2391-2406. |
[13] | Z. Guo, F. Wan and L. Wang, Embeddings of weighted Sobolev spaces and a weighted fourth order elliptic equation, Comm. Contemp. Math., 22 (2020), 1950057, 40 pp. doi: 10.1142/S0219199719500573 |
[14] | Asymptotic symmetry and local behaviors of solutions to a class of anisotropic elliptic equations. Indiana Univ. Math. J. (2011) 60: 1623-1653. |
[15] | Symmetry of extremal functions for the Caffarelli-Kohn-Nirenberg inequalities. Proc. Amer. Math. Soc. (2004) 132: 1685-1691. |
[16] | Liouville-type theorems and bounds of solutions for Hardy-Hénon equations. J. Differential Equations (2012) 252: 2544-2562. |
[17] | Classification of finite Morse index solutions for Hénon type elliptic equation $-\Delta u=|x|^\alpha u_+^p$. Calc. Var. PDEs. (2014) 50: 847-866. |