Weighted fourth order elliptic problems in the unit ball

<p style='text-indent:20px;'>Existence and uniqueness of positive radial solutions of some weighted fourth order elliptic Navier and Dirichlet problems in the unit ball <inline-formula><tex-math id="M1">\begin{document}$ B $\end{document}</tex-math></inline-formula> are studied. The weights can be singular at <inline-formula><tex-math id="M2">\begin{document}$ x = 0 \in B $\end{document}</tex-math></inline-formula>. Existence of positive radial solutions of the problems is obtained via variational methods in the weighted Sobolev spaces. To obtain the uniqueness results, we need to know exactly the asymptotic behavior of the solutions at the singular point <inline-formula><tex-math id="M3">\begin{document}$ x = 0 $\end{document}</tex-math></inline-formula>.</p>

[1]  Zongming Guo,et al.  Asymptotic behavior at the isolated singularities of solutions of some equations on singular manifolds with conical metrics , 2020 .

[2]  Zongming Guo,et al.  Finite Morse index solutions of weighted elliptic equations and the critical exponents , 2015 .

[3]  Q. Phan,et al.  Liouville-type theorems and bounds of solutions of Hardy-Hénon equations , 2012 .

[4]  Salem Rebhi,et al.  Classification of finite Morse index solutions for Hénon type elliptic equation $$-\Delta u = |x|^{\alpha } u_{+}^{p}$$-Δu=|x|αu+p , 2014 .

[5]  D. G. Figueiredo,et al.  Sobolev spaces of symmetric functions and applications , 2011 .

[6]  Zhi-Qiang Wang,et al.  Symmetry of extremal functions for the Caffarelli-Kohn-Nirenberg inequalities , 2004 .

[7]  Zongming Guo,et al.  Existence and regularity of positive solutions of a degenerate elliptic problem , 2018, Mathematische Nachrichten.

[8]  Zongming Guo,et al.  Embeddings of weighted Sobolev spaces and a weighted fourth-order elliptic equation , 2018, Communications in Contemporary Mathematics.

[9]  Existence and nonexistence results for a weighted elliptic equation in exterior domains , 2018, Zeitschrift für angewandte Mathematik und Physik.

[10]  Zongming Guo,et al.  Further study of a weighted elliptic equation , 2017 .

[11]  Zongming Guo,et al.  Monotonicity formula and $$\varepsilon $$ε-regularity of stable solutions to supercritical problems and applications to finite Morse index solutions , 2014 .

[12]  E. N. Dancer,et al.  Finite Morse index solutions of an elliptic equation with supercritical exponent , 2011 .

[13]  P. Caldiroli,et al.  On Caffarelli–Kohn–Nirenberg-type Inequalities for the Weighted Biharmonic Operator in Cones , 2011 .

[14]  Zhi-Qiang Wang,et al.  Asymptotic symmetry and local behaviors of solutions to a class of anisotropic elliptic equations , 2011 .

[15]  K. Chou,et al.  On the Best Constant for a Weighted Sobolev‐Hardy Inequality , 1993 .

[16]  Florin Catrina,et al.  On the Caffarelli-Kohn-Nirenberg inequalities: Sharp constants, existence (and nonexistence), and symmetry of extremal functions † , 2001 .