Radial symmetry and regularity of solutions for poly-harmonic Dirichlet problems. (English) Zbl 1211.35101

Summary: Let \({\mathbf B}=B_1(0)\) be the unit ball in \(\mathbb R^n\) and \(r=|x|\). We study the poly-harmonic Dirichlet problem
\[ \begin{cases} (-\Delta)^mu=f(u) &\text{in }{\mathbf B},\\ u=\frac{\partial u}{\partial r}=\cdots= \frac{\partial^{m-1}u}{\partial r^{m-1}}=0 &\text{on }\partial{\mathbf B}. \end{cases} \]
Using the corresponding integral equation and the method of moving planes in integral forms, we show that the positive solutions are radially symmetric and monotone decreasing about the origin. We also obtain regularity for solutions.


35J40 Boundary value problems for higher-order elliptic equations
35J30 Higher-order elliptic equations
35J35 Variational methods for higher-order elliptic equations
35B65 Smoothness and regularity of solutions to PDEs
Full Text: DOI


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