## Duke Mathematical Journal

### Kelvin transform for Grushin operators and critical semilinear equations

#### Abstract

We study positive entire solutions $u = u(x,y)$ of the critical equation $\begin{equation} \label{sceriffo} \Delta_x u+{(\alpha+1)^2}|x|^{2\alpha} \Delta_y u=-u^{({Q+2})/({Q-2})} \textrm{in}\mathbb{R}^{n}= \mathbb{R}^m\times\mathbb{R}^k, \end{equation}$ where $(x,y)\in \mathbb{R}^m\times \mathbb{R}^k$, $\alpha>0$, and $Q=m+k(\alpha+1)$. In the first part of the article, exploiting the invariance of the equation with respect to a suitable conformal inversion, we prove a “spherical symmetry result for solutions”. In the second part, we show how to reduce the dimension of the problem using a hyperbolic symmetry argument. Given any positive solution $u$ of (1), after a suitable scaling and a translation in the variable $y$, the function $v(x)= u(x,0)$ satisfies the equation $\begin{equation} \label{pistolero} {\mathrm{div}\!_x}(p\nabla_xv )-q v=-p v^{({Q+2})/({Q-2})},|x|\lt 1,\end{equation}$ with a mixed boundary condition. Here, $p$ and $q$ are appropriate radial functions. In the last part, we prove that if $m=k=1$, the solution of (2) is unique and that for $m\ge 3$ and $k=1$, problem (2) has a unique solution in the class of $x$-radial functions

#### Article information

Source
Duke Math. J., Volume 131, Number 1 (2006), 167-202.

Dates
First available in Project Euclid: 15 December 2005

https://projecteuclid.org/euclid.dmj/1134666124

Digital Object Identifier
doi:10.1215/S0012-7094-05-13115-5

Mathematical Reviews number (MathSciNet)
MR2219239

Zentralblatt MATH identifier
1094.35036

Subjects
Primary: 35H20: Subelliptic equations
Secondary: 34B15: Nonlinear boundary value problems

#### Citation

Monti, Roberto; Morbidelli, Daniele. Kelvin transform for Grushin operators and critical semilinear equations. Duke Math. J. 131 (2006), no. 1, 167--202. doi:10.1215/S0012-7094-05-13115-5. https://projecteuclid.org/euclid.dmj/1134666124

#### References

• M. S. Baouendi, Sur une classe d'opérateurs elliptiques dégénérés, Bull. Soc. Math. France 95 (1967), 45--87.
• R. Beals, P. Greiner, and B. Gaveau, Green's functions for some highly degenerate elliptic operators, J. Funct. Anal. 165 (1999), 407--429.
• W. Beckner, On the Grushin operator and hyperbolic symmetry, Proc. Amer. Math. Soc. 129 (2001), 1233--1246.
• J.-M. Bony, Principe du maximum, inégalite de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérés, Ann. Inst. Fourier (Grenoble) 19 (1969), 277--304.
• L. A. Caffarelli, B. Gidas, and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math. 42 (1989), 271--297.
• W. X. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J. 63 (1991), 615--622.
• M. Cowling, A. H. Dooley, A.KoráNyi, and F. Ricci, $H$-type groups and Iwasawa decompositions, Adv. Math. 87 (1991), 1--41.
• B. Franchi and E. Lanconelli, Hölder regularity theorem for a class of nonuniformly elliptic operators with measurable coefficients, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 10 (1983), 523--541.
• B. Franchi, G. Lu, and R. L. Wheeden, A relationship between Poincaré-type inequalities and representation formulas in spaces of homogeneous type, Internat. Math. Res. Notices 1996, no. 1, 1--14.
• N. Garofalo and D. Vassilev, Symmetry properties of positive entire solutions of Yamabe-type equations on groups of Heisenberg type, Duke Math. J. 106 (2001), 411--448.
• B. Gidas, W. M. Ni, and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), 209--243.
• V. V. Grushin [GRU$\check\hbox\fontsize88\selectfonts$IN], A certain class of hypoelliptic operators, Math. USSR-Sb. 12 (1970), 458--476.
• T. Iwaniec and G. Martin, Geometric Function Theory and Non-Linear Analysis, Oxford Math. Monogr., Oxford Univ. Press, New York, 2001.
• D. Jerison and J. M. Lee, The Yamabe problem on CR manifolds, J. Differential Geom. 25 (1987), 167--197.
• —, Extremals for the Sobolev inequality on the Heisenberg group and the CR Yamabe problem, J. Amer. Math. Soc. 1 (1988), 1--13.
• A. KoráNyi, Kelvin transforms and harmonic polynomials on the Heisenberg group, J. Funct. Anal. 49 (1982), 177--185.
• M. K. Kwong and Y. Li, Uniqueness of radial solutions of semilinear elliptic equations, Trans. Amer. Math. Soc. 333 (1992), 339--363.
• Y. Li and L. Zhang, Liouville-type theorems and Harnack-type inequalities for semilinear elliptic equations, J. Anal. Math. 90 (2003), 27--87.
• Y. Li and M. Zhu, Uniqueness theorems through the method of moving spheres, Duke Math. J. 80 (1995), 383--417.
• J. G. Ratcliffe, Foundations of Hyperbolic Manifolds, Grad. Texts in Math. 149, Springer, New York, 1994.