# American Institute of Mathematical Sciences

July  2010, 28(3): 1179-1206. doi: 10.3934/dcds.2010.28.1179

## Energy estimates and 1-D symmetry for nonlinear equations involving the half-Laplacian

 1 ICREA and Universitat Politècnica de Catalunya, Departament de Matemàtica Aplicada 1, Diagonal 647, 08028 Barcelona 2 Universitat Politècnica de Catalunya, Departament de Matemàtica Aplicada 1, Diagonal 647, 08028 Barcelona, Spain

Received  April 2010 Published  April 2010

We establish sharp energy estimates for some solutions, such as global minimizers, monotone solutions and saddle-shaped solutions, of the fractional nonlinear equation $(-\Delta)$1/2 $u=f(u)$ in R n. Our energy estimates hold for every nonlinearity $f$ and are sharp since they are optimal for one-dimensional solutions, that is, for solutions depending only on one Euclidian variable.
As a consequence, in dimension $n=3$, we deduce the one-dimensional symmetry of every global minimizer and of every monotone solution. This result is the analog of a conjecture of De Giorgi on one-dimensional symmetry for the classical equation $-\Delta u=f(u)$ in R n.
Citation: Xavier Cabré, Eleonora Cinti. Energy estimates and 1-D symmetry for nonlinear equations involving the half-Laplacian. Discrete & Continuous Dynamical Systems - A, 2010, 28 (3) : 1179-1206. doi: 10.3934/dcds.2010.28.1179
 [1] Antonio Capella. Solutions of a pure critical exponent problem involving the half-laplacian in annular-shaped domains. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1645-1662. doi: 10.3934/cpaa.2011.10.1645 [2] María del Mar González, Regis Monneau. Slow motion of particle systems as a limit of a reaction-diffusion equation with half-Laplacian in dimension one. Discrete & Continuous Dynamical Systems - A, 2012, 32 (4) : 1255-1286. doi: 10.3934/dcds.2012.32.1255 [3] Zhigang Wu, Hao Xu. Symmetry properties in systems of fractional Laplacian equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1559-1571. doi: 10.3934/dcds.2019068 [4] Miaomiao Niu, Zhongwei Tang. Least energy solutions of nonlinear Schrödinger equations involving the half Laplacian and potential wells. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1215-1231. doi: 10.3934/cpaa.2016.15.1215 [5] Alberto Farina. Some symmetry results for entire solutions of an elliptic system arising in phase separation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (6) : 2505-2511. doi: 10.3934/dcds.2014.34.2505 [6] Lizhi Zhang. Symmetry of solutions to semilinear equations involving the fractional laplacian. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2393-2409. doi: 10.3934/cpaa.2015.14.2393 [7] Tingzhi Cheng. Monotonicity and symmetry of solutions to fractional Laplacian equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3587-3599. doi: 10.3934/dcds.2017154 [8] Peter Poláčik. On uniqueness of positive entire solutions and other properties of linear parabolic equations. Discrete & Continuous Dynamical Systems - A, 2005, 12 (1) : 13-26. doi: 10.3934/dcds.2005.12.13 [9] Christos Sourdis. On the growth of the energy of entire solutions to the vector Allen-Cahn equation. Communications on Pure & Applied Analysis, 2015, 14 (2) : 577-584. doi: 10.3934/cpaa.2015.14.577 [10] Ran Zhuo, Wenxiong Chen, Xuewei Cui, Zixia Yuan. Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 1125-1141. doi: 10.3934/dcds.2016.36.1125 [11] Ran Zhuo, Yan Li. Nonexistence and symmetry of solutions for Schrödinger systems involving fractional Laplacian. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1595-1611. doi: 10.3934/dcds.2019071 [12] Leyun Wu, Pengcheng Niu. Symmetry and nonexistence of positive solutions to fractional p-Laplacian equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1573-1583. doi: 10.3934/dcds.2019069 [13] Rongrong Yang, Zhongxue Lü. The properties of positive solutions to semilinear equations involving the fractional Laplacian. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1073-1089. doi: 10.3934/cpaa.2019052 [14] Phuong Le. Symmetry of singular solutions for a weighted Choquard equation involving the fractional $p$-Laplacian. Communications on Pure & Applied Analysis, 2020, 19 (1) : 527-539. doi: 10.3934/cpaa.2020026 [15] Miaomiao Niu, Zhongwei Tang. Least energy solutions for nonlinear Schrödinger equation involving the fractional Laplacian and critical growth. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3963-3987. doi: 10.3934/dcds.2017168 [16] Vitali Liskevich, Igor I. Skrypnik, Zeev Sobol. Estimates of solutions for the parabolic $p$-Laplacian equation with measure via parabolic nonlinear potentials. Communications on Pure & Applied Analysis, 2013, 12 (4) : 1731-1744. doi: 10.3934/cpaa.2013.12.1731 [17] Gabriele Grillo, Matteo Muratori, Maria Michaela Porzio. Porous media equations with two weights: Smoothing and decay properties of energy solutions via Poincaré inequalities. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3599-3640. doi: 10.3934/dcds.2013.33.3599 [18] Giuseppina Autuori, Patrizia Pucci. Entire solutions of nonlocal elasticity models for composite materials. Discrete & Continuous Dynamical Systems - S, 2018, 11 (3) : 357-377. doi: 10.3934/dcdss.2018020 [19] Antonio Vitolo. On the growth of positive entire solutions of elliptic PDEs and their gradients. Discrete & Continuous Dynamical Systems - S, 2014, 7 (6) : 1335-1346. doi: 10.3934/dcdss.2014.7.1335 [20] Wan-Tong Li, Li Zhang, Guo-Bao Zhang. Invasion entire solutions in a competition system with nonlocal dispersal. Discrete & Continuous Dynamical Systems - A, 2015, 35 (4) : 1531-1560. doi: 10.3934/dcds.2015.35.1531

2018 Impact Factor: 1.143