# 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 and Continuous Dynamical Systems, 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 and 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 and Continuous Dynamical Systems, 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 and Continuous Dynamical Systems, 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 and 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 and Continuous Dynamical Systems, 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 and 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 and Continuous Dynamical Systems, 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 and Continuous Dynamical Systems, 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 and 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 and Continuous Dynamical Systems, 2016, 36 (2) : 1125-1141. doi: 10.3934/dcds.2016.36.1125 [11] Leyun Wu, Pengcheng Niu. Symmetry and nonexistence of positive solutions to fractional p-Laplacian equations. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1573-1583. doi: 10.3934/dcds.2019069 [12] Ran Zhuo, Yan Li. Nonexistence and symmetry of solutions for Schrödinger systems involving fractional Laplacian. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1595-1611. doi: 10.3934/dcds.2019071 [13] Phuong Le, Hoang-Hung Vo. Monotonicity and symmetry of positive solutions to degenerate quasilinear elliptic systems in half-spaces and strips. Communications on Pure and Applied Analysis, 2022, 21 (3) : 1027-1048. doi: 10.3934/cpaa.2022008 [14] Rongrong Yang, Zhongxue Lü. The properties of positive solutions to semilinear equations involving the fractional Laplacian. Communications on Pure and Applied Analysis, 2019, 18 (3) : 1073-1089. doi: 10.3934/cpaa.2019052 [15] Phuong Le. Symmetry of singular solutions for a weighted Choquard equation involving the fractional $p$-Laplacian. Communications on Pure and Applied Analysis, 2020, 19 (1) : 527-539. doi: 10.3934/cpaa.2020026 [16] Francesco Esposito. Symmetry and monotonicity properties of singular solutions to some cooperative semilinear elliptic systems involving critical nonlinearities. Discrete and Continuous Dynamical Systems, 2020, 40 (1) : 549-577. doi: 10.3934/dcds.2020022 [17] Miaomiao Niu, Zhongwei Tang. Least energy solutions for nonlinear Schrödinger equation involving the fractional Laplacian and critical growth. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 3963-3987. doi: 10.3934/dcds.2017168 [18] Xin Yin, Wenming Zou. Positive least energy solutions for k-coupled critical systems involving fractional Laplacian. Discrete and Continuous Dynamical Systems - S, 2021, 14 (6) : 1995-2023. doi: 10.3934/dcdss.2021042 [19] 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 and Applied Analysis, 2013, 12 (4) : 1731-1744. doi: 10.3934/cpaa.2013.12.1731 [20] Gabriele Grillo, Matteo Muratori, Maria Michaela Porzio. Porous media equations with two weights: Smoothing and decay properties of energy solutions via Poincaré inequalities. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3599-3640. doi: 10.3934/dcds.2013.33.3599

2021 Impact Factor: 1.588

## Metrics

• HTML views (0)
• Cited by (44)

• on AIMS