# American Institute of Mathematical Sciences

May  2008, 7(3): 631-643. doi: 10.3934/cpaa.2008.7.631

## Regularity for positive weak solutions to semi-linear elliptic equations

 1 Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China

Received  July 2007 Revised  November 2007 Published  February 2008

In this paper, we first derive a new non-linear type inequality for Newtonian potential and then we study the regularity problem for positive weak solutions to the non-linear Laplace equation:

$-\Delta u= f(u)\quad$ in $\Omega,$

with $f(u)\in L^1(\Omega)$. Here $\Omega$ is a bounded domain in $R^n$, and $f(u)$ is a regular function with respect to $u$. We give an apriori estimate for positive weak solutions. We show that under some appropriate assumptions on the non-linear term $f$, the positive weak solutions are in fact in some local Sobolev space $W_{l o c}^{1,\tau}(\Omega)$. We also derive a very general local monotonicity formula for variational solutions to the equation above with special nonlinear term $f$.

Citation: Li Ma, Lin Zhao. Regularity for positive weak solutions to semi-linear elliptic equations. Communications on Pure & Applied Analysis, 2008, 7 (3) : 631-643. doi: 10.3934/cpaa.2008.7.631
 [1] Hafedh Bousbih. Global weak solutions for a coupled chemotaxis non-Newtonian fluid. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 907-929. doi: 10.3934/dcdsb.2018212 [2] Xiaolong Han, Guozhen Lu. Regularity of solutions to an integral equation associated with Bessel potential. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1111-1119. doi: 10.3934/cpaa.2011.10.1111 [3] Chao Ji. Ground state solutions of fractional Schrödinger equations with potentials and weak monotonicity condition on the nonlinear term. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-19. doi: 10.3934/dcdsb.2019131 [4] Wendong Wang, Liqun Zhang. The $C^{\alpha}$ regularity of weak solutions of ultraparabolic equations. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 1261-1275. doi: 10.3934/dcds.2011.29.1261 [5] Jiahong Wu. Regularity results for weak solutions of the 3D MHD equations. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 543-556. doi: 10.3934/dcds.2004.10.543 [6] Dixiang Cheng, Zhengrong Liu, Xin Huang. Periodic solutions of a class of Newtonian equations. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1795-1801. doi: 10.3934/cpaa.2009.8.1795 [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] Reinhard Farwig, Paul Felix Riechwald. Regularity criteria for weak solutions of the Navier-Stokes system in general unbounded domains. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 157-172. doi: 10.3934/dcdss.2016.9.157 [9] Fengping Yao, Shulin Zhou. Interior $C^{1,\alpha}$ regularity of weak solutions for a class of quasilinear elliptic equations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1635-1649. doi: 10.3934/dcdsb.2016015 [10] Pablo Ochoa, Julio Alejo Ruiz. A study of comparison, existence and regularity of viscosity and weak solutions for quasilinear equations in the Heisenberg group. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1091-1115. doi: 10.3934/cpaa.2019053 [11] Yaoping Chen, Jianqing Chen. Existence of multiple positive weak solutions and estimates for extremal values for a class of concave-convex elliptic problems with an inverse-square potential. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1531-1552. doi: 10.3934/cpaa.2017073 [12] Scipio Cuccagna, Masaya Maeda. On weak interaction between a ground state and a trapping potential. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3343-3376. doi: 10.3934/dcds.2015.35.3343 [13] Miaohua Jiang. Derivative formula of the potential function for generalized SRB measures of hyperbolic systems of codimension one. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 967-983. doi: 10.3934/dcds.2015.35.967 [14] Xijun Hu, Penghui Wang. Hill-type formula and Krein-type trace formula for $S$-periodic solutions in ODEs. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 763-784. doi: 10.3934/dcds.2016.36.763 [15] Emil Novruzov. On existence and nonexistence of the positive solutions of non-newtonian filtration equation. Communications on Pure & Applied Analysis, 2011, 10 (2) : 719-730. doi: 10.3934/cpaa.2011.10.719 [16] David McCaffrey. A representational formula for variational solutions to Hamilton-Jacobi equations. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1205-1215. doi: 10.3934/cpaa.2012.11.1205 [17] Tetsuya Ishiwata. Motion of polygonal curved fronts by crystalline motion: v-shaped solutions and eventual monotonicity. Conference Publications, 2011, 2011 (Special) : 717-726. doi: 10.3934/proc.2011.2011.717 [18] Cheng-Hsiung Hsu, Jian-Jhong Lin. Existence and non-monotonicity of traveling wave solutions for general diffusive predator-prey models. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1483-1508. doi: 10.3934/cpaa.2019071 [19] Tomoyuki Suzuki. Regularity criteria in weak spaces in terms of the pressure to the MHD equations. Conference Publications, 2011, 2011 (Special) : 1335-1343. doi: 10.3934/proc.2011.2011.1335 [20] Adam Andersson, Felix Lindner. Malliavin regularity and weak approximation of semilinear SPDEs with Lévy noise. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-24. doi: 10.3934/dcdsb.2019081

2018 Impact Factor: 0.925