November  2008, 7(6): 1497-1506. doi: 10.3934/cpaa.2008.7.1497

Existence results for nonlinear elliptic equations related to Gauss measure in a limit case

1. 

Seconda Universitá degli Studi di Napoli, Dipartimento di Matematica, Via Vivaldi 43, Caserta, Italy

2. 

Universitá degli Studi di Napoli Parthenope, Dipartimento per le Tecnologie, Is. C4, Centro Direzionale, Napoli, Italy

3. 

Universitá degli Studi di Napoli “Federico II”, Dipartimento di Matematica “R. Caccioppoli”, Complesso Monte S. Angelo, Napoli, Italy

Received  October 2007 Revised  April 2008 Published  August 2008

The aim of this paper is to prove existence results for nonlinear elliptic equations whose the prototype is -div$(|\nabla u|^{p-2}\nabla u\varphi) =g\varphi $ in a open subset $ \Omega $ of $R^n,$ $u=0$ on $\partial \Omega $, where $p\geq 2$, the function $\varphi (x)=(2\pi)^{-\frac{n}{2}}$exp$( -|x|^2 /2) $ is the density of Gauss measure and $g\in L^1$ (log $L)^{\frac{1}{2}}( \varphi, \Omega)$. This condition on the function $g$ is sharp in the class of Zygmund spaces.
Citation: Giuseppina di Blasio, Filomena Feo, Maria Rosaria Posteraro. Existence results for nonlinear elliptic equations related to Gauss measure in a limit case. Communications on Pure & Applied Analysis, 2008, 7 (6) : 1497-1506. doi: 10.3934/cpaa.2008.7.1497
[1]

Xavier Cabré. Elliptic PDE's in probability and geometry: Symmetry and regularity of solutions. Discrete & Continuous Dynamical Systems - A, 2008, 20 (3) : 425-457. doi: 10.3934/dcds.2008.20.425

[2]

Neil S. Trudinger, Xu-Jia Wang. Quasilinear elliptic equations with signed measure. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 477-494. doi: 10.3934/dcds.2009.23.477

[3]

Evgeny Galakhov. Some nonexistence results for quasilinear PDE's. Communications on Pure & Applied Analysis, 2007, 6 (1) : 141-161. doi: 10.3934/cpaa.2007.6.141

[4]

Marco Degiovanni, Michele Scaglia. A variational approach to semilinear elliptic equations with measure data. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1233-1248. doi: 10.3934/dcds.2011.31.1233

[5]

Fatiha Alabau-Boussouira. On the influence of the coupling on the dynamics of single-observed cascade systems of PDE's. Mathematical Control & Related Fields, 2015, 5 (1) : 1-30. doi: 10.3934/mcrf.2015.5.1

[6]

Dario Bambusi, Simone Paleari. Families of periodic orbits for some PDE’s in higher dimensions. Communications on Pure & Applied Analysis, 2002, 1 (2) : 269-279. doi: 10.3934/cpaa.2002.1.269

[7]

Armen Shirikyan, Leonid Volevich. Qualitative properties of solutions for linear and nonlinear hyperbolic PDE's. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 517-542. doi: 10.3934/dcds.2004.10.517

[8]

Ronald Mickens, Kale Oyedeji. Traveling wave solutions to modified Burgers and diffusionless Fisher PDE's. Evolution Equations & Control Theory, 2019, 8 (1) : 139-147. doi: 10.3934/eect.2019008

[9]

Ammari Zied, Liard Quentin. On uniqueness of measure-valued solutions to Liouville's equation of Hamiltonian PDEs. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 723-748. doi: 10.3934/dcds.2018032

[10]

Robert Jensen, Andrzej Świech. Uniqueness and existence of maximal and minimal solutions of fully nonlinear elliptic PDE. Communications on Pure & Applied Analysis, 2005, 4 (1) : 199-207. doi: 10.3934/cpaa.2005.4.187

[11]

Paul H. Rabinowitz. A new variational characterization of spatially heteroclinic solutions of a semilinear elliptic PDE. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 507-515. doi: 10.3934/dcds.2004.10.507

[12]

Huyuan Chen, Hichem Hajaiej, Ying Wang. Boundary blow-up solutions to fractional elliptic equations in a measure framework. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 1881-1903. doi: 10.3934/dcds.2016.36.1881

[13]

G. R. Cirmi, S. Leonardi. Higher differentiability for solutions of linear elliptic systems with measure data. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 89-104. doi: 10.3934/dcds.2010.26.89

[14]

Armen Shirikyan. Ergodicity for a class of Markov processes and applications to randomly forced PDE'S. II. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 911-926. doi: 10.3934/dcdsb.2006.6.911

[15]

Konstantinos Chrysafinos, Efthimios N. Karatzas. Symmetric error estimates for discontinuous Galerkin approximations for an optimal control problem associated to semilinear parabolic PDE's. Discrete & Continuous Dynamical Systems - B, 2012, 17 (5) : 1473-1506. doi: 10.3934/dcdsb.2012.17.1473

[16]

Pierre Frankel. Alternating proximal algorithm with costs-to-move, dual description and application to PDE's. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 545-557. doi: 10.3934/dcdss.2012.5.545

[17]

Diego Castellaneta, Alberto Farina, Enrico Valdinoci. A pointwise gradient estimate for solutions of singular and degenerate pde's in possibly unbounded domains with nonnegative mean curvature. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1983-2003. doi: 10.3934/cpaa.2012.11.1983

[18]

Peter Bella, Arianna Giunti. Green's function for elliptic systems: Moment bounds. Networks & Heterogeneous Media, 2018, 13 (1) : 155-176. doi: 10.3934/nhm.2018007

[19]

Pengyan Wang, Pengcheng Niu. Liouville's theorem for a fractional elliptic system. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1545-1558. doi: 10.3934/dcds.2019067

[20]

Ugur G. Abdulla. Wiener's criterion at $\infty$ for the heat equation and its measure-theoretical counterpart. Electronic Research Announcements, 2008, 15: 44-51. doi: 10.3934/era.2008.15.44

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (14)
  • HTML views (0)
  • Cited by (4)

[Back to Top]