Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Strong comparison principle for solutions of quasilinear equations


Authors: M. Lucia and S. Prashanth
Journal: Proc. Amer. Math. Soc. 132 (2004), 1005-1011
MSC (2000): Primary 35B50
DOI: https://doi.org/10.1090/S0002-9939-03-07285-X
Published electronically: November 19, 2003
MathSciNet review: 2045415
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $\Omega \subset \mathbb{R} ^N$, $N \geq 1$, be a bounded smooth connected open set and $\mathbf{a} : \Omega \times \mathbb{R} ^N \to \mathbb{R} ^N$ be a map satisfying the hypotheses (H1)-(H4) below. Let $f_1,f_2 \in \mathrm{L}_{loc}^{1} (\Omega)$ with $f_2 \geq f_1$, $f_1 \not\equiv f_2$ in $\Omega$ and $u_1, u_2 \in \mathcal{C}^{1,\theta} (\overline \Omega)$ with $\theta \in (0,1]$ be two weak solutions of

\begin{displaymath}(P_i)\quad -\mathrm{div} (\mathbf{a}(x,\nabla u_i)) = f_i \quad \hbox{in } \Omega, \,\quad i=1,2.\end{displaymath}

Suppose that $u_2 \geq u_1$ in $\Omega$. Then we show that $u_2 > u_1$ in $\Omega$ under the following assumptions: either $u_2>u_1$ on $\partial \Omega$, or $u_1=u_2=0$ on $\partial \Omega$ and $u_1 \geq 0$ in $\Omega$. We also show a measure-theoretic version of the Strong Comparison Principle.


References [Enhancements On Off] (What's this?)

  • 1. M. Cuesta and C. P. Takác, A strong comparison principle for positive solutions of degenerate elliptic equations, Differential Integral Equations $\mathbf{13}$ (2000), 721-746. MR 2001h:35008
  • 2. L. C. Evans and R. F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics, 1st edition, CRC Press, Boca Raton, FL, 1992. MR 93f:28001
  • 3. M. Guedda and L. Véron, Quasilinear elliptic equations involving critical Sobolev exponents, Nonlinear Anal. $\mathbf{13}$ (1989), 879-902. MR 90h:35100
  • 4. S. Prashanth, Strong comparison principle for radial solutions of quasi-linear equations, J. Math. Anal. Appl. $\mathbf{258}$ (2001), 366-370. MR 2002a:35010
  • 5. W. Reichel and W. Walter, Radial solutions of equations and inequalities involving the $p$-Laplacian, J. Inequalities and Applications $\mathbf{1}$ (1997), 47-71. MR 2000j:34006
  • 6. P. Tolksdorf, On the Dirichlet problem for quasilinear equations in domains with conical boundary points, Comm. Partial Differential Equations $\mathbf{8}$(7) (1983), 773-817. MR 85g:35053

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 35B50

Retrieve articles in all journals with MSC (2000): 35B50


Additional Information

M. Lucia
Affiliation: Department Mathematics, Hill Center, Rutgers University, 110 Frelinghuysen Rd, Piscataway, New Jersey 08854
Email: mlucia@math.rutgers.edu

S. Prashanth
Affiliation: TIFR Center, IISc. Campus, Post Box No. 1234, Bangalore 560012, India
Email: pras@math.tifrbng.res.in

DOI: https://doi.org/10.1090/S0002-9939-03-07285-X
Received by editor(s): August 20, 2002
Published electronically: November 19, 2003
Additional Notes: The first author was supported by Swiss National Foundation, Contract No. 8220-64676
The second author was supported by Indo-French Project (IFCPAR) No. 1901-2
Communicated by: David S. Tartakoff
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society