## Osaka Journal of Mathematics

### A comparison principle and applications to asymptotically $p$-linear boundary value problems

Dang Dinh Hai

#### Abstract

Consider the problems \begin{equation*} \left\{ \begin{array}{@{}ll@{}} -\Delta_{p}u=f\ \text{in}\ \Omega{,} & u=0\ \text{on}\ \partial \Omega,\\ -\Delta_{p}v=g\ \text{in}\ \Omega{,} & v=0\ \text{on}\ \partial \Omega, \end{array} \right. \end{equation*} where $\Omega$ is a bounded domain in $\mathbb{R}^{n}$ with smooth boundary $\partial \Omega$, $\Delta_{p}z=\mathrm{div}(\lvert\nabla z\rvert^{p-2}\nabla z)$, $p>1$. We prove a strong comparison principle that allows $f-g$ to change sign. An application to singular asymptotically $p$-linear boundary problems is given.

#### Article information

Source
Osaka J. Math., Volume 52, Number 2 (2015), 393-409.

Dates
First available in Project Euclid: 24 March 2015

https://projecteuclid.org/euclid.ojm/1427202893

Mathematical Reviews number (MathSciNet)
MR3326617

Zentralblatt MATH identifier
1325.35042

Subjects
Primary: 35J95 35J70: Degenerate elliptic equations

#### Citation

Hai, Dang Dinh. A comparison principle and applications to asymptotically $p$-linear boundary value problems. Osaka J. Math. 52 (2015), no. 2, 393--409. https://projecteuclid.org/euclid.ojm/1427202893

#### References

• A. Ambrosetti, D. Arcoya and B. Buffoni: Positive solutions for some semi-positone problems via bifurcation theory, Differential Integral Equations 7 (1994), 655–663.
• A. Ambrosetti and P. Hess: Positive solutions of asymptotically linear elliptic eigenvalue problems, J. Math. Anal. Appl. 73 (1980), 411–422.
• A. Ambrosetti, J. Garcia Azorero and I. Peral: Multiplicity results for some nonlinear elliptic equations, J. Funct. Anal. 137 (1996), 219–242.
• D.D. Hai: On an asymptotically linear singular boundary value problems, Topol. Methods Nonlinear Anal. 39 (2012), 83–92.
• D.D. Hai: On a class of singular $p$-Laplacian boundary value problems, J. Math. Anal. Appl. 383 (2011), 619–626.
• D.D. Hai and J.L. Williams: Positive radial solutions for a class of quasilinear boundary value problems in a ball, Nonlinear Anal. 75 (2012), 1744–1750.
• A.C. Lazer and P.J. McKenna: On a singular nonlinear elliptic boundary-value problem, Proc. Amer. Math. Soc. 111 (1991), 721–730.
• G.M. Lieberman: Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal. 12 (1988), 1203–1219.
• M. Lucia and S. Prashanth: Strong comparison principle for solutions of quasilinear equations, Proc. Amer. Math. Soc. 132 (2004), 1005–1011.
• T. Oden: Qualitative Methods in Nonlinear Mechanics, Prentice-Hall, Inc, Englewood Cliffs, NJ, 1986.
• S. Sakaguchi: Concavity properties of solutions to some degenerate quasilinear elliptic Dirichlet problems, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 14 (1987), 403–421.
• P. Tolksdorf: Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations 51 (1984), 126–150.
• \begingroup P. Tolksdorf: On the Dirichlet problem for quasilinear equations in domains with conical boundary points, Comm. Partial Differential Equations 8 (1983), 773–817. \endgroup
• J.L. Vázquez: A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim. 12 (1984), 191–202.
• Z. Zhang: Critical points and positive solutions of singular elliptic boundary value problems, J. Math. Anal. Appl. 302 (2005), 476–483.