## Abstract and Applied Analysis

### Positive solutions of higher order quasilinear elliptic equations

Marcelo Montenegro

#### Abstract

The higher order quasilinear elliptic equation $-\Delta(\Delta_{p}(\Delta u))= f(x,u)$ subject to Dirichlet boundary conditions may have unique and regular positive solution. If the domain is a ball, we obtain a priori estimate to the radial solutions via blowup. Extensions to systems and general domains are also presented. The basic ingredients are the maximum principle, Moser iterative scheme, an eigenvalue problem, a priori estimates by rescalings, sub/supersolutions, and Krasnosel′skiĭ fixed point theorem.

#### Article information

Source
Abstr. Appl. Anal., Volume 7, Number 8 (2002), 423-452.

Dates
First available in Project Euclid: 14 April 2003

https://projecteuclid.org/euclid.aaa/1050348373

Digital Object Identifier
doi:10.1155/S1085337502204030

Mathematical Reviews number (MathSciNet)
MR1930826

Zentralblatt MATH identifier
1044.35021

Subjects
Primary: 35J55 35A05
Secondary: 35J60: Nonlinear elliptic equations

#### Citation

Montenegro, Marcelo. Positive solutions of higher order quasilinear elliptic equations. Abstr. Appl. Anal. 7 (2002), no. 8, 423--452. doi:10.1155/S1085337502204030. https://projecteuclid.org/euclid.aaa/1050348373

#### References

• A. Anane, Simplicité et isolation de la première valeur propre du $p$-Laplacien avec poids [Simplicity and isolation of the first eigenvalue of the $p$-Laplacian with weight ], C. R. Acad. Sci. Paris Sér. I Math. 305 (1987), no. 16, 725–728 (French).
• P. Clément, J. Fleckinger, E. Mitidieri, and F. de Thélin, Existence of positive solutions for a nonvariational quasilinear elliptic system, J. Differential Equations 166 (2000), no. 2, 455–477.
• P. Clément, R. Manásevich, and E. Mitidieri, Positive solutions for a quasilinear system via blow up, Comm. Partial Differential Equations 18 (1993), no. 12, 2071–2106.
• H.-C. Grunau and G. Sweers, Positivity properties of elliptic boundary value problems of higher order, Nonlinear Anal. 30 (1997), no. 8, 5251–5258.
• M. A. Krasnosel'skiĭ, Positive Solutions of Operator Equations, P. Noordhoff Ltd., Groningen, 1964, translated from the Russian by Flaherty, R. E.
• G. Lu, J. Wei, and X. Xu, On conformally invariant equation $(-\Delta)\sp p u-{K}(x)u\sp {{N}+2p/{N}-2p}=0$ and its generalizations, Ann. Mat. Pura Appl. (4) 179 (2001), 309–329. \CMP1+848+7691 848 769
• S. Luckhaus, Existence and regularity of weak solutions to the Dirichlet problem for semilinear elliptic systems of higher order, J. reine angew. Math. 306 (1979), 192–207.
• M. Ôtani, Existence and nonexistence of nontrivial solutions of some nonlinear degenerate elliptic equations, J. Funct. Anal. 76 (1988), no. 1, 140–159.
• L. A. Peletier and R. C. A. M. van der Vorst, Existence and nonexistence of positive solutions of nonlinear elliptic systems and the biharmonic equation, Differential Integral Equations 5 (1992), no. 4, 747–767.
• P. Pucci and J. Serrin, A general variational identity, Indiana Univ. Math. J. 35 (1986), no. 3, 681–703.
• ––––, Critical exponents and critical dimensions for polyharmonic operators, J. Math. Pures Appl. (9) 69 (1990), no. 1, 55–83.
• S. Sakaguchi, Concavity properties of solutions to some degenerate quasilinear elliptic Dirichlet problems, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 14 (1987), no. 3, 403–421.
• J. L. Vázquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim. 12 (1984), no. 3, 191–202.
• X. J. Wang, Sharp constant in a Sobolev inequality, Nonlinear Anal. 20 (1993), no. 3, 261–268.
• J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations, Math. Ann. 313 (1999), no. 2, 207–228.