A priori estimates in L${}^{\infty }$ for non-diagonal perturbed quasilinear systems

Krömer, Stefan

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 8 (2009) no. 3, pp. 417-428

We present a way to derive a priori estimates in $L^{\infty }$ for a class of quasilinear systems containing examples with a leading part which is neither diagonal nor of Uhlenbeck type. Moreover, a perturbation term with natural growth in first order derivatives is allowed.

Classification : 35B45, 35J45

[1] A. Bensoussan, L. Boccardo and F. Murat, On a nonlinear partial differential equation having natural growth terms and unbounded solution, Ann. Inst. H. Poincaré, Anal. Non Linéaire 5 (1988), 347–364. | MR | EuDML | Zbl | Numdam

[2] A. Bensoussan and L. Boccardo, Nonlinear systems of elliptic equations with natural growth conditions and sign conditions, Appl. Math. Optim. 46 (2002), 143–166. | MR | Zbl

[3] J. Chabrowski and K. Zhang, Quasi-monotonicity and perturbated systems with critical growth, Indiana Univ. Math. J. 41 (1992), 483–504. | MR | Zbl

[4] P. Drábek, A. Kufner and F. Nicolosi, “Quasilinear Elliptic Equations with Degenerations and Singularities”, de Gruyter Series in Nonlinear Analysis and Applications, Vol. 5, Walter de Gruyter, Berlin, 1997. | MR

[5] L. C. Evans and R. F. Gariepy, “Measure Theory and Fine Properties of Functions”, Studies in Advanced Mathematics, CRC Press, Boca Raton, 1992. | MR | Zbl

[6] L. C. Evans, “Weak Convergence Methods for Nonlinear Partial Differential Equations”, Regional conference series in mathematics, Vol. 74, AMS, Providence, RI, 1990. | MR

[7] J. Frehse, A note on the Hölder continuity of solutions of variational problems, Abh. Math. Sem. Univ. Hamburg 43 (1975), 59–63. | MR | Zbl

[8] M. Giaquinta, “Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems”, Annals of Mathematics Studies, Vol. 105, Princeton University Press, Princeton, New Jersey, 1983. | Zbl | MR

[9] S. Krömer and M. Lilli, Branches of positive solutions of quasilinear elliptic equations on non–smooth domains, Nonlinear Anal. 64 (2006), 2183–2202. | MR | Zbl

[10] O. A. Ladyzhenskaya and N. N. Ural’Tseva, “Linear and Quasilinear Elliptic Equations”, Mathematics in Science and Engineering, Vol. 46, Academic Press, New York-London, 1968. | MR | Zbl

[11] R. Landes, On the existence of weak solutions of perturbed systems with critical growth, J. Reine Angew. Math. 393 (1989), 21–38. | MR | EuDML | Zbl

[12] R. Landes, Solvability of perturbed elliptic equations with critical growth exponent for the gradient, J. Math. Anal. Appl. 139 (1989), 63–77. | MR | Zbl

[13] R. Landes, On the angle condition for the perturbation of elliptic systems, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 29 (2000), 253–268. | MR | EuDML | Zbl | Numdam

[14] R. Landes, Testfunctions for elliptic systems and maximum principles, Forum Math. 12 (2000), 23–52. | MR | Zbl

[15] R. Landes On the regularity of weak solutions of certain elliptic systems, Calc. Var. Partial Differential Equations 25 (2006), 247–255. | MR | Zbl

[16] B. Pellacci and M. Squassina, Unbounded critical points for a class of lower semicontinuous functionals, J. Differential Equations 201(2004), 25–62. | MR | Zbl

[17] M. Squassina, Existence, multiplicity, perturbation, and concentration results for a class of quasi-linear elliptic problems, Electron. J. Differ. Equ., Monogr. 7 (electronic), 2006. | MR | Zbl