A priori estimates and application to the symmetry of solutions for critical \(p\)-Laplace equations. (English) Zbl 1327.35117
Summary: We establish pointwise a priori estimates for solutions in \(D^{1,p}(\mathbb{R}^n)\) of equations of type \(- \Delta_p u = f(x, u)\), where \(p \in(1, n)\), \(\Delta_p : = \mathrm{div}(| \nabla u |^{p-2} \nabla u)\) is the \(p\)-Laplace operator, and \(f\) is a Caratheodory function with critical Sobolev growth. In the case of positive solutions, our estimates allow us to extend previous radial symmetry results. In particular, by combining our results and a result of L. Damascelli and M. Ramaswamy [Adv. Nonlinear Stud. 1, No. 1, 40–64 (2001; Zbl 0998.35016)], we are able to extend a recent result of L. Damascelli et al. [Adv. Math. 265, 313–335 (2014; Zbl 1316.35135)] on the symmetry of positive solutions in \(D^{1, p}(\mathbb{R}^n)\) of the equation \(- \Delta_p u = u^{p^\ast -1}\), where \(p^\ast:=np/(n-p)\).
MSC:
| 35J75 | Singular elliptic equations |
| 35B06 | Symmetries, invariants, etc. in context of PDEs |
| 35J30 | Higher-order elliptic equations |
| 35B09 | Positive solutions to PDEs |
Keywords:
critical \(p\)-Laplace equations; a priori estimates for solutions; positive solutions; radial symmetryReferences:
| [1] | Alves, C. O., Existence of positive solutions for a problem with lack of compactness involving the \(p\)-Laplacian, Nonlinear Anal., 51, 7, 1187-1206 (2002) · Zbl 1157.35355 |
| [2] | Caffarelli, L. A.; Gidas, B.; Spruck, J., Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math., 42, 3, 271-297 (1989) · Zbl 0702.35085 |
| [3] | Chen, W.; Li, C., Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63, 3, 615-622 (1991) · Zbl 0768.35025 |
| [4] | Cuesta Leon, M. C., Existence results for quasilinear problems via ordered sub- and supersolutions, Ann. Fac. Sci. Toulouse Math. (6), 6, 4, 591-608 (1997) · Zbl 0910.35055 |
| [5] | Damascelli, L.; Pacella, F.; Ramaswamy, M., Symmetry of ground states of \(p\)-Laplace equations via the moving plane method, Arch. Ration. Mech. Anal., 148, 4, 291-308 (1999) · Zbl 0937.35050 |
| [6] | Damascelli, L.; Ramaswamy, M., Symmetry of \(C^1\) solutions of \(p\)-Laplace equations in \(R^N\), Adv. Nonlinear Stud., 1, 1, 40-64 (2001) · Zbl 0998.35016 |
| [7] | Damascelli, L.; Merchán, S.; Montoro, L.; Sciunzi, B., Radial symmetry and applications for a problem involving the \(- \Delta_p(\cdot)\) operator and critical nonlinearity in \(R^n\), Adv. Math., 265, 10, 313-335 (2014) · Zbl 1316.35135 |
| [8] | DiBenedetto, E., \(C^{1 + \alpha}\) local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal., 7, 8, 827-850 (1983) · Zbl 0539.35027 |
| [9] | Druet, O.; Hebey, E.; Robert, F., Blow-up Theory for Elliptic PDEs in Riemannian Geometry, Mathematical Notes, vol. 45 (2004), Princeton University Press · Zbl 1059.58017 |
| [10] | Friedman, A.; Véron, L., Singular solutions of some quasilinear elliptic equations, Arch. Ration. Mech. Anal., 96, 4, 359-387 (1986) · Zbl 0619.35045 |
| [11] | Ghoussoub, N., Duality and Perturbation Methods in Critical Point Theory, Cambridge Tracts in Mathematics, vol. 107 (1993), Cambridge University Press · Zbl 0790.58002 |
| [12] | Gidas, B.; Ni, W.; Nirenberg, L., Symmetry of positive solutions of nonlinear elliptic equations in \(R^n\), (Mathematical Analysis and Applications, Part A. Mathematical Analysis and Applications, Part A, Adv. in Math. Suppl., vol. 7 (1981), Academic Press: Academic Press New York-London), 369-402 · Zbl 0469.35052 |
| [13] | Grafakos, L., Classical Fourier Analysis, Graduate Texts in Mathematics, vol. 249 (2008), Springer: Springer New York · Zbl 1220.42001 |
| [14] | Guedda, M.; Véron, L., Local and global properties of solutions of quasilinear elliptic equations, J. Differential Equations, 76, 1, 159-189 (1988) · Zbl 0661.35029 |
| [15] | Jannelli, E.; Solimini, S., Concentration estimates for critical problems, Ric. Mat., 48, suppl., 233-257 (1999) · Zbl 0944.35022 |
| [16] | Kato, T., Schrödinger operators with singular potentials, (Proceedings of the International Symposium on Partial Differential Equations and the Geometry of Normed Linear Spaces. Proceedings of the International Symposium on Partial Differential Equations and the Geometry of Normed Linear Spaces, Jerusalem, 1972 (1973)), 135-148 · Zbl 0246.35025 |
| [17] | Leoni, G., A First Course in Sobolev Spaces, Graduate Studies in Mathematics, vol. 105 (2009), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 1180.46001 |
| [18] | Li, C., Monotonicity and symmetry of solutions of fully nonlinear elliptic equations on unbounded domains, Comm. Partial Differential Equations, 16, 4-5, 585-615 (1991) · Zbl 0741.35014 |
| [19] | Li, Y.; Ni, W., Radial symmetry of positive solutions of nonlinear elliptic equations in \(R^n\), Comm. Partial Differential Equations, 18, 5-6, 1043-1054 (1993) · Zbl 0788.35042 |
| [20] | Mercuri, C.; Willem, M., A global compactness result for the \(p\)-Laplacian involving critical nonlinearities, Discrete Contin. Dyn. Syst., 28, 2, 469-493 (2010) · Zbl 1193.35051 |
| [21] | Peral, I., Multiplicity of solutions for the p-Laplacian, (Lecture Notes at the Second School on Nonlinear Functional Analysis and Applications to Differential Equations (1997), ICTP: ICTP Trieste) |
| [22] | Poláčik, P.; Quittner, P.; Souplet, P., Singularity and decay estimates in superlinear problems via Liouville-type theorems. I. Elliptic equations and systems, Duke Math. J., 139, 3, 555-579 (2007) · Zbl 1146.35038 |
| [23] | Saintier, N., Asymptotic estimates and blow-up theory for critical equations involving the \(p\)-Laplacian, Calc. Var. Partial Differential Equations, 25, 3, 299-331 (2006) · Zbl 1357.35132 |
| [24] | Sciunzi, B., Classification of positive \(D^{1, p}(R^N)\)-solutions to the critical \(p\)-Laplace equation in \(R^N\), Preprint at · Zbl 1344.35061 |
| [25] | Serrin, J., Local behavior of solutions of quasi-linear equations, Acta Math., 111, 1, 247-302 (1964) · Zbl 0128.09101 |
| [26] | Serrin, J.; Zou, H., Symmetry of ground states of quasilinear elliptic equations, Arch. Ration. Mech. Anal., 148, 4, 265-290 (1999) · Zbl 0940.35079 |
| [27] | Struwe, M., A global compactness result for elliptic boundary value problems involving limiting nonlinearities, Math. Z., 187, 4, 511-517 (1984) · Zbl 0535.35025 |
| [28] | Struwe, M., Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems (1990), Springer-Verlag: Springer-Verlag Berlin · Zbl 0746.49010 |
| [29] | Tolksdorf, P., Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations, 51, 1, 126-150 (1984) · Zbl 0488.35017 |
| [30] | Trudinger, N. S., On Harnack type inequalities and their application to quasilinear elliptic equations, Comm. Pure Appl. Math., 20, 721-747 (1967) · Zbl 0153.42703 |
| [31] | Trudinger, N. S., Remarks concerning the conformal deformation of Riemannian structures on compact manifolds, Ann. Sc. Norm. Super. Pisa Cl. Sci. (3), 22 (1968) · Zbl 0159.23801 |
| [32] | Vázquez, J. L., A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim., 12, 3, 191-202 (1984) · Zbl 0561.35003 |
| [33] | Véron, L., Singular solutions of some nonlinear elliptic equations, Nonlinear Anal., 5, 3, 225-242 (1981) · Zbl 0457.35031 |
| [34] | Yan, S., A global compactness result for quasilinear elliptic equation involving critical Sobolev exponent, Chinese J. Contemp. Math., 16, 3, 227-234 (1995) |
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