×

zbMATH — the first resource for mathematics

Existence and symmetry of least energy solutions for a class of quasi-linear elliptic equations. (English) Zbl 1176.35081
Summary: For a general class of autonomous quasi-linear elliptic equations on \(\mathbb R^n\) we prove the existence of a least energy solution and show that all least energy solutions do not change sign and are radially symmetric up to a translation in \(\mathbb R^n\).

MSC:
35J62 Quasilinear elliptic equations
35J40 Boundary value problems for higher-order elliptic equations
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
35B06 Symmetries, invariants, etc. in context of PDEs
35J35 Variational methods for higher-order elliptic equations
PDF BibTeX XML Cite
Full Text: DOI EuDML arXiv
References:
[1] Arcoya, D.; Boccardo, L., Critical points for multiple integrals of the calculus of variations, Arch. ration. mech. anal., 134, 249-274, (1996) · Zbl 0884.58023
[2] Bensoussan, A.; Boccardo, L.; Murat, F., On a nonlinear partial differential equation having natural growth and unbounded solutions, Ann. inst. H. Poincaré anal. non linéaire, 5, 347-364, (1988) · Zbl 0696.35042
[3] Berestycki, H.; Lions, P.-L., Nonlinear scalar field equations. I. existence of a ground state, Arch. ration. mech. anal., 82, 313-345, (1983) · Zbl 0533.35029
[4] Boccardo, L.; Murat, F., Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations, Nonlinear anal., 19, 581-597, (1992) · Zbl 0783.35020
[5] Boccardo, L.; Murat, F.; Puel, J.P., Existence de solutions non bornées pour certaines équations quasi-linéaires, Portugal. math., 41, 507-534, (1982) · Zbl 0524.35041
[6] Brezis, H.; Browder, F.E., Sur une propriété des espaces de Sobolev, C. R. acad. sci. Paris, 287, 113-115, (1978) · Zbl 0381.46019
[7] Brezis, H.; Lieb, E.H., Minimum action solutions of some vector field equations, Comm. math. phys., 96, 97-113, (1984) · Zbl 0579.35025
[8] Byeon, J.; Jeanjean, L.; Maris, M., Symmetry and monotonicity of least energy solutions, www.arxiv.org · Zbl 1226.35041
[9] Campa, I.; Degiovanni, M., Subdifferential calculus and nonsmooth critical point theory, SIAM J. optim., 10, 1020-1048, (2000) · Zbl 1042.49018
[10] Canino, A., Multiplicity of solutions for quasilinear elliptic equations, Topol. methods nonlinear anal., 6, 357-370, (1995) · Zbl 0863.35038
[11] Canino, A.; Degiovanni, M., Nonsmooth critical point theory and quasilinear elliptic equations, (), 1-50 · Zbl 0851.35038
[12] Corvellec, J.N.; Degiovanni, M.; Marzocchi, M., Deformation properties for continuous functionals and critical point theory, Topol. methods nonlinear anal., 1, 151-171, (1993) · Zbl 0789.58021
[13] Degiovanni, M.; Marzocchi, M., A critical point theory for nonsmooth functionals, Ann. mat. pura appl., 167, 73-100, (1994) · Zbl 0828.58006
[14] Degiovanni, M.; Musesti, A.; Squassina, M., On the regularity of solutions in the pucci – serrin identity, Calc. var. partial differential equations, 18, 317-334, (2003) · Zbl 1046.35039
[15] DiBenedetto, E., \(C^{1, \alpha}\) local regularity of weak solutions of degenerate elliptic equations, Nonlinear anal., 7, 827-850, (1983) · Zbl 0539.35027
[16] do O, J.M.; Medeiros, E.S., Remarks on least energy solutions for quasilinear elliptic problems in \(\mathbb{R}^N\), Electron. J. differential equations, 83, 1-14, (2003)
[17] Ekeland, I., Nonconvex minimization problems, Bull. amer. math. soc., 1, 443-474, (1979) · Zbl 0441.49011
[18] Ferrero, A.; Gazzola, F., On the subcriticality assumptions for the existence of ground states of quasilinear elliptic equations, Adv. differential equations, 8, 1081-1106, (2003) · Zbl 1290.35096
[19] Frehse, J., A note on the Hölder continuity of solutions of variational problems, Abh. math. sem. univ. Hamburg, 43, 59-63, (1975) · Zbl 0316.49008
[20] Gazzola, F.; Serrin, J.; Tang, M., Existence of ground states and free boundary problems for quasilinear elliptic operators, Adv. differential equations, 5, 1-30, (2000) · Zbl 0987.35064
[21] Giacomini, A.; Squassina, M., Multi-peak solutions for a class of degenerate elliptic equations, Asymptotic anal., 36, 115-147, (2003) · Zbl 1137.35362
[22] Ioffe, A., On lower semicontinuity of integral functionals. I, SIAM J. control optim., 15, 521-538, (1977) · Zbl 0361.46037
[23] Ioffe, A., On lower semicontinuity of integral functionals. II, SIAM J. control optim., 15, 991-1000, (1977) · Zbl 0379.46022
[24] Ioffe, A.; Schwartzman, E., Metric critical point theory 1. Morse regularity and homotopic stability of a minimum, J. math. pures appl., 75, 125-153, (1996) · Zbl 0852.58018
[25] Katriel, G., Mountain pass theorems and global homeomorphism theorems, Ann. inst. H. Poincaré anal. non linéaire, 11, 189-209, (1994) · Zbl 0834.58007
[26] Lieb, E.H., On the lowest eigenvalue of the Laplacian for the intersection of two domains, Invent. math., 74, 441-448, (1983) · Zbl 0538.35058
[27] Mariş, M., On the symmetry of minimizers, www.arxiv.org, DOI:10.1007/s00205-008-0136-2, in press · Zbl 1159.49005
[28] Pucci, P.; Serrin, J., A general variational identity, Indiana univ. math. J., 35, 681-703, (1986) · Zbl 0625.35027
[29] Serrin, J., Local behavior of solutions of quasi-linear equations, Acta math., 111, 247-302, (1964) · Zbl 0128.09101
[30] Squassina, M., On the existence of positive entire solutions of nonlinear elliptic equations, Topol. methods nonlinear anal., 17, 23-39, (2001) · Zbl 0997.35019
[31] Squassina, M., Weak solutions to general Euler’s equations via nonsmooth critical point theory, Ann. fac. sci. Toulouse math., 9, 113-131, (2000) · Zbl 0983.35050
[32] Stampacchia, G., Équations elliptiques du second ordre à coefficients discontinus, () · Zbl 0151.15501
[33] Tolksdorf, P., Regularity for a more general class of quasilinear elliptic equations, J. differential equations, 51, 126-150, (1984) · Zbl 0488.35017
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.