Abstract
We show that the problem at critical growth, involving the 1-Laplace operator and obtained by relaxation of \({-\Delta_1 u=\lambda |u|^{-1}u+|u|^{1^*-2} u}\) , admits a nontrivial solution \({u \in BV(\Omega)}\) for any λ ≥ λ1. Nonstandard linking structures, for the associated functional, are recognized.
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Ambrosetti A., Rabinowitz P.H.: Dual variational methods in critical point theory and applications. J. Funct. Anal. 14, 349–381 (1973)
Ambrosio L., Fusco N., Pallara D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press, New York (2000)
Arioli G., Gazzola F.: Some results on p-Laplace equations with a critical growth term. Differ. Int. Equ. 11, 311–326 (1998)
Aubin T.: Problèmes isopérimétriques et espaces de Sobolev. J. Differ. Geom. 11, 573–598 (1976)
Brezis H., Nirenberg L.: Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Comm. Pure Appl. Math. 36, 437–477 (1983)
Campa I., Degiovanni M.: Subdifferential calculus and nonsmooth critical point theory. SIAM J. Optim. 10, 1020–1048 (2000)
Capozzi A., Fortunato D., Palmieri G.: An existence result for nonlinear elliptic problems involving critical Sobolev exponent. Ann. Inst. H. Poincaré Anal. Non Linéaire 2, 463–470 (1985)
Chang K.-C.: Variational methods for nondifferentiable functionals and their applications to partial differential equations. J. Math. Anal. Appl. 80, 102–129 (1981)
Chang K.-C.: Infinite-dimensional Morse Theory and Multiple Solution Problems. Birkhäuser, Boston (1993)
Clarke F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)
Corvellec J.-N., Degiovanni M., Marzocchi M.: Deformation properties for continuous functionals and critical point theory. Topol. Methods Nonlinear Anal. 1, 151–171 (1993)
Degiovanni M., Lancelotti S.: Linking over cones and nontrivial solutions for p-Laplace equations with p-superlinear nonlinearity. Ann. Inst. H. Poincaré Anal. Non Linéaire 24, 907–919 (2007)
Degiovanni M., Lancelotti S.: Linking solutions for p-Laplace equations with nonlinearity at critical growth. J. Funct. Anal. 256, 3643–3659 (2009)
Degiovanni M., Marzocchi M.: A critical point theory for nonsmooth functionals. Ann. Mat. Pura Appl. 167, 73–100 (1994)
Degiovanni M., Marzocchi M., Rădulescu V.D.: Multiple solutions of hemivariational inequalities with area-type term. Calc. Var. Partial Differ. Equ. 10, 355–387 (2000)
Degiovanni M., Schuricht F.: Buckling of nonlinearly elastic rods in the presence of obstacles treated by nonsmooth critical point theory. Math. Ann. 311, 675–728 (1998)
Demengel F.: On some nonlinear partial differential equations involving the “1”-Laplacian and critical Sobolev exponent. ESAIM Control Optim. Calc. Var. 4, 667–686 (1999)
Demengel F.: Some existence’s results for noncoercive “1-Laplacian” operator. Asymptot. Anal. 43, 287–322 (2005)
Egnell H.: Existence and nonexistence results for m-Laplace equations involving critical Sobolev exponents. Arch. Ration. Mech. Anal. 104, 57–77 (1988)
Fadell E.R., Rabinowitz P.H.: Bifurcation for odd potential operators and an alternative topological index. J. Funct. Anal. 26, 48–67 (1977)
Fadell E.R., Rabinowitz P.H.: Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems. Invent. Math. 45, 139–174 (1978)
García Azorero J., Peral Alonso I.: Existence and nonuniqueness for the p-Laplacian: nonlinear eigenvalues. Comm. Partial Differ. Equ. 12, 1389–1430 (1987)
Gazzola F., Ruf B.: Lower-order perturbations of critical growth nonlinearities in semilinear elliptic equations. Adv. Differ. Equ. 2, 555–572 (1997)
Giusti E.: Minimal Surfaces and Functions of Bounded Variation. Birkhäuser Verlag, Basel (1984)
Guedda M., Véron L.: Quasilinear elliptic equations involving critical Sobolev exponents. Nonlinear Anal. 13, 879–902 (1989)
Ioffe A., Schwartzman E.: Metric critical point theory. I. Morse regularity and homotopic stability of a minimum. J. Math. Pures Appl. 75, 125–153 (1996)
Katriel G.: Mountain pass theorems and global homeomorphism theorems. Ann. Inst. H. Poincaré Anal. Non Linéaire 11, 189–209 (1994)
Kawohl B., Schuricht F.: Dirichlet problems for the 1-Laplace operator, including the eigenvalue problem. Commun. Contemp. Math. 9, 515–543 (2007)
Marzocchi M.: Multiple solutions of quasilinear equations involving an area-type term. J. Math. Anal. Appl. 196, 1093–1104 (1995)
Milbers, Z., Schuricht, F.: Existence of a sequence of eigensolutions for the 1-Laplace operator. Technische Universität Dresden. MATH-AN-04-2008 (2008. preprint)
Rabinowitz, P.H.: Minimax Methods in Critical Point Theory with Applications to Differential Equations. American Mathematical Society, Providence (1986)
Rockafellar R.T.: Generalized directional derivatives and subgradients of nonconvex functions. Can. J. Math. 32, 257–280 (1980)
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Communicated by L. Ambrosio.
The research of M. Degiovanni was partially supported by the PRIN project “Variational and topological methods in the study of nonlinear phenomena” and by Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (INdAM).
The research of P. Magrone was partially supported by the PRIN project “Variational methods and nonlinear differential equations”.
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Degiovanni, M., Magrone, P. Linking solutions for quasilinear equations at critical growth involving the “1-Laplace” operator. Calc. Var. 36, 591 (2009). https://doi.org/10.1007/s00526-009-0246-1
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DOI: https://doi.org/10.1007/s00526-009-0246-1