Abstract
In this paper, the authors prove the existence of solutions for degenerate elliptic equations of the form −div(a(x)∇ p u(x)) = g(λ, x, |u|p−2u) in ℝN, where ∇ p u = |∇u|p−2∇ u and a(x) is a degenerate nonnegative weight. The authors also investigate a related nonlinear eigenvalue problem obtaining an existence result which contains information about the location and multiplicity of eigensolutions. The proofs of the main results are obtained by using the critical point theory in Sobolev weighted spaces combined with a Caffarelli-Kohn-Nirenberg-type inequality and by using a specific minimax method, but without making use of the Palais-Smale condition.
Similar content being viewed by others
References
Brezis, H. and Marcus, M., Hardy’s inequalities revisited, Ann. Scuola Norm. Sup. Pisa Cl. Sci., (4), 25(1–2), 1997, 217–237.
Brezis, H. and Nirenberg, L., Remarks on finding critical points, Comm. Pure Appl. Math., 44(8–9) 1991, 939–963.
Brezis, H. and Vázquez, J. L., Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid, 10(2), 1997, 443–469.
Caffarelli, L., Kohn, R. and Nirenberg, L., First order interpolation inequalities with weights, Compositio Math., 53(3), 1984, 259–275.
Caldiroli, P. and Musina, R., On the existence of extremal functions for a weighted Sobolev embedding with critical exponent, Calc. Var. Partial Differential Equations, 8(4), 1999, 365–387.
Caldiroli, P. and Musina, M., On a variational degenerate elliptic problem, NoDEA Nonlinear Differential Equations Appl., 7(2), 2000, 187–199.
Catrina, F. and Wang, Z. Q., On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions, Comm. Pure Appl. Math., 54(2), 2001, 229–258.
Ciarlet, P. G., Linear and Nonlinear Functional Analysis with Applications, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2013.
Dautray, R. and Lions, J. L., Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 1, Physical Origins and Classical Methods, Springer-Verlag, Berlin, 1990.
Davies, E. B., A review of Hardy inequalities, The Maz’ya Anniversary Collection, 2, (Rostock, 1998), Oper. Theory Adv. Appl., 110, Birkhä user, Basel, 1999, 55–67.
Hardy, G. H., Note on a theorem of Hilbert, Math. Z., 6, 1920, 314–317.
Kawohl, B., On a family torsional creep problems, J. Reine Angew. Math., 410, 1990, 1–22.
Kohn, R. and Nirenberg, L., Degenerate elliptic-parabolic equations of second order, Comm. Pure Appl. Math., 20, 1967, 797–872.
Motreanu, D., A saddle point approach to nonlinear eigenvalue problems, Math. Slovaca, 47(4), 1997, 463–477.
Motreanu, D. and Rădulescu, V., Variational and Non-variational Methods in Nonlinear Analysis and Boundary Value Problems, Nonconvex Optimization and Its Applications, 67, Kluwer Academic, Dordrecht, 2003.
Motreanu, D. and Rădulescu, V., Eigenvalue problems for degenerate nonlinear elliptic equations in anisotropic media, Boundary Value Problems, 2, 2005, 107–127.
Oh, Y. G., Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of the class (Va), Comm. Partial Differential Equations, 13(12), 1988, 1499–1519.
Pélissier, M. C. and Reynaud, M. L., Etude d’un modèle mathématique d’écoulement de glacier, C. R. Acad. Sci. Paris Sér. I Math., 279, 1974, 531–534.
Showalter, R. E. and Walkington, N. J., Diffusion of fluid in a fissured medium with microstructure, SIAM J. Math. Anal., 22, 1991, 1702–1722.
Stredulinsky, E. W., Weighted Inequalities and Degenerate Elliptic Partial Differential Equations, Lecture Notes in Mathematics, 1074, Springer-Verlag, Berlin, 1984.
Vazquez, J. L. and Zuazua, E., The Hardy inequality and the asymptotic behavior of the heat equation with an inverse-square potential, J. Funct. Anal., 173(1), 2000, 103–153.
Wang, Z. Q. and Willem, M., Singular minimization problems, J. Differential Equations, 161(2), 2000, 307–320.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Mălin, M., Udrea, C. Degenerate nonlinear elliptic equations lacking in compactness. Chin. Ann. Math. Ser. B 37, 53–72 (2016). https://doi.org/10.1007/s11401-015-0935-3
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11401-015-0935-3