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Existence and nonexistence results for m-Laplace equations involving critical Sobolev exponents. (English) Zbl 0675.35036
The paper is concerned with the boundary value problem: \[ -\nabla \cdot (f| \nabla u|^{m-2}\nabla u)-b\cup^ p=\lambda hu^{m-1}\quad in\quad \Omega;\quad u>0\quad in\quad \Omega,\quad u=0\quad on\quad \partial \Omega \] where \(1<m<n\), \(\Omega \subset R^ n\) smooth, bounded, open and connected and p the critical Sobolev exponent \((p=mn/(n-m))\). The results proved regard the existence and non-existence of solutions \(u\in H_ 0^{1,m}(\Omega).\)
Thus it is proved that there exists \(\lambda^*\) such that for \(\lambda^*<\lambda <\lambda_ 0\) there exist solutions and for \(\lambda >\lambda_ 0\) there are no solutions of the problem where \(\lambda_ 0\) is the first eigenvalue of: \[ -\nabla \{f| \nabla u|^{m-2} \nabla u)=\lambda hu^{m-1}\quad in\quad \Omega,\quad u=0\quad on\quad \partial \Omega \] under certain conditions on the coefficients and on the domain for \(n\geq m^ 2\) sharp existence and non- existence results are obtained for which a Pohozaev identity is used.
Reviewer: B.Vernescu

MSC:
35J65 Nonlinear boundary value problems for linear elliptic equations
35D05 Existence of generalized solutions of PDE (MSC2000)
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
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[1] F. V. Atkinson, L. A. Peletier, & J. Serrin, in preparation.
[2] H. Brezis & L. Nirenberg, Positive Solutions of Nonlinear Elliptic Equations Involving Critical Sobolev Exponents. Communications on Pure and Applied Mathematics 36, 437-477, 1983. · Zbl 0541.35029 · doi:10.1002/cpa.3160360405
[3] E. DiBenedetto, 76-01 Local Regularity of Weak Solutions of Degenerate Elliptic Equations. Nonlinear Analysis, Theory, Methods & Applications 8, 827-850, 1983. · Zbl 0539.35027 · doi:10.1016/0362-546X(83)90061-5
[4] H. Egnell, Semilinear Elliptic Equations Involving Critical Sobolev Exponents. Archive for Rational Mechanics and Analysis 104, 27-56, 1988. · Zbl 0674.35033 · doi:10.1007/BF00256931
[5] M. Guedda & L. Veron, Quasilinear Elliptic Equations Involving Critical Sobolev Exponents. To appear in Nonlinear Analysis, Theory, Methods & Applications.
[6] D. Gilbarg & N. S. Trudinger, Elliptic Partial Differential Equations of Second Order. Springer Verlag 1977. · Zbl 0361.35003
[7] G. H. Hardy, J. E. Littlewood, & G. Pólya, Inequalities. Cambridge University Press 1934. · Zbl 0010.10703
[8] O. A. Ladyzenskaya & N. N. Ural’tzeva, Linear and Quasilinear Elliptic Equations. Academic Press, New York 1968.
[9] G. Talenti, Best Constant in Sobolev Inequality. Annali di Matematica Pura ed Applicata 110, 353-372, 1976. · Zbl 0353.46018 · doi:10.1007/BF02418013
[10] P. Tolksdorf, On the Dirichletproblem for Quasilinear Equations in Domains with Conical Boundary Points. Communications in Partial Differential Equations 8, 773-817, 1983. · Zbl 0515.35024 · doi:10.1080/03605308308820285
[11] P. Tolksdorf, Regularity for a More General Class of Quasilinear Elliptic Equations. Journal of Differential Equations 51, 126-150, 1984. · Zbl 0522.35018 · doi:10.1016/0022-0396(84)90105-0
[12] N. Trudinger, Remarks Concerning the Conformal Deformation of Riemannian Structures on Compact Manifolds. Annali della Scuola Normale Superiore di Pisa 22, 265-274, 1968. · Zbl 0159.23801
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