tailieunhanh - EIGENVALUE PROBLEMS FOR DEGENERATE NONLINEAR ELLIPTIC EQUATIONS IN ANISOTROPIC MEDIA ˘ DUMITRU

EIGENVALUE PROBLEMS FOR DEGENERATE NONLINEAR ELLIPTIC EQUATIONS IN ANISOTROPIC MEDIA ˘ DUMITRU MOTREANU AND VICENTIU RADULESCU Received 23 September 2004 and in revised form 26 November 2004 We study nonlinear eigenvalue problems of the type − div(a(x)∇u) = g(λ,x,u) in RN , where a(x) is a degenerate nonnegative weight. We establish the existence of solutions and we obtain information on qualitative properties as multiplicity and location of solutions. Our approach is based on the critical point theory in Sobolev weighted spaces combined with a Caffarelli-Kohn-Nirenberg-type inequality. A specific minimax method is developed without making use of Palais-Smale condition. 1. Introduction We. | EIGENVALUE PROBLEMS FOR DEGENERATE NONLINEAR ELLIPTIC EQUATIONS IN ANISOTROPIC MEDIA DUMITRU MOTREANU AND VICENỊIU RADULESCU Received 23 September 2004 and in revised form 26 November 2004 We study nonlinear eigenvalue problems of the type - div a x Vu g A x u in RN where a x is a degenerate nonnegative weight. We establish the existence of solutions and we obtain information on qualitative properties as multiplicity and location of solutions. Our approach is based on the critical point theory in Sobolev weighted spaces combined with a Caffarelli-Kohn-Nirenberg-type inequality. A specific minimax method is developed without making use of Palais-Smale condition. 1. Introduction We are concerned in this paper with the existence of critical points to Euler-Lagrange energy functionals generated by nonlinear equations involving degenerate differential operators. Precisely we study the existence of nontrivial weak solutions to degenerate elliptic equations of the type - div fl x V g A x u x e u where A is a real parameter u is a bounded or unbounded domain in RN N 2 and a is an nonnegative measurable weight function that is allowed to have essential zeroes at some points. Problems like this have a long history see the pioneering papers 3 16 17 18 22 and come from the consideration of standing waves in anisotropic Schrodinger equations see . 23 . Such problems in anisotropic media can be regarded as equilibrium solutions of the evolution equations ut SP A u Vu in u X 0 T where u u x t is the state of a certain system. For instance in describing the behavior of a bacteria culture the state variable u represents the number of mass of the bacteria. It is worth to stress that the study of nontrivial solutions of the problem SP A u Vu 0 in u is motivated by important phenomena. For example consider a fluid which flows irrotationally along a flat-bottomed canal. Then the flow can be modelled by an equation of the form SP A u Vu 0 with A 0 0 0. One possible motion is