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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 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 a(x)∇u = g(λ,x,u),

x ∈ Ω,

(1.1)

where λ is a real parameter, Ω 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 Schr¨odinger equations (see, e.g., [23]). Such problems in anisotropic media can be regarded as equilibrium solutions of the evolution equations ut = Ᏺ(λ,u, ∇u) in Ω × (0,T),

(1.2)

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 Ᏺ(λ,u, ∇u) = 0 in Ω 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 Ᏺ(λ,u, ∇u) = 0, with Ᏺ(λ,0,0) = 0. One possible motion is a uniform stream Copyright © 2005 Hindawi Publishing Corporation Boundary Value Problems 2005:2 (2005) 107–127 DOI: 10.1155/BVP.2005.107

108

Nonlinear elliptic equations in anisotropic media

(corresponding to the trivial solution u = 0), but it is of course the nontrivial solutions which are of physical interest. Other problems of this type are encountered in various reaction-diffusion processes (cf. [1, 2]). A model equation that we consider in this paper is
 − div a(x)∇u = f (x,u)u − λu,

x ∈ Ω ⊂ RN ,

(1.3)

where a is a nonnegative weight and λ is a real parameter. The behavior of solutions to the above equation depends heavily on the sign of λ. Here we focus on the attractive case λ > 0 which, from an analytical point of view, seems to be the richest one. The main interest of these equations is due to the presence of the singular potential a(x) in the divergence operator. Problems of this kind arise as models for several physical phenomena related to equilibrium of continuous media which may somewhere be “perfect insulators” (cf. [13, page 79]). These equations can be often reduced to elliptic equations with Hardy singular potential (see [23]). For further resu