On the Existence of Weak Solutions for a Degenerate and Singular Elliptic System in ℝ N

Abstract

This paper deals with the existence of weak solutions to a class of degenerate and singular elliptic systems in R N , N 2 of the form, 2) are allowed to have "essential" zeroes at some points in R N . Our proofs rely essentially on the critical point theory tools combined with a variant of the Caffarelli-Kohn-Nirenberg inequality in Nonlinear Differ.

Existence and multiplicity results for a class of degenerate quasilinear elliptic systems

“…We observe that there exists a vast literature on non-uniformly nonlinear elliptic problems in bounded or unbounded domains. Many authors studied the existence of solutions for such problems (equations or systems), for instance see [5,6,7,8,9,10,14,17,18,19,21,22]. In a recent paper Caldiroli et al [5] considered the Dirichlet elliptic problem −div(h(x)∇u) = λu + g(x, u) in Ω, (1.3) where Ω is a (bounded or unbounded) domain in R N (N ≥ 2), and h is a nonnegative measurable weighted function that is allowed to have "essential" zeroes at some points in Ω, i.e., the function h can have at most a finite number of zeroes in Ω.…”

“…The results in [5] were used by Zographopoulos [21], Zhang et al [18] and Chung et al [7,8,9] to study the existence of solutions for a class of degenerate elliptic systems.…”

EXISTENCE RESULT FOR NONUNIFORMLY DEGENERATE SEMILINEAR ELLIPTIC SYSTEMS INN

“…In that paper, using variational methods the author proved the existence of a weak solution in a subspace of the Sobolev space H 1 ‫ޒ(‬ N , ‫ޒ‬ 2 ). This was extended by N. T. Chung [6], in which the author considered the situation that h i ∈ L 1 loc ‫ޒ(‬ N ), h i (x) 1 for a.e. x ∈ ‫ޒ‬ N with i = 1, 2.…”

“…Note that by hypothesis (H), the problem which was considered here contains the situations in [6] and [7]. We also do not require the coercivity for the functions a(x) and b(x) as in [12].…”