Electronic Journal of Qualitative Theory of Differential Equations

In this paper, by using the Galerkin method and the generalized Brouwer's theorem, some problems of the higher eigenvalues are studied for a class of singular quasiliner elliptic equations in the weighted Sobolev spaces. The existence of weak solutions is obtained for this problem.


Introduction
In this paper, we consider the existence of weak solutions in weighted Sobolev spaces for the singular quasilinear elliptic equation where that consider the quasilinear elliptic equations in weighted Sobolev spaces, because the compact embedding theorem cannot be obtained easily.
In 2001, V.L. Shapiro [5] established a new weighted compact Sobolev embedding theorem, and proved a series of existence problems for weighted quasilinear elliptic equations and parabolic equations.
In 2005, working in Sobolev space H 1 p,ρ (Ω, Γ) only for the first eigenvalue, A. Rumbos and V. L. Shapiro [6] on the basis of [7] by using the generalized Landesman-Lazer conditions [8] discussed the existence of the solutions for weighted quasilinear elliptic equations x on Ω, where The problems what we discussed have physical background.In fact, equation (1.1) is one of the most useful sets of Navier-Stokes equations, which describe the motion of viscous fluid substances liquids and gases.
The purpose of this paper is to obtain an existence result of the weakly result for problem (1.1).
Our results are bases on the Galerkin-type techniques [9] and the generalized Brouwer's theorem [10] and other methods.
We say functional G satisfies G * − conditions if the following two facts obtain: (2.9) We need the following lemmas in section three.

Main Results and Their Proofs
In this section, we prove that problem (1.1) has at least one solution, which is the main result of this paper stated as Theorem 3.3.
In order to prove the problem (1.1) has a weak solution, we first discuss it in a finite dimension space S n , where S n is the subspace of H 1 p,q,ρ (Ω, Γ) spanned by ϕ 1 , ϕ 2 , • • • , ϕ n , then we extend the result to the infinite dimension space H 1 p,q,ρ (Ω, Γ).Theorem 3.1 Let Ω denote a bounded domain in R N (N ≥ 1), p i (x) and ρ(x) ∈ C 0 (Ω) be positive functions, q(x) ∈ C 0 (Ω) be a nonnegative function and assume that (2.1) holds.Let EJQTDE, 2012 No. 71, p. 4 Γ ⊂ ∂Ω be a fixed closed set (it may be the empty set), the operator L be given by (1.2) and assume (a-1)-(a-3), that V L (Ω, Γ) conditions hold.Suppose that f (x, u) satisfies (f-1)-(f-3), that the functional G satisfies (G-1).Then if n ≥ n 0 = J + J 1 + 1, there exists u * n ∈ S n with the property that Proof We only consider the situation for J > 1.The case J = 1 has already been treated in [6].We set where n ≥ n 0 , and 2) and (3.3), we see that We define It follows from (3.2) and (3.3) that First of all, Secondly, from (2.6), Hölder inequality and Minkowski inequality, we get ρ for fixed n, we have where K 0 , K ′ 0 , K 3 are positive constants.We observe from (3.5), (3.6), (3.7) and (3.8) that there exists t > 0, such that By the generalized Brouwer's theorem [10] , there exists and the proof of Theorem 3.1 is complete by the definition of S n .
Theorem 3.2 The sequence {u * n } obtained in Theorem 3.1 is uniformly bounded in H 1 p,q,ρ with respect to the norm ||u|| p,q,ρ = u, u p,q,ρ .
Proof From Theorem 3.1, for u * n ∈ S n , we have where 0 < γ ′ < γ, γ = For ease of notation, we represent the sequence {u * n } n≥n0 by {u n } n≥n0 .In order to prove Theorem 3.2, we need to prove that there exists constant K 4 for above u n ∈ S n , such that (3.10) Suppose to the contrary that (3.10) does not hold, then there exists subsequence (for ease of notation, we still denoted by {u n }), such that Taking v = u n in (3.9), from Lemma 2.2, G ∈ (H 1 p,q,ρ (Ω, Γ)) * and the methods of (3.7), there exists K 5 > 0, such that EJQTDE, 2012 No. 71, p. 6 Hence dividing both sides of above inequality by ||u n || 2 p,q,ρ and taking the limit as n → ∞, then there exists positive integer n 1 (n 1 ≥ n 0 ), when n ≥ n 1 that From (3.11), we have and when n ≥ n 1 that Set In the following, for ∀n ≥ n 1 , we propose to show the fact In matter of fact, we obtain from (3.9) and (3.14) that From (2.6) and (3.14), we conclude that Taking δ = γ − γ ′ , from γ = (λ J+J1 − λ J )/2, we also obtain from (3.16), (3.17), It is clear that for fixed n, ∃γ ′′ > 0, such that where ∀n ≥ n 1 , K 0 , K 7 and γ * are positive constants.Dividing both sides of (3.20) by ||u n || 2 ρ and taking the limit as n → ∞, from (3.12) and (3.13), we obtain (3.15).