A Class of Degenerate Nonlinear Elliptic Equations in Weighted Sobolev Space

under suitable hypotheses on the functions g 1 , g 2 , and h. The present work is inspired by a semilinear problem in bounded domain given in the book by Zeidler [1]. In general, the Sobolev spacesW(Ω) without weights occur as spaces of solutions for elliptic and parabolic PDEs. For degenerate problemswith various types of singularities in the coefficients it is natural to look for solutions in weighted Sobolev spaces; for example, see [2–8]. Section 2 deals with preliminaries and some basic results. Section 3 contains the main result and is about the existence of a weak solution to (3) in a suitable weighted Sobolev space.


Introduction
Let Ω ⊂ R  be a bounded domain with boundary Ω.Let  be an operator in divergence form: (  ()    ()) with   =    , with coefficients   / ∈  ∞ (Ω) which are symmetric and satisfy the degenerate ellipticity condition: ()     ≤ Λ          2  () , a.e. ∈ Ω, for all  ∈ R  , and  is an  2 -weight ( > 0, Λ > 0).Let  ∈ R and / ∈  2 (Ω, ) and let ℎ be a real valued continuous function defined on R×R  .In this paper, we study the existence of weak solution of the BVP: under suitable hypotheses on the functions  1 ,  2 , and ℎ.The present work is inspired by a semilinear problem in bounded domain given in the book by Zeidler [1].In general, the Sobolev spaces  , (Ω) without weights occur as spaces of solutions for elliptic and parabolic PDEs.For degenerate problems with various types of singularities in the coefficients it is natural to look for solutions in weighted Sobolev spaces; for example, see [2][3][4][5][6][7][8].Section 2 deals with preliminaries and some basic results.Section 3 contains the main result and is about the existence of a weak solution to (3) in a suitable weighted Sobolev space.

Preliminaries
We need the following preliminaries for the ensuing study.
Let Ω ⊂ R  be a bounded domain (open connected set).Let  : R  → R + be a locally integrable function with 0 <  < ∞ a.e.We say that  belongs to the Muckenhoupt class   , 1 <  < ∞, or that  is an   -weight, if there is a constant  =  , such that for all balls  in R  , where | ⋅ | denotes the -dimensional Lebesgue measure in R  .We assume that  ∈   , 1 <  < ∞.
We will denote by   (Ω, )(1 ≤  < ∞) the usual Banach space of measurable real valued functions, , defined in Ω for which

ISRN Mathematical Analysis
For  ≥ 1 and a positive integer , the weighted Sobolev space  , (Ω, ) is defined by with the associated norm In order to avoid too many suffices, at each step, a generic constant is denoted by  or  Ω .We need the following result.
Proposition 2. Let  ∈   , 1 <  < ∞, and let Ω be a bounded open set in R  .If   →  in   (Ω, ), then there exist a subsequence {   } and a function V ∈   (Ω, ) such that Proof.The proof of this theorem follows in the lines of Theorem 2.8.1 in [12].
Let ( | ) denote the value of linear functional  at . Definition 3. Let ,  :  →  * be the operators on the real separable reflexive Banach space .Then, (i)  satisfies condition () if (ii)  is demicontinuous if and only if   →  as  → ∞ implies   ⇀  as  → ∞, (iii)  +  is asymptotically linear if  is linear and In Section 3, we use the following result.
For a detailed proof of the above theorem, we refer to [13] or to [1,Theorem 29.].
We need the following hypotheses for further study.

Main Results
The main result of this section is to establish the existence of a solution for the degenerate nonlinear elliptic BVP (3), when  > 0 is not an eigenvalue of with certain restrictions.Also, two results are established related to the cases when  1 does not change sign.
Step 1.We note that the operator  is linear.It follows from ( 17) and (20) that the operators ,  are bounded.
Step 3. Next, we claim that + is asymptotically linear.Since ℎ is bounded, we observe that, for all  ∈ which implies ‖  ‖ ≤   , where which shows that  +  is asymptotically linear.
Remark 7. In the following results, we dispense with condition (25), when  1 does not change sign.The two results are related to the cases when  1 > 0 with  < 0 and  1 < 0 with  > 0.
Proof.The proof of this result is in the same lines to that of Theorem 8 and hence omitted.
Remark 10.In Theorems 8 and 9, we have studied the BVP (3) both positive and negative values of .In Theorem 9, with positive value of  we do not need the extra condition (25) at the cost of  1 negative.