Constant Sign Solutions for Variable Exponent System Neumann Boundary Value Problems with Singular Coefficient

and Applied Analysis 3 W 1,p(⋅) (Ω) and p(x)-Laplacian which we will use later (for details, see [6, 10, 12, 13]). Write C+ (Ω) = {h | h ∈ C (Ω) , h (x) ≥ 1 for x ∈ Ω} , h + = ess sup x∈Ω h (x) , h − = ess inf x∈Ω h (x) , for any h ∈ L (Ω) ; S (Ω) = {u | u is a measurable real-valued function in Ω} , L p(⋅) (Ω) = {u ∈ S (Ω) | ∫ Ω |u(x)| p(x) dx < ∞} . (12) We introduce the norm on L(Ω) by |u|p(⋅) = inf {λ > 0 | ∫ Ω 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 u(x) λ 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 p(x) dx ≤ 1} , (13) and (L(Ω), | ⋅ |p(⋅)) becomes a Banach space; we call it variable exponent Lebesgue space. Proposition 1 (see [6]). (i) The space (L(Ω), | ⋅ |p(⋅)) is a separable, uniform convex Banach space, and its conjugate space is L 0 (⋅) (Ω), where (1/p(x)) + (1/p(x)) ≡ 1. For any u ∈ L p(⋅) (Ω) and V ∈ L 0 (⋅) (Ω), one has 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 ∫ Ω uV dx 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 ≤ ( 1

When (⋅) ≡  (a constant), ()-Laplacian is the usual -Laplacian.The ()-Laplacian is nonhomogeneity.Because of its nonhomogeneity, the ()-Laplacian possesses more complicated nonlinearity than the -Laplacian.Many results and methods for -Laplacian problems do not hold for ()-Laplacian problems anymore.For the following examples.
It is well known that the fact that   > 0 is very important in the study of -Laplacian problems.For instance, in [34], the first eigenvalue and first eigenfunction are used to discuss the existence of positive solutions of -Laplacian problems successfully.But the ()-Laplacian does not have the first eigenvalue and first eigenfunction in general.
(2 0 ) The norm in  (⋅) (Ω) is of Luxemburg type (we will explain later in Section 2).It is easy to see that ∫ Ω || ()  = || () (⋅) for some  ∈ Ω. Hence the integral and the norm cannot keep the constant exponent relationship.It implies that we will have more difficulties in the study of ()-Laplacian problems.For example, it is very difficult to get the best Sobolev imbedding constant when we deal with the critical Sobolev exponent problems.Even if the best Sobolev imbedding constant could be obtained, it is also very hard to be applied to study the critical exponent problems.
(3 0 ) In [35], the authors applied the homogeneous transformation method to discuss the existence of positive solutions for a class of superlinear semipositon systems.Nonetheless, the ()-Laplacian is nonhomogeneity; this method is very hard to be used on the ()-Laplacian problems.
Regarding the existence of solutions of (P), if (, ⋅, ⋅) then the corresponding functional of (P) is coercive; if (, ⋅, ⋅) satisfies the super-( + ,  + ) growth condition (subcritical), that is, the following Ambrosetti-Rabinowitz condition: where positive constants  1 and  2 satisfy then the corresponding functional of (P) satisfies Palais-Smale conditions (see [15,25]).If (, ⋅, ⋅) satisfies the subcritical growth condition, but it satisfies neither the sub-( − ,  − ) growth condition nor the super-( + ,  + ) growth condition, then it would be difficult to testify that the corresponding functional is coercive or satisfying Palais-Smale conditions; the results in this case are rare.
In this paper, we deal with the existence of constant sign solutions of the problem (P), when the corresponding functional neither is coercive nor satisfies Ambrosetti-Rabinowitz condition.For example, we discuss the existence of solutions of (P), when  satisfies sub-((), ()) growth condition near the origin in local; that is, the following condition or  satisfies super-((), ()) growth condition in local (subcritical growth); that is, the following condition: where positive functions  1 (⋅) and  2 (⋅) satisfy In particular, we get the existence of eight constant sign solutions of (P).
This paper is divided into four sections.In Section 2, we introduce some basic properties of the variable exponent Lebesgue-Sobolev spaces.In Section 3, several properties of ()-Laplacian are presented.In Section 4, we give the existence results of constant sign solutions of problem (P).

Preliminary Results and Notations
Throughout this paper, the letters ,   ,   ,  = 1, 2, . .., denote positive constants which may vary from line to line but are independent of the terms which will take part in any limit process.
Proof.Similar to the proof of [41], we omit it here.

Properties of Operators
In this section, we will discuss the properties of ()-Laplacian and Nemytsky operator.

Existence and Multiplicity of Solutions
In this section, using the critical point theory, we will discuss the existence and multiple existence of constant sign solutions of problem (P).Definition 12.One calls (, V) ∈  is a weak solution of (P) if It is easy to see that the critical point of  is a solution of (P).
(A 1 ) where 1 ≤  Remark.(i) Similarly, we can give the definitions of  satisfying sub-((), ()) growth condition near the origin, super-((), ()) growth condition near the origin or near the infinity in the second, the third, and the fourth quadrant of , respectively.
(ii) We say  satisfies sub-((), ()) growth condition near the origin, super-((), ()) growth condition near the infinity in , if  satisfies corresponding growth condition in every quadrant of .
(iii) We say  satisfies some growth condition in local, if it satisfies some growth condition in a quadrant.
Note that (50) is satisfied.Without loss of generality, we may assume that max Take  0 , V 0 ∈  2 0 (Ω 0 ) which are nontrivial nonnegative.It is easy to see that Thus (, V) has at least one nontrivial critical point ( * , V * ) in the first quadrant of  with ( * , V * ) < 0. Thus, ( * , V * ) is a nontrivial constant sign solution of (P).According to condition (S), it is easy to see that  * and V * are all nontrivial.Theorem 15.If  > 0 is large enough and  satisfies (A 0 ), (S), and sub-((), ()) growth condition near the origin in , then problem (P) has at least four nontrivial constant sign solutions.
Proof.(i) Similar to the proof of Theorem 14, we can see that (P) has a nontrivial constant sign (  , V  ) in the th quadrant of , such that (  , V  ) < 0,  = 1, 2, 3, 4. According to condition (S),   and V  are both nontrivial.Thus (P) has at least four constant sign solutions.

Case (II)
Theorem 16.If  > 0 is large enough and  satisfies (A 0 ), (S), and the super-((), ()) growth condition near the infinity in the first quadrant of , then (P) has a nontrivial constant sign solution in the first quadrant of .
The corresponding functional is where We will prove that  + satisfies the conditions of Mountain Pass Lemma. It From Lemma 11, we can see that  + (, V) satisfies (PS) condition in .

Case (III)
Theorem 18.If  > 0 is large enough and  satisfies (A 0 ), (S), sub-((), ()) growth condition near the origin, and super-((), ()) growth condition near the infinity in the first quadrant of , then (P) has two nontrivial constant sign solutions in the first quadrant of .