Study on the Existence and Uniqueness of Solution of Generalized Capillarity Problem

and Applied Analysis 3 space X is embedded compactly in space Y and “X ↪→ Y ′′ denote that space X is embedded continuously in space Y. A mapping T : D T X → X∗ is said to be hemicontinuous on X if w − limt→ 0T x ty Tx, for any x, y ∈ X. Let J denote the duality mapping from X into 2 ∗ defined by J x { f ∈ X∗ : (x, f) ‖x‖ · ∥∥f∥∥,∥∥f∥∥ ‖x‖}, x ∈ X, 1.5 where ·, · denotes the generalized duality pairing between X and X∗. It is wellknown that J is a single-valued mapping since X∗ is strictly convex. Let A : X → 2 be a given multivalued mapping. We say that A is boundedlyinversely compact if for any pair of bounded subsets G and G′ of X, the subset G ⋂ A−1 G′ is relatively compact in X. The mapping A : X → 2 is said to be accretive if v1 − v2, J u1 − u2 ≥ 0, for any ui ∈ D A and vi ∈ Aui, i 1, 2. The accretive mapping A is said to be m-accretive if R I μA X, for some μ > 0. Let B : X → 2X be a given multi-valued mapping. The graph of B, G B , is defined by G B { u,w | u ∈ D B , w ∈ Bu}. Then B : X → 2X is said to be monotone if G B is a monotone subset of X ×X∗ in the sense that u1 − u2, w1 −w2 ≥ 0, 1.6 for any ui,wi ∈ G B , i 1, 2. The monotone operator B is said to be maximal monotone if G B is maximal among all monotone subsets of X × X∗ in the sense of inclusion. The mapping B is said to be strictly monotone if the equality in 1.6 implies that u1 u2. The mapping B is said to be coercive if limn→ ∞ xn, x∗ n /‖xn‖ ∞ for all xn, x∗ n ∈ G B such that limn→ ∞‖xn‖ ∞. Definition 1.1. The duality mapping J : X → 2X is said to be satisfying Condition I if there exists a function η : X → 0, ∞ such that ‖Ju − Jv‖ ≤ η u − v , for ∀ u, v ∈ X. I Definition 1.2. Let A : X → 2 be an accretive mapping and J : X → X∗ be a duality mapping. We say that A satisfies Condition ∗ if, for any f ∈ R A and a ∈ D A , there exists a constant C a, f such that ( v − f, J u − a ) ≥ C(a, f), for any u ∈ D A , v ∈ Au. ∗ Lemma 1.3 Li and Guo 9 . LetΩ be a bounded conical domain inR . Then we have the following results. a If mp > N, then W Ω ↪→ CB Ω ; if mp < N and q Np/ N − mp , then W Ω ↪→ L Ω ; ifmp N and p > 1, then for 1 ≤ q < ∞,Wm,p Ω ↪→ L Ω . b If mp > N, then W Ω ↪→↪→ CB Ω ; if 0 < mp ≤ N and q0 Np/ N −mp , then W Ω ↪→↪→ L Ω , 1 ≤ q < q0. 4 Abstract and Applied Analysis Lemma 1.4 Pascali and Sburlan 10 . If B : X → 2X is an everywhere defined, monotone and hemicontinuous operator, then B is maximal monotone. If B : X → 2X is maximal monotone and coercive, then R B X∗. Lemma 1.5 Pascali and Sburlan 10 . If Φ : X → −∞, ∞ is a proper convex and lowersemicontinuous function, then ∂Φ is maximal monotone from X to X∗. Lemma 1.6 Pascali and Sburlan 10 . IfA and B are two maximal monotone operators in X such that intD A ⋂ D B / ∅, then A B is maximal monotone. Proposition 1.7 Calvert and Gupta 1 . Let X L Ω and Ω be a bounded domain in R . For 2 ≤ p < ∞, the duality mapping Jp : L Ω → Lp Ω defined by Jpu |u|p−1 sgnu‖u‖2−p p , for u ∈ L Ω , satisfies Condition I ; for 2N/ N 1 < p ≤ 2 and N ≥ 1, the duality mapping Jp : L Ω → Lp Ω defined by Jpu |u|p−1sgnu, for u ∈ L Ω , satisfies Condition I , where 1/p 1/p ′ 1. Lemma 1.8 see Calvert and Gupta 1 . Let Ω be a bounded domain in R and g : Ω × R → R be a function satisfying Caratheodory’s conditions such that i g x, · is monotonically increasing on R; ii the mapping u ∈ L Ω → g x, u x ∈ L Ω is well defined, where 2N/ N 1 < p < ∞ and N ≥ 1. Let Jp : L Ω → Lp Ω , 1/p 1/p ′ 1 be the duality mapping defined by Jpu ⎧⎨ ⎩ |u|p−1 sgnu, if 2N N 1 < p ≤ 2, |u|p−1 sgnu‖u‖2−p p , if 2 ≤ p < ∞, 1.7 for u ∈ L Ω . Then the mapping B : L Ω → L Ω defined by Bu x g x, u x , for any x ∈ Ω satisfies Condition ∗ . Theorem 1.9 Calvert and Gupta 1 . Let X be a real Banach space with a strictly convex dual X∗. Let J : X → X∗ be a duality mapping on X satisfying Condition I . Let A,B1 : X → 2 be accretive mappings such that i either both A, B1 satisfy Condition ∗ or D A ⊂ D B1 and B1 satisfies Condition ∗ , ii A B1 is m-accretive and boundedly inversely compact. If B2 : X → X be a bounded continuous mapping such that, for any y ∈ X, there is a constant C y satisfying B2 u y , Ju ≥ −C y for any u ∈ X. Then. a R A R B1 ⊂ R A B1 B2 . b int R A R B1 ⊂ intR A B1 B2 . Abstract and Applied Analysis 5 2. The Main Results 2.1. Notations and Assumptions of 1.4 Next in this paper, we assume 2N/ N 1 < p < ∞, 1 ≤ q, r < ∞ if p ≥ N, and 1 ≤ q, r ≤ Np/ N − p if p < N, where N ≥ 1. We use ‖ · ‖p, ‖ · ‖q, ‖ · ‖r , and ‖ · ‖1,p,Ω to denote the norms in L Ω , L Ω , L Ω and W1,p Ω . Let 1/p 1/p′ 1, 1/q 1/q′ 1, and 1/r 1/r ′ 1. In 1.4 ,Ω is a bounded conical domain of a Euclidean space R with its boundary Γ ∈ C1, c.f. 4 . We suppose that the Green’s Formula is available. Let | · | denote the Euclidean norm in R , 〈·, ·〉 the Euclidean inner-product and θ the exterior normal derivative of Γ. λ is a nonnegative constant. Let φ : Γ × R → R be a given function such that, for each x ∈ Γ, i φx φ x, · : R → R is a proper, convex, lower-semicontinuous function with φx 0 0. ii βx ∂φx : subdifferential of φx is maximalmonotonemapping onRwith 0 ∈ βx 0 and for each t ∈ R, the function x ∈ Γ → I μβx −1 t ∈ R is measurable for μ > 0. Let g : Ω × R → R be a given function satisfying Caratheodory’s conditions such that for 2N/ N 1 < p < ∞ and N ≥ 1, the mapping u ∈ L Ω → g x, u x ∈ L Ω is defined. Further, suppose that there is a function T x ∈ L Ω such that g x, t t ≥ 0, for |t| ≥ T x , x ∈ Ω.and Applied Analysis 5 2. The Main Results 2.1. Notations and Assumptions of 1.4 Next in this paper, we assume 2N/ N 1 < p < ∞, 1 ≤ q, r < ∞ if p ≥ N, and 1 ≤ q, r ≤ Np/ N − p if p < N, where N ≥ 1. We use ‖ · ‖p, ‖ · ‖q, ‖ · ‖r , and ‖ · ‖1,p,Ω to denote the norms in L Ω , L Ω , L Ω and W1,p Ω . Let 1/p 1/p′ 1, 1/q 1/q′ 1, and 1/r 1/r ′ 1. In 1.4 ,Ω is a bounded conical domain of a Euclidean space R with its boundary Γ ∈ C1, c.f. 4 . We suppose that the Green’s Formula is available. Let | · | denote the Euclidean norm in R , 〈·, ·〉 the Euclidean inner-product and θ the exterior normal derivative of Γ. λ is a nonnegative constant. Let φ : Γ × R → R be a given function such that, for each x ∈ Γ, i φx φ x, · : R → R is a proper, convex, lower-semicontinuous function with φx 0 0. ii βx ∂φx : subdifferential of φx is maximalmonotonemapping onRwith 0 ∈ βx 0 and for each t ∈ R, the function x ∈ Γ → I μβx −1 t ∈ R is measurable for μ > 0. Let g : Ω × R → R be a given function satisfying Caratheodory’s conditions such that for 2N/ N 1 < p < ∞ and N ≥ 1, the mapping u ∈ L Ω → g x, u x ∈ L Ω is defined. Further, suppose that there is a function T x ∈ L Ω such that g x, t t ≥ 0, for |t| ≥ T x , x ∈ Ω. 2.2. Existence and Uniqueness of the Solution of 1.4 Definition 2.1 Calvert and Gupta 1 . Define g x lim inft→ ∞ g x, t and g− x lim supt→−∞ g x, t . Further, define a function g1 : Ω × R → R by g1 x, t ⎧⎪⎪⎪⎨ ⎪⎪⎩ ( inf s≥t g x, s ) ∧ t − T x , ∀t ≥ T x , 0, ∀t ∈ −T x , T x , ( sup s≤t g x, s ) ∨ t T x , ∀t ≤ −T x . 2.1 Then for all x ∈ Ω, g1 x, t is increasing in t and limt→±∞ g1 x, t g± x . Moreover, g1 : Ω×R → R satisfies Caratheodory’s conditions and the functions g± x are measurable onΩ. And, if g2 x, t g x, t − g1 x, t then g2 x, t t ≥ 0, for |t| ≥ T x , x ∈ Ω. Proposition 2.2 see Calvert and Gupta 1 . For 2N/ N 1 < p < ∞ and N ≥ 1, define the mapping B1 : L Ω → L Ω by B1u x g1 x, u x , for all u ∈ L Ω and x ∈ Ω, then B1 is a bounded, continuous, and m-accretive mapping. Moreover, Lemma 1.8 implies that B1 satisfies Condition ∗ . 6 Abstract and Applied Analysis Define B2 : L Ω → L Ω by B2u x g2 x, u x , where g2 x, t g x, t −g1 x, t , then B2 satisfies the inequality: ( B2 ( u y ) , Jpu ) ≥ −C(y), 2.2 for any u, y ∈ L Ω , where C y is a constant depending on y and Jp : L Ω → Lp Ω denotes the duality mapping, where 1/p 1/p′ 1. Lemma 2.3 Wei and Agarwal 7 . The mapping Φp : W1,p Ω → R defined by