Solution of Nonlinear Elliptic Boundary Value Problems and Its Iterative Construction

and Applied Analysis 3 2. Preliminaries Now, we list some of the knowledge we need in the sequel. Let X be a real Banach space with a strictly convex dual space X∗. We use ·, · to denote the generalized duality pairing between X and X∗. We use “→ ” to denote strong convergence. Let “X ↪→ Y” denote the space X embedded continuously in space Y . For any subset G of X, we denote by intG its interior. Function Φ is called a proper convex function on X 11 if Φ is defined from X to −∞, ∞ , not identically ∞ such that Φ 1 − λ x λy ≤ 1 − λ Φ x λΦ y , whenever x, y ∈ X and 0 ≤ λ ≤ 1. Function Φ : X → −∞, ∞ is said to be lower semicontinuous on X 11 if lim infy→xΦ y ≥ Φ x , for any x ∈ X. Given a proper convex function Φ on X and a point x ∈ X, we denote by ∂Φ x the set of all x∗ ∈ X∗ such that Φ x ≤ Φ y x − y, x∗ , for every y ∈ X. Such elements x∗ are called subgradients of Φ at x, and ∂Φ x is called the subdifferential of Φ at x 11 . A single-valued mapping T : D T X → X∗ is said to be hemicontinuous 11 if w − limt→ 0T x ty Tx, for any x, y ∈ X. A multivalued mapping A : X → 2X is said to be monotone 10 if its graph G A is a monotone subset of X ×X∗ in the sense that u1 − u2, w1 −w2 ≥ 0, 2.1 for any ui,wi ∈ G A , i 1, 2. The mapping A is said to be strictly monotone if the equality in 2.1 implies that u1 u2. The monotone operator A is said to be maximal monotone if G A is maximal among all monotone subsets of X × X∗ in the sense of inclusion. The mapping A is said to be coercive 10 if limn→ ∞ xn, x∗ n /‖xn‖ ∞ for all xn, x∗ n ∈ G A such that limn→ ∞‖xn‖ ∞. A point x ∈ D A is said to be a zero point of A if 0 ∈ Ax, and we denote by A−1 0 {x ∈ X : 0 ∈ Ax} the set of zero points of A. Lemma 2.1 Adams 12 . LetΩ be a bounded conical domain inR . Ifmp > N, thenW Ω ↪→ CB Ω ; if mp < N and q Np/ N −mp , then W Ω ↪→ L Ω ; if mp N and p > 1, then for 1 ≤ q < ∞, W Ω ↪→ L Ω . Lemma 2.2 Pascali and Sburlan 10 . If B : X → 2X is an everywhere defined, monotone, and hemicontinuous operator, then B is maximal monotone. Lemma 2.3 Pascali and Sburlan 10 . If Φ : X → −∞, ∞ is a proper convex and lower semicontinuous function, then ∂Φ is maximal monotone from X to X∗. Lemma 2.4 Pascali and Sburlan 10 . If B1 and B2 are two maximal monotone operators in X such that intD B1 ⋂ D B2 / ∅, then B1 B2 is maximal monotone. Lemma 2.5 Pascali and Sburlan 10 . If A : X → 2X is maximal monotone and coercive, then R A X∗. 4 Abstract and Applied Analysis Definition 2.6 Kamimura and Takahashi 13 . Let X be a real smooth Banach space. Then the Lyapunov functional φ : X ×X → R is defined as follows: φ ( x, y ) ‖x‖ − 2(x, Jy) ∥∥y∥∥2, ∀x, y ∈ X, 2.2 where J : X → 2X is the duality mapping defined by Jx {f ∈ X∗ : x, f ‖x‖‖f‖, ‖f‖ ‖x‖}, for x ∈ X. Lemma 2.7 Kamimura and Takahashi 13 . LetX be a real reflexive, strictly convex, and smooth Banach space, let C be a nonempty closed and convex subset of X, and let x ∈ X. Then there exists a unique element x0 ∈ C such that φ x0, x min { φ z, x : z ∈ C}. 2.3 Define a mapping ΠC from X onto C by ΠCx x0 for all x ∈ X. Then ΠC is called the generalized projection mapping fromX onto C. It is easy to see thatΠC coincides with the metric projection PC in a Hilbert space. 3. Main Results 3.1. Notations and Assumptions of 1.4 In the following of this paper, unless otherwise stated, we will assume that 2N/ N 1 < p < ∞, 1 ≤ q < ∞ if p ≥ N, and 1 ≤ q ≤ Np/ N − p if p < N, for N ≥ 1. Let 1/p 1/p′ 1. We use ‖ · ‖p, ‖ · ‖p′ , and ‖ · ‖1,p,Ω to denote the norm of spaces L Ω , Lp Ω , and W1,p Ω , respectively. In nonlinear boundary value problem 1.4 , Ω is a bounded conical domain of an Euclidean space R with its boundary Γ ∈ C1 see Wei and He 1 . We will assume that Green’s formula is available. f x ∈ Lp Ω is a given function. 0 ≤ C x ∈ L Ω , ε is a nonnegative constant and θ denotes the exterior normal derivative of Γ. Let φ : Γ × R → R be a given function such that, for each x ∈ Γ, φx φ x, · : R → R is a proper, convex, and lower semicontinuous function with φx 0 0. Let βx be the subdifferential of φx, that is, βx ≡ ∂φx. Suppose that 0 ∈ βx 0 and for each t ∈ R, the function x ∈ Γ → I λβx −1 t ∈ R is measurable for λ > 0. Suppose that g : Ω×RN 1 → R is a given function satisfying the following conditions, which can be found in Zeidler 14 . a Carathéodory’s conditions: x −→ g x, r is measurable on Ω ∀r ∈ R 1, r −→ g x, r is continuous on R 1 for almost all x ∈ Ω. 3.1 b Growth condition: g x, r1, . . . , rN 1 ≤ h1 x b N 1 ∑