Nonhomogeneous Nonlinear Dirichlet Problems with a p-Superlinear Reaction

and Applied Analysis 3 2. Mathematical Background and Hypotheses Let X be a Banach space, and let X∗ be its topological dual. By 〈·, ·〉 we denote the duality brackets for the pair X∗, X . Let φ ∈ C1 X . We say that φ satisfies the Cerami condition if the following is true: “every sequence {xn}n≥1 ⊆ X, such that {φ xn }n≥1 is bounded and 1 ‖xn‖ φ′ xn −→ 0 in X∗, 2.1 admits a strongly convergent subsequence.” This compactness-type condition is in general weaker than the usual Palais-Smale condition. Nevertheless, the Cerami condition suffices to have a deformation theorem, and from it the minimax theory of certain critical values of φ is derive see, e.g., Gasiński and Papageorgiou 3 . In particular, we can state the following theorem, known in the literature as the “mountain pass theorem.” Theorem 2.1. If φ ∈ C1 X satisfies the Cerami condition, x0, x1 ∈ X are such that ‖x1−x0‖ > > 0, and max { φ x0 , φ x1 } < inf { φ x : ‖x − x0‖ } η , c inf γ∈Γ max 0≤t≤1 φ ( γ t ) , 2.2 where Γ { γ ∈ C 0, 1 ;X : γ 0 x0, γ 1 x1 } , 2.3 then c ≥ η and c is a critical value of φ. In the analysis of problem 1.5 in addition to the Sobolev spaceW 0 Ω , we will also use the Banach space C1 0 ( Ω ) { u ∈ C1 ( Ω ) : u|∂Ω 0 } . 2.4 This is an ordered Banach space with positive cone C { u ∈ C1 0 ( Ω ) : u z ≥ 0 ∀z ∈ Ω } . 2.5 This cone has a nonempty interior, given by int C { u ∈ C : u z > 0 ∀z ∈ Ω, ∂u ∂n z < 0 ∀z ∈ ∂Ω } , 2.6 where n · denotes the outward unit normal on ∂Ω. 4 Abstract and Applied Analysis In what follows, by λ̂1 we denote the first eigenvalue of −Δp, W 0 Ω , where Δp denotes the p-Laplace operator, defined by Δpu div ( ‖∇u‖p−2∇u ) ∀u ∈ W 0 Ω . 2.7 We know see, e.g., Gasiński and Papageorgiou 3 that λ̂1 > 0 is isolated and simple i.e., the corresponding eigenspace is one-dimensional and λ̂1 inf {‖∇u‖p ‖u‖pp : u ∈ W 0 Ω , u / 0 } . 2.8 In this variational characterization of λ̂1, the infimum is realized on the corresponding onedimensional eigenspace. From 2.8 , we see that the elements of the eigenspace do not change sign. In what follows, by û1 we denote the L-normalized i.e., ‖û1‖p 1 positive eigenfunction corresponding to λ̂1 > 0. The nonlinear regularity theory for the p-Laplacian equations see, e.g., Gasiński and Papageorgiou 3, p. 737 and the nonlinear maximum principle of Vázquez 4 imply that û1 ∈ int C . Now, let φ ∈ C1 X and let c ∈ R. We introduce the following notation: φ { x ∈ X : φ x ≤ c, Kφ { x ∈ X : φ′ x 0, K φ { x ∈ Kφ : φ x c } . 2.9 Let Y1, Y2 be a topological pair with Y2 ⊆ Y1 ⊆ X. For every integer k ≥ 0, by Hk Y1, Y2 we denote the kth relative singular homology group for the pair Y1, Y2 with integer coefficients. The critical groups of φ at an isolated point x0 ∈ Kφ with φ x0 c i.e., x0 ∈ K φ are defined by Ck ( φ, x0 ) Hk ( φ ∩U, φ ∩U \ {x0} ) ∀k ≥ 0, 2.10 where U is a neighbourhood of x0, such that Kφ ∩ φ ∩ U {x0}. The excision property of singular homology implies that this definition is independent of the particular choice of the neighbourhood U. Suppose that φ ∈ C1 X satisfies the Cerami condition and φ Kφ > −∞. Let c < infφ Kφ . The critical groups of φ at infinity are defined by Ck ( φ,∞ Hk ( X,φ ) ∀k ≥ 0. 2.11 The second deformation theorem see, e.g., Gasiński and Papageorgiou 3, p. 628 guarantees that this definition is independent of the particular choice of the level c < infφ Kφ . Abstract and Applied Analysis 5 Suppose that Kφ is finite. We set M t, x ∑ k≥0 rank Ck ( φ, x ) t ∀t ∈ R, x ∈ Kφ, P t,∞ ∑ k≥0 rank Ck ( φ,∞tk ∀t ∈ R. 2.12and Applied Analysis 5 Suppose that Kφ is finite. We set M t, x ∑ k≥0 rank Ck ( φ, x ) t ∀t ∈ R, x ∈ Kφ, P t,∞ ∑ k≥0 rank Ck ( φ,∞tk ∀t ∈ R. 2.12 The Morse relation says that ∑ x∈Kφ M t, x P t,∞ 1 t Q t , 2.13 where Q t ∑ k≥0 βkt k 2.14 is a formal series in t ∈ R with nonnegative integer coefficients βk ∈ N. Now we will introduce the hypotheses on the maps a y and f z, ζ . So, let h ∈ C1 0, ∞ be such that 0 < th′ t h t ≤ c0 ∀t > 0, 2.15 for some c0 > 0 and c1t p−1 ≤ h t ≤ c2 ( 1 |t|p−1 ) ∀t > 0, 2.16 for some c1, c2 > 0. The hypotheses on the map a y are the following: H a : a y a0 ‖y‖ y, where a0 t > 0 for all t > 0 and i a ∈ C R ;R ∩ C1 R \ {0};RN , ii there exists c3 > 0, such that ∥ ∥∇ay∥ ≤ c3 h (∥ ∥y ∥ ∥ ) ∥ ∥y ∥ ∥ ∀y ∈ R \ {0}, 2.17