Noncoercive Perturbed Densely Defined Operators and Application to Parabolic Problems

Abstract

Let 𝑋 be a real locally uniformly convex reflexive separable Banach space with locally uniformly convex dual space 𝑋∗. Let 𝑇: 𝑋 ⊇ 𝐷(𝑇) → 2𝑋∗ be maximal monotone and 𝑆 : 𝑋 ⊇ 𝐷(𝑆) → 𝑋∗ quasibounded generalized pseudomonotone such that there exists a real reflexive separable Banach space 𝑊 ⊂ 𝐷(𝑆), dense and continuously embedded in 𝑋. Assume, further, that there exists 𝑑 ≥ 0 such that ⟨𝘷∗ + 𝑆𝑥, 𝑥⟩ ≥ −d‖𝑥‖² for all 𝑥 ∈ 𝐷(𝑇) ∩𝐷(𝑆) and 𝘷∗ ∈ 𝑇𝑥. New surjectivity results are given for noncoercive, not everywhere defined, and possibly unbounded operators of the type 𝑇+𝑆. A partial positive answer for Nirenberg’s problem on surjectivity of expansive mapping is provided. Leray-Schauder degree is applied employing the method of elliptic superregularization. A new characterization of linear maximal monotone operator 𝐿 : 𝑋 ⊇ 𝐷(𝐿) → 𝑋 ∗ is given as a result of surjectivity of 𝐿 + 𝑆, where 𝑆 is of type (𝑀) with respect to 𝐿.These results improve the corresponding theory for noncoercive and not everywhere defined operators of pseudomonotone type. In the last section, an example is provided addressing existence of weak solution in 𝑋 = 𝐿𝑝(0, 𝑇;𝑊₀¹,𝑝 (Ω)) of a nonlinear parabolic problem of the type 𝑢𝑡− Σ𝑛𝑖=1(𝜕/𝜕𝑥𝑖)𝑎𝑖 (𝑥, 𝑡, 𝑢, ∇𝑢) = 𝑓(𝑥, 𝑡), (𝑥, 𝑡) ∈ 𝑄; 𝑢(𝑥, 𝑡) = 0, (𝑥, 𝑡) ∈ 𝜕Ω × (0, 𝑇); 𝑢(𝑥, 0) = 0, 𝑥 ∈ Ω, where 𝑝 > 1, Ω is a nonempty, bounded, and open subset of R𝑁, 𝑎𝑖: Ω × (0,𝑇) × ℝ × ℝ𝑁 → ℝ (𝑖 = 1, 2, . . . , 𝑛) satisfies certain growth conditions, and 𝑓 ∈ 𝐿𝑝' (𝑄), 𝑄 = Ω × (0,𝑇), and 𝑝' is the conjugate exponent of 𝑝.