Solvability of Abstract Semilinear Equations by a Global Diffeomorphism Theorem

In this work we develop a method towards unique solvability of abstract semilinear equations. We use a global diffeomorphism theorem for which we provide a simplified proof. Applications to second order partial differential equations are given. Some additional technical tools about the properties of the Niemytskij operator are also given.


Introduction
The aim of this work is to consider unique solvability of semilinear abstract equations with further applications to elliptic Dirichlet boundary value problems. In order to fulfill this task we derive some new methodology based on the global diffeomorphism theorem given in [8,Theorem 3.1] for which we provide a somewhat simplified and more intuitive proof. Up to now Theorem 2 and related global implicit function theorems from [9] have been applied to various first order integro-differential problems which cover also the so called fractional case (with the fractional derivative and also the fractional Laplacian) and correspond to Urysohn and Volterra type equations, see [4,10,12]. There was also an attempt to examine second order Dirichlet problems for O.D.E. in [2], but for certain specific problems and without any abstract scheme which allows for considering boundary value problems in some unified manner. Results for the continuous problem in [2] are related to the existence result obtained in [19]; although the methods are different, both yield the existence with similar assumptions. This suggests that it is possible to obtain an abstract framework based on a global invertibility result. Our applications are meant for partial differential equations and thus do not have their counterparts in [19,21].
By · A we denote the associated norm, i.e. the graph norm of A. Let (B, · B ) be a real Banach space and let be an operator which is not necessarily linear. In this framework we shall study in D(A) the following equation In order to consider (1) we will make the following assumptions: Our main abstract result reads as follows.

Theorem 1.
Assume that (A1)-(A2) and (N1)-(N2) are satisfied. Then equation (1) has a unique solution in D(A). Moreover, the operator F given by In the above Theorem we may replace assumption (A1) by the following one: with purely discrete spectrum.
If the above holds then it follows by [20,Proposition 5.12] that the embedding Remark. While all spaces which we consider are over R, the theory developed in [20] works for spaces over C. Nevertheless, results which we use (namely: Propositions 3.10, 5.12 and 10.19) extend to the setting of a space over R using direct calculation based on Brezis book [5]. Moreover, the theorem of Kato-Rellich works in spaces over R. The proof of this fact could be found in [7]. We will further apply our abstract tools in order to study the following nonlinear Dirichlet problem Here Ω ⊂ R m is an open and bounded set of class C 2 and f : Ω × R → R is a C 1 -Caratheodory function, i.e. for a.e.
Our results towards the abstract approach were inspired by some recent abstract approaches developed in [6,17] which were based on the variational framework due to [17] and which utilize relations between critical points to action functionals and fixed points of certain mappings. Nevertheless, our approach towards solvability is different and relies on different abstract tools.
The paper is organized as follows. Firstly we provide some remarks on the main abstract tool. Next we consider the solvability of abstract equations which is then followed by applications to elliptic problems. Finally we provide an appendix in which we address differentiability of some Niemytskij type operator in our applied case. We provide the proof since we have not found such a result in the literature.

Remarks on a Global Diffeomorphism Theorem
Theorem 2 proposes an approach to the existence of unique solutions to nonlinear equations which is variational in spirit, i.e. concerns the use of certain functionals which are different from the classical energy (Euler type) action functional. Moreover, it allows one to obtain uniqueness of solutions without any notion of convexity, again contrary to what is known in the application of a direct method, see for example [ Then the operator F is a diffeomorphism.
The proof of Theorem 2 is based on the application of the celebrated Mountain Pass Theorem due to Ambrosetti and Rabinowitz, see [1], and relies in checking that the functional ϕ satisfies the mountain geometry. Precisely the fact that F is onto is reached through the classical Ekeland's Variational Principle. The injectivity part is obtained by contradiction by using the Mountain Pass Theorem. The most difficult part of the proof is the estimation of ϕ on some sphere around 0. However, we will show using some ideas from [18] that the proof can be performed in a different and more readable manner thus simplifying the arguments from [8].
For the proof of Theorem 2 we need the following Theorem 3 [18,Theorem 2]. Let X be a Banach space and let J : X → R be a C 1 functional satisfying the Palais-Smale condition with 0 X its strict local minimum. If there exists e = 0 X such that J(e) J(0 X ), then there is a critical pointx of J, with J(x) > J(0 X ), which is not a local minimum.
Proof of Theorem 2. The proof that operator F is ,,onto" is taken from the original proof of Theorem 2 and we provide it for reader's convenience. Fix y ∈ H. As F is of class C 1 , ϕ y (x) = 1 2 F (x) − y 2 is of the same type and its differential ϕ y (x) at x ∈ X is given by for all h ∈ X. Clearly, ϕ y is bounded from below and it satisfies the Palais-Smale condition, by D1. Hence, ϕ y has a critical point (see [13,Chapter 3,Corollary 3.3]). In other words, there exists Now we show that F is "one-to-one". Aiming for a contradiction, suppose that there exist x 1 , x 2 ∈ X such that x 1 = x 2 and F (x 1 ) = F (x 2 ). Define e := x 1 − x 2 and put ψ : X → R by formula Then ψ is of class C 1 and ψ(0 X ) = ψ(e) = 0. Moreover, 0 X is a strict local minimum of ψ, since otherwise, in any neighbourhood of 0 X we would have a nonzero x with F (x + x 2 ) − F (x 1 ) = 0 H and this would contradict the fact that F defines a local diffeomorphism. Therefore we can apply Theorem 3 and, consequently, there existsx ∈ X such that ψ(x) > 0 and ψ (x) = 0 X * . Hence for all h ∈ X. Again, by surjectivity of F (x+x 2 ), we have F (x+x 2 )−F (x 1 ) = 0 and so ψ(x) = 0, which contradicts ψ(x) > 0. The obtained contradiction finishes the proof.
We would like to note that our arguments comply with the proof of the well known Hadamard's Theorem, see Theorem 5.4 from [11], see also [16].

Proof of the Main Abstract Tool
Now, we can present the proof of the main Theorem. Proof of Theorem 1. By (A2) we have Au α u for u ∈ D(A) and so Let X := (D(A), · A ) and let the operator N : is a compact embedding given by (A1). Then N ∈ C 1 (X, H) and the operator N (u) is symmetric, compact and linear for all u ∈ X, by (N1). Since i(u) = u, any solution of equation is also a solution of Eq. (1). Let us define F : X → H by Fix y ∈ H and consider the mapping ϕ y : X → R given by Then ϕ y ∈ C 1 (X, R) and F ∈ C 1 (X, H). Moreover, their derivatives are given, respectively, by the following formulas In order to be able to use Theorem 2 we must show that ϕ y satisfies the Palais-Smale condition and F (u) is bijective for all u ∈ X.
By applying (N2) we see that for every u ∈ X. This implies that ϕ y is coercive. Thus any Palais-Smale sequence can be assumed to be weakly convergent. Now we show that the functional ϕ y satisfies the Palais-Smale condition on X. Assume that (u n ) n∈N ⊂ X is such that: Since ϕ y is coercive, (PS1) shows that (u n ) n∈N is bounded in X, and then possibly up to a subsequence, it is weakly convergent to some u 0 ∈ X. From (A1) there exists another subsequence, denoted by (u n ) n∈N , which is convergent in (B, · B ). So, by our assumptions we have Now, a direct calculation yields where Then, using observations made above, we obtain On the other hand, by (PS2) and the weak convergence of (u n ) n∈N to u 0 in X, we have as n → ∞. Combining the above observations, we can now show that equality (6) implies Au n − Au 0 → 0 as n → ∞ which means, by (3), that (u n ) n∈N converges strongly to u 0 in X. This shows that ϕ y satisfies the Palais-Smale condition. Now, we show that F (u) is bijective for any u ∈ X. Fix u ∈ X. Since A is a self-adjoint operator and since N (u) is a symmetric, compact and linear operator, it follows by the RKNG Theorem in real Hilbert space, see [7], that F (u) is self-adjoint operator. Using (A2) and (N2) we get Therefore we have for all h ∈ H. Then, as F (u) is linear, it is injective. By Proposition 3.10 from [20] it follows that F (u) is also surjective, and this is why it is bijective. Now we can apply Theorem 2 in order to obtain a unique u * ∈ X such that 0 = F (u * ) = Au * − N (u * ). Moreover, using the same Theorem we see that F is a diffeomorphism.

Applications
As an application of Theorem 1 we discuss the solvability of problem (2). Firstly we show how to construct a suitable abstract framework. We take H = L 2 (Ω) and as A, the operator Au = −Δu with D (A) = H 1 0 (Ω) ∩ H 2 (Ω), see [20,Proposition 10.19] . By the Poincaré inequality and by the Green's formula we have where c Ω is a constant in Poincaré inequality and where · | · and · denote the scalar product and the norm in H, respectively. We note that on space D(A) the graph norm of A and norm · H 2 (Ω) are equivalent, see [20, p. 240]. Therefore, if we put We will need the following assumptions on f : x ∈ Ω and every u ∈ R; (if m 4) there exist a 1 ∈ L 2 (Ω) and b 1 ∈ (0, c Ω ) such that |f (x, u)| a 1 (x)+ b 1 |u| for a.e. x ∈ Ω and every u ∈ R; (P2m) (if m 3) there exist a 2 ∈ L 2 (Ω) and g ∈ C(R, R) such that |f u (x, u)| a 2 (x)g(u) for a.e. x ∈ Ω and every u ∈ R; (if m 4) there exist a 2 ∈ L q (Ω) and b 2 > 0 such that |f u (x, u)| a 2 (x) + b 2 |u| r for a.e. x ∈ Ω and every u ∈ R, where r = pm−2 2 and q = 2pm pm−2 ; (P3) there exists b 3 ∈ (0, c 2 Ω ) such that f u (x, u) < b 3 for a.e. x ∈ Ω and every u ∈ R.
Under the above assumptions the operator N f : B m (Ω) → L 2 (Ω) given by the formula N f (u)(x) = f (x, u(x)) for x ∈ Ω is of class C 1 with N f (u)(h) = N f u (u)(h) for all u, h ∈ B m (Ω). For the case m 4 see [15,Proposition 2.78] and for the case m 3 see "Appendix". Clearly, N f (u) is a symmetric operator for all u ∈ B m (Ω) .
As an example the following problems where Ω ⊂ R 3 is an open bounded set of class C 2 , constants C 1 and C 2 only depend on Ω, φ ∈ L 2 (Ω) and | · | denotes the Euclidean norm, have unique solutions in H 1 0 (Ω) ∩ H 2 (Ω).