Solvability of abstract semilinear equations by a global diffeomorphism theorem

In this work we proivied a new simpler proof of the global diffeomorphism theorem from [9] which we further apply to consider unique solvability of some abstract semilinear equations. Applications to the second order Dirichlet problem driven by the Laplace operator are given.


Introduction
The idea of applying global invertibility results to boundary value problems and integral, integro-differential equations has been known in the literature for some time now. There is a variational tool concerning global invertibility which we are going to use. Theorem 1. [9,Theorem 3.1]Let X be a real Banach space and let H be a real Hilbert space. Suppose that F : X → H is a C 1 mapping such that: D1 for every y ∈ H functional ϕ y : X → R given by ϕ y (x) := 1 2 F(x) − y 2 satisfies Palais-Smale condition, i.e. every sequence (x n ) n∈N ⊂ X such that (ϕ y (x n )) n∈N is bounded and ϕ ′ y (x n ) → 0 X * admits convergent subsequence; D2 for every x ∈ X an operator F ′ (x) is bijective.
Then F is diffeomorphism.
The proof of Theorem 1 relies 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 speaking the fact that f is onto is reached through the classical Ekeland's Variational Principle. The injectivity part is obtained by contradiction assuming to the contrary and arguing by the application of 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 readible manner thus simplifying the arguments from [9]. Theorem 1 proposes some approach towards the existence of solutions to nonlinear equations which is variational in spirit, i.e. concerns the usage of certain functional which is at the same different from the classical energy (Euler type) action functional. Moreover, it allows for obtaining uniqueness of a solutions without any notion of convexity, again contrary to what is known in the application of a direct method, see for example [13,Corollary 1.3]. However up to now Theorem 1 and related global implicit function theorem from [10] have been applied to various first order integro-differential problems which cover also the so called fractional case (with the fractional derivative) and correspond to Urysohn and Volterra type equations, see [4,11,12]. Some comments on the global invertibility results from [9], relation with other approaches and possible applications are contained in [8]. There was also an attempt to examine second order Dirichlet problem for O.D.E. in [2], but for some specific problem and without any abstract scheme allowing for considering boundary value problems in some unified manner. Results for 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 possibly the abstract framework here is to be obtained with some different global invertibility result. Our applications are meant for partial differential equations and thus do not have their counterparts in [19].
In this work we aim at proposing some abstract approach in order to examine solvability of some semilinear equations pertaining to second order Dirichlet problems for both ordinary and partial differential equations using the approach suggested by Theorem 1. Our results towards 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 utilized relations between critical points to actions functional andÂ. . . xed points to certain mappings. Nevertheless, our approach towards solvability is different and relies on different abstract tools. Morevoer, the setting is now somehow different since for the sake of global invertiblity densly defined operators are insuficient. In fact one need to consider the domain of the operator with its natural topology induced by a suitable norm.

Problem formulation and main results
Let (H, · | · ) be a real Hilbert space with a norm denoted by · and let A be a self-adjoint operator on H with the domain D(A). Recall that (D(A), · | · A ) is a real Hilbert space, where · | · A = · | · + A· | A· . By · A we denote its norm, i.e. the graph norm of A. Let (B, · B ) be a real Banach space and let N : (B, · B ) → (H, · | · ) be an operator which is not necessary linear. In this framework we shall study in D(A) the following equation In order to consider (1) we will make the following assumptions: (N2) there exist constants 0 < β < 1, 0 < γ < α, δ > 0, such that: Our main result reads as follows. In this Theorem we may replace assumption (A1) by the following one: Remark. While all spaces which we consider are real, the theory developled in [20] works for complex spaces. Nevetheless, results which we use (namely: Proposition 3.10, Proposition 5.12, Proposition 10.19) can be clearly taken to the setting of a real space using the spirit of a book by Brezis [5]. Moreover Kato-Rellich Theorem for the setting of a real space is contained in [7].

Proofs
For the proof of Theorem 1 we need the following Theorem.
Theorem 3. [18, Theorem 2] Let X be a Banach space and let J : X → R be a C 1 functional satisfying 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 point x of J, with J(x) > J(0 X ), witch is not a local minimum.
Proof of Theorem 1. Firstly, we show that operator F is ,,onto". 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 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, in consequence, 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. Obtained contradiction ends the proof. Now, we can present the proof of the main Theorem.
Proof of Theorem 2. 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 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 equation (1).
Fix y ∈ H and consider the mapping ϕ y : X → R given by Then ϕ y ∈ C 1 (X, R), F ∈ C 1 (X, H) and its derivatives are given, respectively, by the following formulas In order to be able to use Theorem 1 we must show that ϕ y satisfies 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 (PS) sequence can be assumed to be weakly convergent.
Now we show that the functional ϕ y satisfies (PS) 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 after a subsequence, it is weakly convergent to some u 0 ∈ X. From (A1) there exists another subsequence, denote it once again by (u n ) n∈N , wich 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 be the weak convergence of (u n ) n∈N to u 0 in X, we have Coining the above observations together, we can now show that equality (5) implies Au n − Au 0 → 0 as n → ∞ which means, by (2), that (u n ) n∈N converges strongly to u 0 in X. This shows that ϕ y satisfies (PS) condition. Now, we show that F ′ (u) is bijective for any u ∈ X. Fix u ∈ X. Since A is self-adjoint operator and since N ′ (u) is a symmetric compact linear operator, it follows that F ′ (u) is self-adjoint operator, by [7, RKNG Theorem in real Hilbert space]. Using (A2) and (N2) we get Hence, equivalently for all h ∈ H. Then, as F ′ (u) is linear, it is injective. Applying [20, Proposition 3.10], F ′ (u) is also surjective, and so bijective. Now we can apply Theorem 1 and obtain a unique u * ∈ X such that 0 = F(u * ) = Au * − N(u * ).
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. x ∈ Ω, f (x, ·) is of class C 1 and for all u ∈ R, f (·, u), f ′ u (·, u) are measurable. [20,Proposition 10.19]. By the Poincaré inequality

An unbounded linear operator
and Green's formula we have where c Ω is a constant in Poincaré inequality and · | · and · denote the scalar product and the norm in H, respectively. On D(A) the graph norm of A and norm · H 2 (Ω) are equivalent, see [20, p. 240]. Therefore if we put where p m > 2 for m = 4 and p m ∈ 2, 2m m−4 for m > 4, we obtain the compact embedding (D(A), · | · A ) ֒→ (B m (Ω), · B m ), see [15,Theorem 1.51]. For m 3 let c m > 0 be such that u ∞ c m Au for all u ∈ D(A), where · ∞ denotes the supremum norm.
Under the above assumptions the operator N f : Assumption (P3) provides that for every u, h ∈ B m (Ω) there is As a conclusion, we obtained Theorem 4. Assume that f : Ω × R → R is a C 1 -Caratheodory function such that (P1m), (P2m) and (P3) hold. Then problem (7) has an unique solution in H 1 0 (Ω) ∩ H 2 (Ω).