On the nonlinear perturbations of self-adjoint operators


               <jats:p>Using elements of the theory of linear operators in Hilbert spaces and monotonicity tools we obtain the existence and uniqueness results for a wide class of nonlinear problems driven by the equation <jats:inline-formula>
                     <jats:alternatives>
                        <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_anona-2022-0235_eq_001.png" />
                        <m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
                           <m:mi>T</m:mi>
                           <m:mi>x</m:mi>
                           <m:mo>=</m:mo>
                           <m:mi>N</m:mi>
                           <m:mrow>
                              <m:mo>(</m:mo>
                              <m:mrow>
                                 <m:mi>x</m:mi>
                              </m:mrow>
                              <m:mo>)</m:mo>
                           </m:mrow>
                        </m:math>
                        <jats:tex-math>Tx=N\left(x)</jats:tex-math>
                     </jats:alternatives>
                  </jats:inline-formula>, where <jats:inline-formula>
                     <jats:alternatives>
                        <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_anona-2022-0235_eq_002.png" />
                        <m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
                           <m:mi>T</m:mi>
                        </m:math>
                        <jats:tex-math>T</jats:tex-math>
                     </jats:alternatives>
                  </jats:inline-formula> is a self-adjoint operator in a real Hilbert space <jats:inline-formula>
                     <jats:alternatives>
                        <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_anona-2022-0235_eq_003.png" />
                        <m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
                           <m:mi class="MJX-tex-caligraphic" mathvariant="script">ℋ</m:mi>
                        </m:math>
                        <jats:tex-math>{\mathcal{ {\mathcal H} }}</jats:tex-math>
                     </jats:alternatives>
                  </jats:inline-formula> and <jats:inline-formula>
                     <jats:alternatives>
                        <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_anona-2022-0235_eq_004.png" />
                        <m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
                           <m:mi>N</m:mi>
                        </m:math>
                        <jats:tex-math>N</jats:tex-math>
                     </jats:alternatives>
                  </jats:inline-formula> is a nonlinear perturbation. Both potential and nonpotential perturbations are considered. This approach is an extension of the results known for elliptic operators.</jats:p>


Introduction
Let be a real Hilbert space. Take a densely defined linear operator ( ) ⊃ ⟶ T T : and a nonlinear operator (| | ) ⊃ ⟶ / N T : 1 2 . It makes sense to consider the solvability of the following equation: in case N interacts with the spectrum of T . When T is positive definite and when the growth of N is governed by its first eigenvalue leading to the coercivity of − T N (see, for example, [1]), then the celebrated Strong Monotonicity Principle is applicable. If the additional assumption of potentiality is imposed on N , then in the presence of the coercivity of the related Euler action functional the direct variational method can be used leading also to the variational characterization of the solution (see [2]). In addition to those, the dual least action principle concerning problems driven by positive definite self-adjoint operator T and its potential perturbation N was considered in [3], where again the growth conditions were strictly related to the first eigenvalue of T . The situation complicates if the growth conditions interact with the spectrum of this operator in a more subtle manner, which does not lead to the coercivity of the relevant operator. This is the case when the growth conditions concern behavior of the nonlinear term somehow in-between the eigenvalues. In such a situation one cannot directly use any method related to the coercivity. In order to overcome this difficulty based on the methods of Hilbert space theory we introduce a related operator and next a new real problem, which can be treated by the Strong Monotonicity Principle and which provides the existence result for an original problem as well. Consequently, we obtain some approach towards the unique solvability of nonlinear equations that cannot be directly treated by the aforementioned methods related to the coercivity. In the presence of the potentiality of the operators involved we may obtain a type of min-max characterization of a solution. The idea lying behind our approach can be best illustrated in the finite dimensional case as follows. Consider the matrices In the infinite dimensional case, the spectrum of a differential operator, like for example the negative Laplacian, is unbounded from above which makes the situation more complicated and leads either to a direct minimization or to a version of a saddle point result in the potential case. There has already been some attempt to deal with situation when growth conditions involve the spectrum of a differential operator in [4]. The special case of elliptic problems for both Dirichlet and Neuman boundary conditions via arguments related to the global invertibility was considered. The method which we propose here allows one to have some unified approach towards certain type of nonlinear problems together with the variational characterization of a solution in the potential case.
The article is organized as follows. In Section 2, we invoke some definitions and facts which will be employed throughout this article. We start from elements of the theory of Hilbert space linear operators, go through the concepts from the theory of monotone operators, give some tools from the analysis in Euclidean spaces, and conclude with some background on Sobolev spaces. In Section 3, we provide motivation for our research based on the classical nonlinear boundary value problem with the Dirichlet conditions. Next, in a real Hilbert space we formulate its abstract counterpart which is associated with a fixed self-adjoint operator acting in . In Proposition 3.1, we give an equivalent condition for an element of to be a solution of the problem under consideration. Further we proceed to the unique solvability of the potential problem, which is the subject of Theorem 3.8. We complete the existence result in Theorem 3.9 with the min-max characterization. For the nonpotential case we get Theorem 3.11, which is an abstract analog of the "freezing method." Finally, Section 4 contains applications of our existence and uniqueness tools towards nonlinear boundary value problems driven the Neumann Laplacian.

Preliminaries
In all that follows, is a real Hilbert space and * stands for its dual space.

Linear operators in Hilbert spaces
Most of the basic results from the theory of Hilbert space operators can be transferred, with proper caution, from the complex to the real case. The reader is referred, for example, to [5], where special attention is drawn to the subtleties arising when working in a real Hilbert space.
By a (linear) operator T in we understand a linear mapping ( ) with the domain ( ) T . We denote by ( ) the Banach algebra of all bounded operators from the whole space into itself. Below we mainly deal with unbounded operators in , which makes the situation more delicate because their domains are in general strictly contained in . For operators S and T in , is a closed subspace of the product Hilbert space ⊕ , or equivalently ( ) T is a Hilbert space under the graph norm ‖⋅‖ T given by If T is a densely defined operator in (i.e., the closure of ( ) T equals to ), then we denote by * T its adjoint (being also a linear operator in ). When = * T T we say that the operator T is self-adjoint. Analogously as in the complex case, each self-adjoint operator T in can be written as the spectral with respect to a unique spectral measure E (see [5,Theorem 7.17]), i.e., where ( ) σ T stands for the spectrum of T . This representation together with some other basic properties of spectral integrals will be utilized in Section 3. Let us add that by ( ) ρ T we will mean the resolvent set of T , that is, T . In this case, we can find a unique positive self-adjoint operator S in satisfying the equality = S T

Monotone operators
Following [6] or [7] we provide necessary background information on the theory of monotone operators, which pertains to the existence of nonlinear equations. We consider a (nonnecessarily linear) mapping ⟶ * A : called further an operator. We say that A is demicontinuous if for every sequence ( ) ⊂ x n and ∈ An operator A is called strongly monotone (or α-strongly monotone) if there exists > α 0 such that f o r a l l, . 2 In turn A is said to be relaxed monotone (or β-relaxed monotone) if there exists ∈ β such that A y x y β x y x y , f o ra l l, . 2 Recall that the Gâteaux derivative of a mapping ⟶ F : , where is a real Hilbert space, at the point ∈ x is the functional ( ) ′ ∈ * F x satisfying Perturbations of self-adjoint operators  1119 F is called Gâteaux differentiable if it has the Gâteaux derivative at each point. Moreover, we denote by ″ F the second derivative of F , that is, the Gâteaux derivative of ′ F . An operator A is called potential if there exists a Gâteaux differentiable functional ⟶ : , called the potential of A, such that ′ = A. Note that for a given operator A, a potential of A (if exists) is uniquely determined up to a constant. The Gâteaux differentiability of a functional in general does not imply any type of its continuity. However, every monotone and potential operator is necessarily demicontinuous (see [6 Lemma 5.4]). We generalize this observation to the case of relaxed monotone and potential operators.
Lemma 2.1. Every relaxed monotone and potential operator is demicontinuous.
Proof. Let A be a β-relaxed monotone and potential operator. Then we easily see that the operator − βj A is monotone and potential, where ⟶ * j : is the normalized duality mapping between and * , which is a Gâteaux derivative of ‖⋅‖ we see that j is continuous and strongly monotone with constant 1. Hence, by [6, Lemma 5.4], the operator − βj A is demicontinuous. Therefore, since j is continuous, the operator Next we recall main existence and uniqueness tools used in this article.
is a demicontinuous and strongly monotone operator, then it is a bijection.
It is worth noting that the proof of the above theorem presented in [6] works in separable Hilbert spaces only. However, this result can be extended to the general case following ideas described in [7, Section 7].

Analysis on the Euclidean space
In this section, we present the results that, for a given mapping, relate the relaxed monotonicity to the Lipschitz condition. These tools will be extended to the infinite dimensional case in Section 3. The symbols ( ) C m , ( ) C m 1 , and ( ) ∞ C m stand for the spaces of all, respectively, continuous, continuously differentiable, and smooth functionals on m . We denote the support of a functional . The Euclidean space m will be considered with the standard inner product ⟨⋅ ⋅⟩ , . , the following conditions are equivalent: Proof. To show that (1) ⇒ (2) it suffices to estimate the expression while to get the implication (2) ⇒ (1) one can apply the standard estimation techniques to Let us fix a sequence of mollifiers ( ) ρ n , that is, the functions such that is an open ball centered at the origin with radius where ⋆ stands for a convolution operator. Proposition 4.21 of [8] provides an almost uniform convergence [8,Proposition 4.20]). Let us recall that for a self-adjoint operator . In order to extend this observation to the case of nonlinear mappings, we have to assume some analog of the symmetry = * L L . For operators of the class C 1 , such a role plays the symmetry of the derivative operator at each point, which is equivalent to the operator's potentiality. This is the subject of the following lemma.
) and let f n be defined by (2.1). By direct calculations, for all ∈ x y , m and every ∈ n , we obtain Hence, by Lemma 2.3, we obtain that for all ∈ n However, since ( ) ″ f x n is symmetric as the second derivative of a functional f n , we get ( ) almost uniformly, we obtain the assertion. □

Functional framework
Let Ω be an open bounded subset of m with a boundary of the class C 2 (see [9] for details). We denote by ) the first Sobolev space (resp. the second Sobolev space) consisting of all functions from ( ) L Ω 2 , whose all first (resp. second) order weak derivatives belong to ( ) we mean the closure of ( ) Ω Ω a n d Δ Ω : 0o n Ω. x Ω and all , , are continuous. We define (pointwisely a.e.) the Niemytskii operator N f associated with f by the formula , , .

Main results
The motivation for our research comes from the following classical problem with the Dirichlet boundary conditions: where Ω is an open and bounded subset of the Euclidean space. In order to settle the question of its solvability by using variational and monotonic methods we need to consider its solvability in a weak sense. Thus, we transfer our considerations from a closed subset of ( ) , where the operator −Δ is defined in a strong sense, to the space ( ) H Ω 0 1 which in fact is the domain of the operator ( ) − / Δ 1 2 . Having finally obtained a weak solution u to problem (3.1), that is, an element we use the regularization techniques to show that u belongs to the class H 2 . The self-adjointness of −Δ is a key factor to improve the regularity of the solution u. Such an approach to the solvability of problem (3.1) in the abstract settings is the subject of our discussion in the present section. We additionally note that the right hand side of (3.1) is typically defined on the space ( ) H Ω entails the sublinearity of f with respect to the second variable, which in turn can be regarded as a restrictive condition.

Regularity of a solution
Take a self-adjoint operator T in and an operator (| | ) ⟶ / N T : 1 2 . Let us consider the following equation: T . We define a bilinear form t (| | ) (| | ) × ⟶ / / T T :  Proof. A look at the definition of the adjoint of an operator and the fact that | | / T 1 2 is self-adjoint reveals that , f o ra l l . 1 2 By the density of (| | ) / T 1 2 in , we obtain ( ) = Tx N x . This means that x is a solution to (3.2).
It seemingly differs from the standard weak solvability condition: Note, however, that for every which by the density of ( ) leads to the equivalence of (3.5) and (3.4).

The case of potential perturbations
Take a self-adjoint operator ( ) ⊃ ⟶ T T : , the spectral measure E of T , and a Gâteaux differentiable functional (| | ) ⟶ / T : 1 2 . We assume that for every

6)
for every (| | ) ∈ / y T 1 2 . Note the above assumption means that for every x has a representation with respect to the inner product in , while the differentiability of ensures only that such a representation exists with respect to the inner product in (| | ) / T 1 2 . The meaningness of these assumptions is well described by the following remark.
Then for every has an integer representation of the form ( ( )) ⋅ ⋅ f u , .
To be more precise, we have We put x y x y 2 2 (3.7) Remark 3.4. Note that if α or β is finite, then ′ is demicontinuous by Lemma 2.1. Moreover, ′ is continuous on every finite dimensional subspace of (| | ) / T 1 2 because the norm and weak topologies coincide therein.  y x y  c x y for all x y  T  , , , 1 2 then N is c-Lipschitz.
Proof. Take (| | ) ∈ / x y T , 1 2 and consider any unitary isomorphism s p a n , 0, 1, 2 , depending on the choice of vectors x and y) and an auxiliary functional ⟶ f : m given by , , 2 for all ∈ z w , m . Hence, we can use Lemma 2.1 to obtain the continuity of ′ f . Applying Lemma 3.5, we obtain Remark 3.6. Let us note that in our model case (3.1) the function h is the Euler action functional defined by coincide. This is again due to the density of ( ) and due to the equality where > δ ω , 0. Note that, for arbitrarily chosen constants δ and ω, the norm ‖⋅‖ δ ω , is equivalent to the graph norm of | | / T 1 2 . The space (| | ) / T 1 2 becomes a Hilbert space when equipped with norm (3.8).
for ∈ t . The function d α β , plays only a technical role and enables us to avoid more complex calculations.
Remark 3.7. Note that for every closed subset C of on which the function d α β , is positive there exist constants δ and ω satisfying

9)
Indeed, it suffices to take (( ) ) = > γ α β C dist , , 0, and put The constants selected in this way may not be optimal.
Perturbations of self-adjoint operators  1125 Proof. By the assumptions, the function d α β , is positive on ( ) σ T . We take any > δ ω , 0 satisfying condition (3.9). Let us define the operator y Uh y  x y  T  , , , , , f o ra l l , , 1 2 where , . We show that for all , then = U I and direct calculations yield We act analogously if . If both α and β are finite, then we have Moreover, by assumptions we obtain is potential, we can use Lemma 3.5 to deduce that − , where X and Y are linear subspaces satisfying { } ∩ = X Y 0 . Let us recall that the min-max inequality says that x y x y sup inf inf sup .
We say that h is convex (resp. concave) along a space X if for every ∈ y Y a functional ( ) ⋅+ ⟶ y X : h is convex (resp. concave). For a given ∈ ξ , we define Theorem 3.9. Assume that the distance between ( ) α β , and ( ) σ T is positive and let ∈ h . Then the unique solution to (3.10) satisfies the condition Proof. We define the operator A as in the proof of Theorem 3.8. Take ∈ x X α and (| | ) ∈ / z T 1 2 . Let us calculate which means that h is concave along X α . Analogously, it follows that h is convex along Y β . Therefore, since x h is a critical point of h , there is x y x x y inf sup sup inf .
The min-max inequality gives the assertion. □

The case of nonpotential perturbations
We start with providing the model problem which best illustrates our ideas. Let us consider where × ⟶ f : Ω is such a Carathéodory function, that the related Niemytskii operator Such a form of N allows us to consider separate assumptions on the term involving u (the potential part) and on the term involving ∇u (the nonpotential part).
The abstract approach which we present now is related to the so-called "freezing method" (see, for example, [13] or [14] for its applications). This method relies on fixing the term that is responsible for the fact that the system is nonpotential, then on obtaining the solution operator to the new problem via some variational approach and finally on reaching the solution to the original problem via suitable fixed point theorem applied to this solution operator. Here we also follow such a path with one exception that instead of a fixed point we apply the strongly monotone principle.
Remark 3.10. With reference to problem (3.13) we freeze the gradient on right hand side and obtain now problem with a fixed parameter v which is now potential. The solution operator to (3.15) depends on v and the assumptions which we impose make it strongly monotone and continuous. Hence, the strongly monotone principle applies.
We consider the operator , which is an abstract counterpart of the operator N defined in (3.14) and corresponds to assumption (3.6). We assume that for every (| | ) ∈ / z T 1 2 there exists a Gâteaux differentiable functional (| | ) ⟶ / T : , for all 1 2 and, analogously as before, we denote Therefore, by Theorem 3.11, we obtain the assertion. □ Clearly, we can also consider gradient depending terms "between" eigenvalues. For the sake of simplicity, we provide now an explicit example for the ordinary differential equation.