On variational nonlinear equations with monotone operators

Abstract: Using monotonicity methods and some variational argument we consider nonlinear problems which involve monotone potential mappings satisfying condition (S) and their strongly continuous perturbations. We investigate when functional whose minimum is obtained by a direct method of the calculus of variations satis es the Palais-Smale condition, relate minimizing sequence and Galerkin approximaitons when both exist, then provide structure conditions on the derivative of the action functional under which bounded Palais-Smale sequences are convergent. Finally, we make some comment concerning the convergence of Palais-Smale sequence obtained in the mountain pass theorem due to Rabier.


Introduction
Let E be a real, separable, re exive Banach space and let J : E → R be a Gâteaux di erentiable functional. When we are interested in solving equation several methods can be used. We mention the following: i) the direct variational approach for which we require the action functional J to be sequentially weakly lower semicontinuous and coercive; ii) the Ekeland Variational Principle for which to be applied we need to assume that J is bounded from below and satis es the Palais-Smale condition; iii) the Minty-Browder Theorem (monotonicity approach) for which we require operator J to be coercive and monotone (then its potential J is necessarily demicontinuous). The application of the direct method of the calculus of variations provides that the argument of a minimum is approximated by the weakly convergent minimizing sequence. In case of the application of the Ekeland Variational Principle the argument of a minimum is approximated by the strongly convergent sequence of almost critical points. The monotonicity approach on the other hand provides that the sequence of Galerkin type approximations to equation (1.1) converges strongly -under the additional assumption that operator J satis es condition (S). Thus it becomes interesting to know whether there are some natural conditions on J which would provide that, disregarding of the approach which we use, we can approximate the solution by some strongly convergent sequence corresponding to the method that we use. We give positive answer to this question thereby proving that the minimizing sequence obtained by the direct variational approach is strongly convergent. This observation has several other implications like the result concerning the conver-gence of bounded Palais-Smale sequences and the nature of the sequence of Galerkin type approximations in case the solution is additionally the argument of a minimum. Moreover, we show that, under some conditions, existence results denoted above as i) and ii) overlap which means that the minimizing sequence obtained by a direct variational method is strongly convergent and consists of almost critical points. We obtain our results linking monotonicity and variational approaches.
Concerning the existing research connected to the area of our investigations it is known for semilinear problems involving the Laplacian, see for example Proposition 2.2 in [7], that if a derivative of a functional is a sum of an invertible linear bounded operator and a continuous and compact operator, then a bounded Palais-Smale sequence is necessarily convergent. The simpli cation of checking of the Palais-Smale condition which we provide works for boundary value problems related to the p−Laplacian and the p (x) −Laplacian as well, see [11] for the background on the variable exponent laplacian. We use also this observation to derive a corollary from the version of the mountain pass theorem introduced by Rabier in [10]. Our improvement is that the Palais-Smale sequence is convergent at the expense of some additional assumptions. Some applications to potential problems are also given.
There have been already some results in the area of nonlinear abstract equations, see for example [2,3] but the papers mentioned concerned mainly existence results for problem involving a duality mapping relative to function t → t p− for a suitable p.
Paper is organized as follows. Firstly we recall some background from the monotonicity theory. Then we provide some results for uniformly monotone operators due to the fact that we use a bit di erent de nition than this in [6]. In the next section, the main topics are addressed with some additional comment on minimizing sequences and relations to Galerkin type approximations. As applications of our results we provide a corollary from Rabier theorem then, see [10] for the version of a mountain pass lemma. Some Dirichlet boundary value problem is also considered.

Preliminaries
For some background from the theory of monotone operators we refer to [6] and also [9]. Symbol ·, · stands for a duality pairing between E * and E in what follows. The norm in E is denoted by · and in E * by · * . Operator A : E → E * is called: i) uniformly monotone if there exists an increasing function ρ : [ , +∞) → [ , +∞) such that ρ ( ) = and for When E is strictly convex, it follows that a d−monotone operator is strictly monotone, i.e. for all u, v ∈ E, Operator A : E → E * satis es condition (S), if: While a uniformly monotone operator satis es condition (S), a d−monotone operator does so in case E is additionally uniformly convex. Strongly continuous (strong -to -weak) perturbations of operators satisfying condition (S) also ful ll this condition. Note that a strongly continuous operator is necessarily compact, that is continuous and sending bounded sets into compact ones.
As far as the continuity is concerned there are a few notions which are equivalent for monotone operators. Operator A : E → E * is called: For any operator each former continuity notion implies the latter. We say that A : E → E * is coercive when Moreover, operator A is also hemicontinuous and demicontinuous.
Operator A : E → E * is called potential, if there exists a Gâteaux di erentiable functional f : E → R, called the potential of A, such that f = A. For a radially continuous potential operator A : E → E * its potential f : E → R satis es that When A : E → E * is potential and monotone then its potential f : E → R is sequentially weakly lower semincontinous and operator A is demicontinous.
We will use the following corollary from the Minty-Browder Theorem: Theorem 2. Assume that operator A : E → E * is radially continuous, strictly monotone and coercive. Then A is invertible and A − : E * → E is strictly monotone, bounded and demicontinuous. If additionally A satis es condition (S), then A − is continuous.

Remark 3. We see that a continuous and strictly monotone operator which ful lls condition (S) is a homeomor-
phism between E and E * . As an example of such a mapping we can invoke a normalized duality mapping in case when both spaces E and E * are uniformly convex.
We will need also some variational arguments, see [5]. We say that a Gâteaux di erentiable functional J satis es the Palais-Smale condition, denoted (PS) condition, if any sequence (un) ⊂ E such that i) |J(un)| ≤ M for all n ∈ N and some xed M > , ii) lim n→+∞ J (un) = in E * admits a norm convergent subsequence.
Theorem 4 (Ekeland Variational Principle -di erentiable form). Let functional J : E → R be Gâteaux di erentiable, lower semicontinuous and bounded from below. Then there exists a minimizing sequence (un) consisting of almost critical points, i.e. such that Theorem 5. Let J : E → R be a Gâteaux di erentiable and a lower semicontinuous functional which satis es the (PS) condition. Suppose in addition that J is bounded from below. Then J is coercive and the in mum of J is achieved at some u ∈ E which is a critical point of J.

Auxiliary results for uniformly monotone operators
In [6] it is proved that a uniformly monotone operator is coercive, however a de nition of a uniformly monotone operator is a bit di erent, i.e. it is said that if there exists an increasing function ρ : With our de nition similar result holds.

Lemma 6. A uniformly monotone operator
Proof. We follow [6]. Indeed, take u ∈ E, u ≠ , and de ne There is a su cient conditions for an operator to be uniformly monotone: Then A is uniformly monotone.
Proof. Observe that for any u, v ∈ E it holds Remark 8. The assumptions of the above lemma may seem awkward, but we recall that for mappings between Banach spaces the Gâteaux di erentiability might be hard to be achieved, see for example Theorem 2.7 in [5].
Unfortunately it seems that one cannot nd su cient condition for A to be d−monotone in terms of the derivative of A.

Remark 9.
We note that the potential of a monotone operator which equals 0 at 0 is bounded from below. From the example of A (x) = exp (x) we see that the potential of a monotone mapping need not be (weakly) coercive.
Problems mentioned above do not appear when A is coercive. In fact we have the following known result for which we provide a simple proof. Proof. For any t ∈ ( , ) and any u ∈ E we have by the coercivity of A: for some weakly coercive function ρ : [ , +∞) → R. Therefore by (2.3) it follows from (3.4) by integration that Therefore we see that u ρ (s) ds → +∞ as u → +∞.
Corollary 11. Assume that A : E → E * is potential and d−monotone with respect to some coercive function or else that A is uniformly monotone. Then its potential f given by (2.3) is coercive.

. The direct method and the convergence of the minimizing sequence
We consider the existence of a solution to the following problem where A is a potential mapping and where element h ∈ E * is assumed to be xed in what follows. Functional J : E → R is de ned by The followings two lemmas are known.

Lemma 12. Assume that A : E → E * is a radially continuous and potential operator. Then a minimizer u ∈ E of the action functional (4.6) is a solution to (4.5).
Lemma 13. Assume that operator A : E → E * is monotone, coercive and potential. Then there is a solution u to (4.5) which minimizes functional (4.6), i.e. there is a sequence (un) ⊂ E such that un u and for which A solution is unique in case operator A is strictly monotone.
In order to formulate conditions leading to the conclusion that the minimizing sequence is strongly convergent we need some structure condition on operator A in (4.5). This is however common in case of the application of the direct method, see for example [4,8]. The main result now follows:

Theorem 14. Let operator A : E → E * be monotone, radially continuous, potential and satisfying condition (S). Let operator T : E → E * be potential and strongly continuous. Let also A + T be coercive. Then there is a solution u to
which minimizes functional J : E → R de ned by Proof. We see that functional v → T (sv) , v ds is weakly continuous. Indeed, since T is strongly continuous, it follows that for any weakly convergent sequence vn v it holds T (vn) → T (v ) which implies that: is sequentially weakly l.s.c., it follows that J is sequentially weakly l.s.c. Now, by Lemma 10 it follows that functional J is coercive. Since J is also Gâteaux di erentiable, it has an argument of a minimum u which is a critical point, i.e. a solution to (4.7).
Since obviously functional J is bounded from below, it follows by Theorem 4 that there is a minimizing sequence (un) such that conditions (4.9) are satis ed. By the coercivity of J we can assume, taking a subsequence if necessary, that un u . Moreover, by Theorem 4, we see that the minimizing sequence can be chosen so that it consists of almost critical points. Now we prove that un → u . Recall that Note that we do not assume the coercivity of J in the above. However, the monotonicity of A is crucial in the proof. Thus it is interesting if there is a counterpart of the above result without the monotonicity. The assertion is obvious when J is C since in this case A is continuous. However, when A is monotone we can still consider its strongly continuous perturbations under assumption that a minimizer exists. Then any sequence approximating a minimizer is minimizing.
Proposition 19. Assume that operator A : E → E * is monotone, potential and T : E → E * is potential and strongly continuous. Let u be a solution to (4.7) minimizing (4.8), i.e. A (u ) + T (u ) = and J (u ) = min u∈E J (u). If (un) ⊂ E is such that un → u , then Proof. By the convexity of the potential of A we see that Since A is potential and monotone, it is demicontinuous. Since A is demicontinuous and monotone, it is locally bounded. This means that there is a constant M > such that for all n ∈ N it holds Moreover, since T is strongly continuous, we see that (4.10) holds. Then it follows that The above estimation means that J (un) → J (u ) as n → +∞.
We see from Theorem 14 that the sequence of Galerkin type approximation for equation (4.7) constitutes a candidate ( nite dimensional) minimizing sequence, possibly up to a suitably chosen subsequence. Namely, we have Theorem 20. Assume that operator A : E → E * is radially continuous, potential, monotone and satis es condition (S). Let operator T : E → E * be strongly continuous, potential and such that A + T is coercive. Then there is at least one solution u to (4.7). For any n ∈ N there exists at least one n−Galerkin type solution un to (4.7) and un k → u for some subsequence (un k ) of (un). Moreover, (un k ) is a minimizing sequence for J.
Proof. We need to demonstrate that (un k ) is a minimizing sequence. This follows directly from Proposition 19.

Applications . Applications to the mountain geometry
Theorem 14 provides some clue as far as checking the (PS) condition is concerned in case the action functional has some special structure.
Theorem 21. Assume that operator A : E → E * is monotone, continuous, potential and satis es condition (S). Let operator T : E → E * be potential and strongly continuos. Then any bounded (PS) sequence for functional J given by (4.8) admits a strongly convergent subsequence.
Proof. Let (un) be a bounded (PS) sequence for J. We assume (un) to be weakly convergent to some u , up to a subsequence, and proceed as in the proof of Theorem 14.
The applicable version of the above is as follows Corollary 22. Assume either that E is additionally a uniformly convex space and operator A : E → E * is d−monotone, continuous and potential or else that operator A is uniformly monotone. Assume that operator T : E → E * is potential and strongly continuos. Then any bounded (PS) sequence for functional J given by (4.8) admits a strongly convergent subsequence.
The above arguments allow us to make a comment on a Rabier Theorem in which bounded Palais-Smale sequences in problems without parameter are obtained, see [10]. We recall the following result: I(g (t)), (5.11) where Then functional I possesses a bounded Palais-Smale sequence at level c.
Using our observation we see that

. Applications to boundary value problems
It is apparent what types of problems can be considered as examples. Let Assume that Ω ⊂ R N is a bounded region with locally Lipschitz boundary. Let q ∈ , p * . We may consider the following Dirichlet problem for all a ≥ b and for a.e. y ∈ Ω. F2 f : Ω × R → R is an L −Caratheodory function with f (y, ) = for a.e. y ∈ Ω, h ∈ L q (Ω), h ≠ . We see that in this case E = W ,p (Ω) is a uniformly convex, separable real Banach space. With assumption is a classical Euler action functional corresponding to (5.12) which is exactly (4.8). We put A = A + T. Then problem (5.12) can be considered in the equivalent form understood in the sense of space E * .
From the Rellich-Kondrashov Theorem we know that for q ∈ , p * the embedding W ,p (Ω) ⊂ L q (Ω) is compact. Let C be the constant from the Poincaré inequality u L p ≤ C ∇u L p , u W ,p := ∇u L p .
To assumptions F1, F2 we add the following: F3 there exists a constant a < γ (C ) −p such that f (y, x) x ≥ −a |x| p− for all x ∈ R and a.e. y ∈ Ω. We have the following: