A SCHECHTER TYPE CRITICAL POINT RESULT IN ANNULAR CONICAL DOMAINS OF A BANACH SPACE AND APPLICATIONS

. Using Ekeland’s variational principle we obtain a critical point theorem of Schechter type for extrema of a functional in an annular conical domain of a Banach space. The result can be seen as a variational analogue of Krasnoselskii’s ﬁxed point theorem in cones and can be applied for the existence, localization and multiplicity of the positive solutions of variational problems. The result is then applied to p -Laplace equations, where the geometric condi- tion on the boundary of the annular conical domain is established via a weak Harnack type inequality given in terms of the energetic norm. This method can be applied also to other homogeneous operators in order to obtain existence, multiplicity or inﬁnitely many solutions for certain classes of quasilinear equations.


1.
Introduction. The bounded critical point method is a very useful tool to study the existence and localization of solutions of nonlinear equations. Some references are as follows [9,10,12,15,21]. We particularly mention Schechter's theory [25,26] which yields critical points of a C 1 functional in a ball of a Hilbert space, by taking into account boundary conditions of Leray-Schauder type. A result of this type is the following: Theorem 1.1 (Schechter). If X is a Hilbert space with inner product ·, · and norm · , R > 0 and F : X → R is a C 1 functional bounded from below on the ball B R = {x ∈ X : x ≤ r} with F (x) , x ≥ a > −∞ for all x ∈ X with x = R and for some real number a, then there exists a sequence (x n ) such that either F (x n ) → inf F (B R ) and F (x n ) − F (x n ) , x n R 2 x n → 0.

HANNELORE LISEI, RADU PRECUP AND CSABA VARGA
If in addition, any sequence as above has a convergent subsequence, and the following boundary condition holds then there exists x ∈B R such that F (x) = 0 and F (x) = inf F (B R ).
It deserves to be noted that if F (x) has the representation F (x) = x − T (x) , then the critical points of F are exactely the fixed points of T and the boundary condition (3) becomes the Leray-Schauder condition for the operator T, namely T (x) = λx for x = R and λ > 1 (for this aspect, see [18]).
Recently, the second author gave in [19] and [20] similar results for the localization of critical points in an annular conical set of a Hilbert space, by analogy to Krasnoselskii's fixed point theorem in cones [13], and having as main motivation the possibility of the investigation of multiple, possibly infinitely many solutions of nonlinear problems. The approach in [19] and [20] was based on Ekeland's variational principle and, as in [25] and [26], was essentially tributary to the special geometry of Hilbert spaces. It is the aim of the present paper to give a version in Banach spaces, and show its applicability to boundary value problems involving the p-Laplacian. We note that, due to the nonlinearity of the duality mapping, the extension from Hilbert to Banach spaces is not immediate and requires a major refining of the reasoning and a formally different statement of the results (compare (2) and (16) below). However, the extension will be done in such a way that it reduces to the well-known results in case of Hilbert spaces.
In the papers [1,7,14,16,23] using a variational principle of B. Ricceri, see [22], the authors prove the existence of infinitely many solutions for linear and quasilinear equations, while in the papers [7] and [24] the authors use Marcus-Mizel type result [17] to study the existence of infinitely many solutions of certain equations. We mention here that our method can be applied to such type of problems and it is different from the methods mentioned in the above cited papers. Our method can also be applied to study the existence, multiplicity or infinitely many solutions for anisotropic equations, see [2,3,8].
The applicability of the abstract result is then illustrated by the two-point Dirichlet boundary problem for p-Laplace equations. In this case, the geometric condition on the boundary of the annular conical domain where the solution is sougth is established via a weak Harnack type inequality for positive p-superharmonic functions. The essential feature of this inequality consists in its expression in terms of the energetic norm. Once the existence of a solution is establised in an annular set, we shall be able to obtain a finite or infinite number of solutions for problems with oscillating nonlinearities.
We conclude this introduction by a weak version of Ekeland's variational principle [6], enough for our approach.
2. Main abstract result. Let X be a real Banach space, X * its dual, ·, · denotes the duality between X * and X, and let the norms on X and X * be denoted by the same symbol · . We shall denote by J the duality mapping corresponding to the normalization function ϕ (t) := t p−1 (t ∈ R + ) , where p > 1, i.e. the set-valued operator J : X → P(X * ) defined by Obviously, for every x ∈ X and λ ∈ R.
It is known, see [5,Theorem 3,p. 31], that if that X * is strictly convex, then J is single-valued and so Also, if in addition X is reflexive and locally uniformly convex, then J is demicontinuous and bijective and its inverseJ is bounded, continuous and monotone. In what follows we shall assume that the following condition holds: Assumption (A1). X and X * are locally uniformly convex reflexive Banach spaces and J is locally strongly monotone, i.e., there is β > 1 such that for each ρ > 0 there exists a constant a = a (ρ) > 0 with for all x, y ∈ X satisfying x ≤ ρ and y ≤ ρ.
Let K be a wedge of the Banach space X, i.e. a closed convex subset of X such that K = {0} and λK ⊂ K for every λ ∈ R + . Notice that K can be a cone, i.e. may have the property K ∩ (−K) = {0}, and also can be the whole space X.
We shall localize critical points x of F by means of a functional G which verifies suitable assumptions (see for instance assumption (A2)). More exactly, for two fixed numbers r, R with 0 < r < R, we shall look for x ∈ K such that F (x) = 0 and r ≤ G (x) ≤ R. Hence we seek critical points of F in the annular conical set

Denote by
the two parts of the boundary of K r,R which are assumed non-void. Our second assumption is as follows: and inf As for the functional F, we shall assume: Assumption (A3). F : X → R is a C 1 functional which is bounded from below on K r,R , and F maps bounded sets into bounded sets.
We introduce some auxiliary mappings: where µ will be chosen in a suitable way (see (13)) and λ is such that and there exists a = a (R) > 0 such that Next, from (8), (9) and (5), Notice that (5) applied since bothJ (µJx) ,J (µJx − Dx) are bounded independently on x ∈ N R ∪ N r as a consequence of (6), (7) and of the fact that J,J, F and G map bounded sets into bounded sets.
Lemma 2.2. Assume that (A1), (A2) and (A3) are satisfied. Let x ∈ K and z ∈ X be such that Then y ∈ K r,R for t > 0 small enough, in each of the following situations: Proof. In case (a), the conclusion follows from (10), the continuity of G, and the strict inequalities r < G (x) < R. Assume now that condition (b) holds. From the definition of the Fréchet derivative of G, for each ε > 0, there exists δ ε > 0 such that for each t ∈ (0, δ ε ) we have Hence Since G (x), z > 0, we may take ε := G (x), z to obtain Hence, for t sufficiently small such that r ≤ R − 2t G (x), z and t ∈ (0, δ ε ), we have r ≤ G(x − tz) ≤ R. This together with (10) shows that y ∈ K r,R for t > 0 small enough. Finally, if (c) holds, then (11) gives In this case we may take ε := − G (x), z and obtain Hence, for t sufficiently small such that The next lemma is about the condition (10). It requires some compatibility conditions with respect to the wedge K.
for all x ∈ K, and for each ρ > 0, there exists µ ρ > 0 such that if x ∈ K and for some µ (depending on x) with |µ| ≤ µ ρ .
Notice that (A4) is trivially satisfied in case that K is the whole space X. (i) One has that x−t x −J (Jx − F (x)) ∈ K r,R for all t > 0 sufficiently small, in each of the following conditions: then for every ε > 0, one has x − t (εx + E (x)) ∈ K r,R for all t > 0 sufficiently small. (iii) If x ∈ N r , then for every ε > 0, one has x − t (−εx + E (x)) ∈ K r,R for all t > 0 sufficiently small.
Proof. (i) First note that using (12), the representation and the convexity of K yield that x − t x −J (Jx − F (x)) ∈ K for every t ∈ (0, 1) . Then the conclusion of (i) follows from Lemma 2.2.
(iii) We proceed as at the case (ii) and find that and we obtain as above that y ∈ K for all small enough t > 0. Now we are ready to state and prove our main result of this section.
Theorem 2.4. Assume that (A1), (A2), (A3) and (A4) are satisfied. Then there exists a sequence (x n ) ⊂ K r,R such that and one of the following statements holds: where µ n = µ (x n ) is chosen accordingly to (13); If in addition, F satisfies a Palais-Smale type compactness condition guarantying that any sequence as above has a convergent subsequence, and the following boundary conditions hold then there exists x ∈ K r,R such that F (x) = inf F (K r,R ) and F (x) = 0.
Proof. We shall apply Ekeland's variational principle for M := K r,R (we use here that K is closed and G is continuous, hence K r,R is a closed subset of the Banach space X) endowed with the metric d(x, y) := x − y , for the function F (which from (A3) is C 1 and bounded from below), and for ε := 1 n (n ∈ N \ {0}) . It follows that there exists a sequence (x n ) in K r,R such that and F (x n ) ≤ F (y) + 1 n x n − y for every y ∈ K r,R .
Since (x n ) belongs to K r,R , we distinguish three cases: There exists a subsequence of (x n ), still denoted by (x n ), in one of the following situations: (i 1 ) r < G(x n ) < R for all n; (i 2 ) x n ∈ N R and G (x n ), x n −J [Jx n − F (x n )] > 0 for all n; (i 3 ) x n ∈ N r and G (x n ), x n −J [Jx n − F (x n )] < 0 for all n.
Case 2. There exists a subsequence of (x n ), still denoted by (x n ), such that x n ∈ N R and G (x n ), x n −J [Jx n − F (x n )] ≤ 0 for all n.
Case 3. There exists a subsequence of (x n ), still denoted by (x n ), such that x n ∈ N r and G (x n ), x n −J [Jx n − F (x n )] ≥ 0 for all n. Assume Case 1. According to Lemma 2.3 (i), for each n, we have y := x n − t x n −J [Jx n − F (x n )] ∈ K r,R , for all t > 0 sufficiently small. Thus we may apply (20) and deduce Divide by t and let t go to zero to obtain It follows that From (5), Using this inequality in (21) we deduce that Hence x n −J [Jx n − F (x n )] → 0 as n → ∞ and so, property (a) holds in Case 1. Assume Case 2. Now Lemma 2.3 (ii) guarantees that for each n and any ε > 0, y := x n − t (εx n + E (x n )) ∈ K r,R for all t > 0 sufficiently small. Then (20) implies Letting ε → 0 and using Lemma 2.1 we deduce Let us consider the continuous linear operator Since (x n ) ⊂ N R and the level set N R is bounded, it follows that (x n ) is a bounded sequence. By the assumption on G it follows that (G (x n )) is also bounded. In addition We have Hence there exists α R > 0 (independent on n) such that P n x ≤ α R x , for all x ∈ X and n ≥ 1.
Assume now that the additional hypotheses of the theorem are satisfied. The (PS) condition guarantees the existence of a subsequence of (x n ) , which is still denoted by (x n ) , such that x n → x as n → ∞, for some element x ∈ K r,R . Clearly, (15) gives F (x) = inf F (K r,R ). In case of the property (a), if we denote y n := x n −J (Jx n − F (x n )) , then y n → 0 as n → ∞, and from letting n → ∞ and using the continuity of F and the demicontinuity of J, we obtain F (x) = 0 and the proof is finished. Assume that the property (b) holds. Then, if we pass to the limit we obtain where µ is the limit of some convergent subsequence of (µ n ) . Notice that such a subsequence exists since according to (A4), |µ n | ≤ µ ρ , where ρ is a bound for the sequence ( x n ) . Next from (24) In case that η = 0, (25) shows that F (x) = 0 and we are done. Assume η = 0. From (25), This together with (23) gives we may infer that η > 0. Then x ∈ N R , η > 0 and F (x) + ηG (x) = 0, which contradicts (17). Thus the case η = 0 can not occur. The case of the property (c) is similar.
3. Application. In this section we present an application of Theorem 2.4 for the localization in annular conical domains of the positive solutions of the two-point boundary value problem where p > 1, f is a continuous function on R, which is nonnegative and nondecreasing on R + . Hence all possible nonnegative solutions are concave functions on [0, 1] .
Let us consider the cone of all nonnegative functions in W 1,p 0 (0, 1) which are symmetric with respect to the middle of the interval [0, 1] , namely We can immediately see that the assumption (A2) holds. As concerns assumption (A3), note that F is bounded from below on the intersection of K with each ball of W 1,p 0 (0, 1). Indeed, if u ∈ K and u 1,p ≤ ρ, then for all t ∈ [0, 1] , where 1/p + 1/q = 1. Next, since f is nonnegative on R + , g is nondecreasing on R + and thus .
Hence the assumption (A3) also holds. In order to check assumption (A4), we first show that the condition (12) is satisfied. Indeed, if u ∈ K and we let v :=J (Ju − F (u)) , then Jv = Ju − Ju + f (u) , that is Jv = f (u) . Since f (u) ≥ 0, one has v ≥ 0. On the other hand, the symmety of u with respect to 1/2 is obviously passed to f (u) , and then to v. The last assertion follows from the fact that if h is symmetric with respect to 1/2, and v (t) solves Jv = h, then by a direct computation, we have that v (1 − t) also solves it. Then, the uniqueness of the solution yields v (t) = v (1 − t) , i.e. v is symmetric with respect to 1/2. Therefore v ∈ K as desired.
Next we show that the condition (13)  which as above yields the conclusion v ∈ K. On the other hand, for each ρ > 0, there is c (ρ) > 0 with |η (u)| ≤ c (ρ) for every u ∈ K with u 1,p ≤ ρ. Then |µ| = |1 + η (u)| ≤ 1 + c (ρ) =: µ ρ for all u ∈ K with u 1,p ≤ ρ. Thus, assumption (A4) is satisfied. Before we state and proof the main result of existence and localization for the problem (27), we give the weak Harnack type inequality for p-superharmonic symmetric functions on [0, 1] , which is essential for the estimations from below on the part G (u) = r of the boundary of K r,R .