Critical Point Result of Schechter Type in a Banach Space

Using Ekeland's variational principle we give a critical point theorem of Schechter type for extrema on a sublevel set in a Banach space. This result can be applied to localize the solutions of PDEs which contain nonlinear homogeneous operators .


Introduction
Finding the set of solutions of certain PDEs is closely related to the investigation of the critical points of a certain functional defined on an appropriate Hilbert or Banach space.Mountain pass theorems, saddle point theorems, linking theorems, mountain cliff theorems give sufficient conditions for the existence of a minimizer for a certain differentiable functional defined on the whole space or on a bounded region (for example, see [3,[16][17][18][19]21]).
In [13][14][15] R. Precup studies critical point theorems of Schechter type for C 1 functionals on a closed ball and also on a closed conical shell in a Hilbert space by using Palais-Smale type compactness conditions and also Leray-Schauder conditions on the boundary.These results can be used successfully to localize the solutions of PDEs involving the Laplace operator.
In our paper we improve the above mentioned Schechter type results (on a ball) for sublevel sets in locally uniformly convex Banach spaces and then apply our result for localizing the solutions for p-Laplace type equations on bounded, and also on unbounded domains.
The paper is structured as follows: Section 2 contains certain preliminaries concerning duality mappings on Banach spaces and the assumptions for the critical point problem which we are investigating.Section 3 states the main result of our paper.Section 4 presents two examples of localizing the solutions for problems containing the p-Laplacian.

Preliminaries
Let X be a real Banach space, X * its dual, •, • denotes the duality between X * and X.The norm on X and on X * is denoted by • .
The duality mapping corresponding to the normalization function ϕ is the set valued operator J ϕ : X → P (X * ) defined by Assumption (A1): X and X * are locally uniformly convex reflexive Banach spaces.
Observe that X * is strictly convex, because a locally uniformly convex Banach space is also strictly convex, see [5,Theorem 3,p. 31].Then it follows that card(J The following result holds.
Theorem 2.1.[6, Theorem 5, p. 345] Let X be a reflexive, locally uniformly convex Banach space and J ϕ : X → X * .Then J ϕ is bijective and its inverse J −1 ϕ is bounded, continuous and monotone.Moreover, it holds J where χ : X → X * * is the canonical isomorphism between X and X * * and J * ϕ −1 : X * → X * * is the duality mapping on X * corresponding to the normalization function ϕ −1 .We consider J : X * → X defined by J = J −1 ϕ .By Theorem 2.1 it follows that J is bounded, continuous and monotone.For w ∈ X * denote v = J −1 ϕ w and compute and the operator H maps bounded sets into bounded sets.

Assumption (A3):
We introduce some auxiliary mappings: Lemma 2.2.Assume that (A1) holds, and that F : X R → R and H : X → R are C 1 functions.For all u ∈ N R the following properties hold: (1) Proof.Let u ∈ N R be arbitrary.We compute Observe that By using the statement (1) of this lemma, by (2.2) and (2.1) we have 3 Main result Theorem 3.1.Assume that (A1), (A2) and (A3) are satisfied.Then, there exists a sequence (x n ) n ⊂ X R such that F(x n ) → inf F(X R ) and one of the following statements hold If, in addition, there exists a ∈ R + such that and F satisfies a Palais-Smale type compactness condition (i.e.any sequence satisfying (a) or (b) has a convergent subsequence) and the following boundary condition holds then there exists x ∈ X R such that Proof.By Ekeland's variational principle, see [8, Theorem 1, p. 444], applied for X R (we use here that H is continuous, hence X R is a closed set), the distance d(x, y) = x − y , the function F (which is continuous and bounded by below, see (A3)), ε = 1 n and for u ∈ X R such that it follows that there exists a sequence (x n ) n in X R such that and Since (x n ) n belongs to X R , we distinguish two cases: (1) there exists a subsequence of (x n ) n , still denoted by (x n ) n , such that H(x n ) < R for each n ∈ N; (2) there exists a subsequence of (x n ) n , still denoted by Case (1) Fix n ∈ N. Let t > 0 and z ∈ X such that z = 1.Since H is a continuous function and H(x n ) < R, we have that there exits δ > 0 (small enough) such that H(x n − tz) < R for each t ∈ (0, δ).Hence x n − tz ∈ X R \ {x n } for each t ∈ (0, δ) and by (3.2) it holds By taking t 0 it follows But z ∈ X with z = 1 was arbitrary chosen, hence F (x n ) ≤ 1 n , which yields F (x n ) → 0 as n → ∞.Hence we constructed a sequence (x n ) n which satisfies the statement (a) of this theorem.
then by t 0 we get 2) we obtain for t sufficiently small This inequality and (3.4) imply Further we have two possible cases.

Case (2a)
There exists a subsequence of (x n ) n , which we still denote by (x n ) n , such that By the property (2.1) of J it follows that which yields and we obtained a sequence (x n ) n which satisfies the statement (a) of this theorem.

Case (2b)
There exists a subsequence of (x n ) n , which we still denote by Hence, Denote the kernel of H (x n ) by K n = {x ∈ X : H (x n ), x = 0} and the projection mapping Since v ∈ X → H (x n ), v is linear and continuous, it follows that P n is also linear and continuous.
Since (x n ) n ⊂ N R and the level set N R is bounded, it follows that (x n ) n is a bounded sequence.By the assumption on H it follows that (H (x n )) n is also bounded and there exists We write Hence there exists α R > 0 (independent of n) such that Then by (3.6) we have This yields D(x n ) → 0 as n → ∞.Hence we constructed a sequence (x n ) n which satisfies statement (b) of this theorem.If, in addition, F satisfies the (PS) type compactness condition.
Case (a) F (x n ) → 0 as n → ∞ and there exist x ∈ X R and a subsequence (x n k ) k such that Applying the operator J we get which yields Since we are investigating Case (b), it follows that ( H (x n ), JF (x n ) ) n is a bounded sequence in R, hence there exist b ∈ R, b ≤ 0, and a subsequence, denoted again by (x n k ) k , such that Since H (x), x > 0 (assumption on H and the fact that x ∈ N R as the limit of (x n k ) k ), we obtain F (x), x ≤ 0.
which contradicts the assumption (3.1) from the statement of this theorem.

Example 1
Consider the Sobolev space W 1,p 0 (Ω), where Ω is a bounded domain in R k with Lipschitz continuous boundary and 1 < p < ∞, equipped with the norm .
Consider the following Dirichlet problem involving the p-Laplacian: We introduce F : W 1,p 0 (Ω) → R defined by We have (see [9, Theorem 7]) The critical points of F are the solutions of (4.3).
(A4) Assumptions for R: denote by C an upper bound for C q and suppose that one of the following three assumptions is satisfied.
(1) If p > q: let R > 0 be a solution of the inequality in R .
(2) If p = q: assume 1 > C p a L ∞ (Ω) and let R be such that and let R > 0 to be a .
Proposition 4.1.The following relation holds where R satisfies one of the three conditions mentioned in (A4).
Proof.We reason by contradiction: assume that there exist u ∈ N R and µ > 0 such that F (u) + µH (u) = 0, which implies By our assumptions Using (4.1) and (4.4) we get p , and we obtain The assumptions in (A4) imply that (4.5) cannot be satisfied.
Proposition 4.2.Suppose that R satisfies one of the three conditions mentioned in (A4).Then F satisfies the following Palais-Smale type compactness condition: if (u n ) n is a sequence from X R such that one of the following statements hold then (u n ) n admits a convergent subsequence.
Proof.Since the sequence (u n ) n is bounded in W 1,p 0 (Ω) (it belongs to X R ) and since the embedding W 1,p 0 (Ω) → L q (Ω) is compact for q ∈ (1, p * ), there exist u ∈ W 1,p 0 (Ω) and a subsequence of (u n ) n , which we denote again by (u n ) n , which converges weakly in W 1,p 0 (Ω) to u and strongly in L q (Ω) to u.Then by Hölder's inequality we have The (S + ) property of H = J ϕ (see [6,Theorem 10]) implies (u n ) n converges strongly to u.

Case (b):
We denote If µ = 0, the above convergence implies As in Case (a) it follows that there exist u ∈ W 1,p 0 (Ω) and a subsequence of (u n ) n , which we denote again by (u n ) n , which converges strongly in W This contradicts the assumption on R from (A4).Hence, the case µ = 0 is not possible.For µ = 0 we have by (4.7) The (S + ) property of H = J ϕ implies (u n ) n converges strongly to u.The convergence (4.7) and the strong convergence But H (u), JF (u) ≤ 0 and F (u), JF (u) ≥ 0, therefore µ < 0 and the relation F (u) − µH (u) = 0 contradicts the statement of Proposition 4.1.Hence, the case µ = 1 is not possible.
For µ = 1: by (4.7) it follows that Using (2.1) we have and by the assumptions of Case (b) we have As in Case (a) it follows that there exist u ∈ W 1,p 0 (Ω) and a subsequence of (u n ) n , which we denote again by (u n ) n , which converges strongly in W 1,p 0 (Ω) to u.This yields u ∈ N R , because (u n ) n belongs to the closed set N R .Since F and H are continuous, we have by the convergence (4.7) that F (u) − H (u) = 0, F (u), JF (u) = H (u), JF (u) .But H (u), JF (u) ≤ 0 (by the assumption of Case (b) and by the strong convergence u n → u) and F (u), JF (u) ≥ 0, hence F (u) = 0, which implies H (u) = 0 and then u = 0.But 0 / ∈ N R , contradicts u ∈ N R .Hence the case µ = 1 is not possible.
We apply Theorem 3.1 in order to localize the solution of (4.3).
Theorem 4.3.Suppose that R satisfies one of the three conditions mentioned in (A4).Then, equation (4.3) admits a weak solution u ∈ X R , which minimizes F on X R .
In what follows we discuss situations when the best Sobolev constant C q admits an upper estimate which can be computed: Denote the first eigenvalue of the p-Laplace operator by λ p (Ω) = min Hence the best embedding constant of W 1/p , while for q < p the best embedding constant of W 1,p 0 (Ω) → L q (Ω) verifies (via Hölder's inequality) 1/p (here |Ω| denotes the Lebesgue measure, i.e. the k-dimensional volume, of the set Ω).In order to obtain upper bounds for C q (q ≤ p) we need lower bounds for λ p (Ω).
By using the Faber-Krahn inequality [2, Theorem 1] it holds where Ω * is the k-dimensional ball centered at the origin having the same volume as Ω.So it has the radius r = 1 By [10] we have for the ball Ω * = B r ⊂ R k of radius r the inequality Then the best Sobolev constant has the following upper estimate, which can be computed: .
For k = 1 and Ω = (0, T) ⊂ R the value of the first eigenvalue is known (see [7]) For the case k = 1 and Ω = (0, T) the sharp Poincaré inequality is known (see [20], p. 357): for each p > 1, q > 1 and u ∈ W 1,p 0 (0, T) it holds where the embedding constant is given by q , p = p p−1 and B is the Beta function.

Example 2
For 1 < p < ∞ we define the following subspace of radially symmetric functions of The space W In the context of Section 2 and Section 3 we consider X = W 1,p r (R k ) endowed with the above norm • , X is a closed subspace of W 1,p (R k ).Hence it is also uniformly smooth and by [4, Theorem 2.10] it follows that its dual X * is uniformly convex.
Let J ϕ : X → X * be the duality mapping corresponding to the weight function ϕ(t) = t p−1 , t ∈ R + where p ∈ (1, +∞) (see [3,Proposition 2.2.4]).It is well known that the duality mapping J ϕ satisfies the following conditions: Moreover, the functional H : X → R defined by H(u) = 1 p u p is convex and Fréchet differentiable with H = J ϕ .We take J = J −1 ϕ .It is known [11,Théorème II. 1] that the embedding W 1,p r (R n ) → L q (R n ) is compact for q ∈ (p, p * ) (where k ≥ 2, p * = kp k−p if p < k and p * = ∞, if p ≥ k) there exists C q > 0 (the best embedding constant) such that u L q (R k ) ≤ C q u for each u ∈ W  where a ∈ L ∞ (R k ), b ∈ L q q−1 (R k ) are positive functions and q ∈ (p, p * ) and f (x, •) = f (x , •) for all x, x ∈ R n , |x| = |x | ( f is radially symmetric in the first variable).
Consider the following problem involving the p-Laplacian: − ∆ p u + |u| p−2 u = f (x, u) a.e.x ∈ R k .(4.9) We call u ∈ W 1,p (R k ) a weak solution of (4.9) if for each v ∈ W 1,p (R k ) it holds where h : Ω × R → R is h (x, t) = t 0 f (x, s) ds.We have F (u) = H (u) − N f (u) .
Let G = O(R k ) be the set of all rotations on R k .Observe that the elements of G leave R k invariant, i.e. g(R k ) = R k for all g ∈ G. G induces an isometric linear action over W 1,p (R k ) by (gu)(z) = u(g −1 z), g ∈ G, u ∈ W 1,p (R k ), a.e.z ∈ R k . .
Obviously this implies u L p * (R k ) ≤ C R u for each u ∈ W 1,p (R k ).
For any q ∈ (p, p * ) there exists θ ∈ (0, 1) such that q = θ p + (1 − θ)p * , then by Hölder's inequality for each u ∈ W 1,p (R k ).Then the Sobolev constant has the following upper estimate, which can be computed: for q ∈ p, kp k − p and 1 < p < k.