Schwarz Symmetric Solutions for a Quasilinear Eigenvalue Problem

In the present paper we extend some recent results of R. Filippucci, P. Pucci and Cs. Varga to continuous functionals. As an application we prove the existence of at least three different solutions of a quasilinear eigenvalue problem, for every λ in some interval, which solutions are invariants by Schwarz symmetrization.


Introduction
There is a rich literature on the study of symmetric solutions of the PDE's.A very important paper is due to Gidas, Ni, Nirenberg [9], where they prove symmetry, and some related properties, of positive solutions of second order elliptic equations.Their methods employ various forms of the maximum principle, and a device of moving parallel planes to a critical position.After this work appeared many papers where the solutions of PDE's have different symmetries for e.g.radial symmetry (see e.g.Pacella, Salazar [8], Squassina [11]), axial symmetry or have some symmetry properties with respect to certain group actions.
In articles [13,14] Van Shaftingen developed an abstract method for the study of symmetrization.In [15] Van Shaftingen and Willem study different symmetry properties (spherical cap, Schwarz, polarization) of least energy positive or nodal solutions of semilinear elliptic problems with Dirichlet or Neumann boundary conditions.
Filippucci, Pucci, Varga in [6] using the symmetric version of Ekeland's variational principle (Van Schaftingen [13]) and the symmetric Mountain Pass theorem (Squassina [10]) establish the existence of two nontrivial (weak) solutions of abstract eigenvalue problems.In order to show the existence of three different symmetric solutions of an abstract eigenvalue problem, they prove a symmetric version of the Pucci and Serrin three critical points theorem.Then, as a consequence of the main results, they show the existence of two nontrivial nonnegative solutions of quasilinear elliptic Dirichlet problems either in a ball of R N , or in an annulus of R N , both centered at 0. The obtained solutions are invariant by k-spherical cap symmetrization (1 < k < N).
In the present paper we extend some of these results to continuous functionals.More precisely, let Ω be a ball in R N (N ≥ 3) and f : Ω × R → R be a Carathéodory function, which satisfies a natural growth condition, given in Section 2.
Consider the following quasilinear elliptic eigenvalue problem a ij (u)D i uD j u = λ f (x, u), where λ > 0 is a real parameter, a ij : R → R is of class C 1 with a ij (x) = a ji (x) and by D i we denote the partial derivative with respect to x i .We also assume that there exist C, ν > 0 such that for and all s ∈ R, ξ ∈ R N we have be the corresponding energy functional, where F(x, t) = t 0 f (x, s)ds.Under the above conditions the energy functional is continuous.However, we cannot expect that E λ to be of class C 1 or even locally Lipschitz continuous, so, the classical critical point theory cannot be applied.To overcome this difficulty, we define the derivative of the function only in some special direction.Such techniques has been used for quasilinear problems by several authors (see e.g.Canino [1]; Liu, Guo [3] and the references therein).
The aim of our paper is to prove the existence of at least three different solutions of the quasilinear eigenvalue problem (1.1) for every λ in some interval.Moreover, we prove that the obtained solutions are symmetric invariants by Schwarz symmetrization.A comprehensive survey of results about existence, multiplicity, perturbation from symmetry and concentration phenomena for the quasilinear elliptic equations can be found in the monograph of Squassina [12].
Our paper is organized as follows.In Section 2 we present the necessary symmetrization tools.We begin with the abstract framework of symmetrization following Van Schaftingen [13] and in Section 2.2 we obtain symmetric critical point results for E-differentiable continuous functions f : X → R, where E is a dense subspace of X.
In order to demonstrate the main results of the present paper, in Section 3 we study first an abstract eigenvalue problem where λ > 0 is a real parameter, and we give some information about the symmetry of solutions, when the underlying domain is a ball of R N and f : Ω × R → R is a Carathéodory function, which satisfies natural growth conditions given in Section 3. In Theorem 3.1 we guarantee three symmetric invariant critical points for the continuous functional f .Using the aforementioned theorem, in Section 4 we justify the existence of three different, Schwartz symmetric critical point of the E-differentiable energy functional E λ in (1.2).

Abstract framework of symmetrization
In this subsection we recall some symmetrizations notions from Van Schaftingen [13] and Squassina [10].
Let us begin with some notion of symmetrizations.
Definition 2.1.The Schwarz symmetrization of a set A ⊂ R N is the unique open ball centered at the origin A * , such that L N (A * ) = L N (A), where L N denotes the N-dimensional outer Lebesgue measure.
Definition 2.2.Let f : A → R a function and c ∈ R. Then we define the following set The Schwarz symmetrization of a measurable nonegative function f : Given x in R N and a polarizer H the reflection of x with respect to the boundary of H is denoted by x H .
The polarization of a function u : R N → R + by a polarizer H is the function u H : R N → R + defined by The polarization C H ⊂ R N of a set C ⊂ R N is defined as the unique set which satisfies χ C H = (χ C ) H , where χ denotes the characteristic function.The polarization u H of a positive function u defined on C ⊂ R N is the restriction to C H of the polarization of the extension ũ : R N → R + of u by zero outside C. The polarization of a function which may change sign is defined by u H := |u| H , for any given polarizer H.
In the following we present some crucial abstract symmetrization and polarization results.We begin with the following main assumption.
Let X and V be two real Banach spaces, with X ⊂ V and let S ⊆ X.

Main assumptions.
Let H be a path-connected topological space and denote by h : S × H → S, (u, H) → u H , the polarization map.Let : S → S, u → u , be any symmetrization map.Assume that the following properties hold.
1) The embedding X → V is continuous; 2) h is continuous; 3) (u ) H = (u H ) = u and (u H ) H = u H for all u ∈ S and H ∈ H ; 4) for all u ∈ S there exists a sequence (H We assume that there exists a Lipschitz continuous map Θ : (X, and both maps h : S × H → S and : S → S can be extended to h : X × H → S and : X → V by setting u = (Θ(u)) H and u = (Θ(u)) for every u ∈ X and H ∈ H .The previous properties, in particular 4) and 5), and the definition of Θ easily yield that for all u, v ∈ X and for all H ∈ H . Now, we describe the set H * for the Schwarz symmetrization following the papers [6,13,14].
Consider the set of polarizers We endow the set H with a topology.To this aim let i : R N → R N be an isometry, that is, |i(x) − i(y)| = |x − y|, for every x, y ∈ R N and let I be the set of isometries on R N , that is where ∆ is the symmetric difference between two sets.The distance between H 1 and H 2 is defined by Remark 2.6.The metric space (H, d) is a separable, locally compact by Proposition 2.36 of [14].
We recall now, the notion of extended polarizer as given in Definition 2.17 of [13].
Definition 2.7 (Extended polarizer).The set of polarizers H is compactified by an addition of two polarizers at infinity defined by u Therefore, the set H * = H ∪ {H +∞ } is homeomorphic with S N \ {a point}, which is homeomorphic to R N .In conclusion, the space H * is a path-connected locally compact topological space and so H * satisfies all the properties required in the Main assumptions and will be used throughout the paper.
We recall here examples relative to the Schwarz symmetrization proved by Van Schaftingen in [13].
Example 2.8 (Schwarz symmetrization for non-negative functions).Let , with p = N p/(N − p), S be the set of non-negative functions of W 1,p 0 (Ω), * denotes the Schwarz symmetrization and H * be defined as above.Then the assumptions stated in the Main assumptions are satisfied, see [13].
denotes the Schwarz symmetrization and H * be defined as above for Schwarz symmetryzation, but h(u, H) = |u| H in the Main assumptions.Then all the assumptions stated in the Main assumptions are satisfied, see [13].In this case Θ(u) = max{0, u}.

Symmetric critical point results for E-differentiable functions
First, we recall from Guo, Liu [3] some notions and results of nonsmooth critical point theory.
Let X be a Banach space and E be a dense subspace of X.Let f : X → R be a continuous functional.Definition 2.10.A continuous functional f is said to be E-differentiable if (1) for all u ∈ X and ϕ ∈ E the derivative of f in direction ϕ at u exists and will be denoted by D f (u), ϕ : (2) the map (u, ϕ) → D f (u), ϕ satisfies: Definition 2.12.Let c be a real number.We say that an E-differentiable functional f satisfies the concrete Palais-Smale condition at level c (shortly In the following we recall the notion of the weak slope of a continuous functional from the paper of Canino [1].Definition 2.13.Let f : X → R be a continuous functional and let u ∈ X.We denote by |d f |(u) the supremum of the σ's in [0, ∞[ such that there exist δ > 0 and a continuous map The extended number |d f |(u) is called the weak slope of f at u.
We prove the next important lemma, which is used several times in the following.Lemma 2.14.Let f be an E-differentiable functional.Then for every u ∈ X we have Then by the definition of the slope, there exists v ∈ E, with v = 1 and Since D f (u), v is continuous in u, we can choose a δ > 0 such that for every w ∈ B(u, δ) we have (2.4) For δ = δ/2 we define the following continuous map It is trivial that G(w, t) − w = t.On the other hand, since D f (u), v is linear in v, by (2.4) we have that After a rearrangement we obtain Remark 2.15.Using Lemma 2.14, it is easy to verify that if f satisfies the (CPS) c condition, then f satisfies the (PS) c condition as well, for every real number c.
Using Lemma 2.14 and the Theorem 3.9 of Squassina in [11], we have the following lemma.
Lemma 2.16.Let X be a complete metric space and f : X → R a continuous functional.Let D and S denote the closed unit ball and sphere in R N , respectively, and Γ 0 ⊂ C(S, X).Let us define Then, for every ε ∈ (0, (c − a)/2), every δ > 0 and γ ∈ Γ such that Theorem 2.17 (Existence of a quasi-critical sequence).Let E be a dense subspace of X and f be a continuous E-differentiable functional defined on the Banach space X.We assume that f possesses two different local minima u 0 and u 1 in X.We define f (γ(t)).
Then there exist a sequence {u n } ∈ X \ {u 0 , Proof.Since u 0 and u 1 are distinct local minima of f in X, there exists r 0 > 0, with 2r 0 < We use the notations f (u 0 ) = c 0 and f (u 1 ) = c 1 and assume, without loss of generality, that c 0 ≥ c 1 .We distinguish two cases.
Then for r ∈ (0, r 0 ) and for every n ∈ N, there exist a sequence {z n } ∈ X with We fix now r ∈ (0, r 0 ) and choose n > 0 such that 0 < r − 2 n < r + 2 n < r 0 .We assume Then inf u∈U f (u) = c 0 .Now, we apply the Ekeland variational principle (Theorem 1.1 in I. Ekeland [2]) to f | U and u = z n .Then there exists a sequence {u n } ∈ U such that The first assertion follows from (i).
Using the relations (ii) and z n − u 0 = r, we get which means that u n ∈ int U .Let w be any vector in E, with ||w|| = 1, t ∈ R + and put v = u n + tw.Clearly if t > 0 is small enough, then v ∈ U and from (iii) it follows that Lemma 2.18.Let (X, V, , H , S) satisfy the Main assumptions.Assume that f : X → R is a continuous E-differentiable functional bounded from below such that f (u H ) ≤ f (u) for all u ∈ S and H ∈ H . (2.5) and for all u ∈ X there exists ξ ∈ S, with f (ξ) ≤ f (u).
If f satisfies the (CPS) inf f condition, then there exists v ∈ X, such that f (v) = inf f and v = v in V.
Proof.Put inf f = d.For the minimizing sequence (u n ) n we consider the following sequence: Since the embedding X → V is continuous by Main assumption 1), we have that v n → v in V, and using the second inequality of (2.2) we obtain v Theorem 2.19 (Existence of a third symmetric critical point).We assume that (X, V, * , H * , S = X) satisfy the Main assumptions.Let the functional f satisfy the (CPS) condition in X and verify the polarization condition (2.5).Suppose that the local minima u 0 and u 1 of f in X also verify a polarization condition: u H 0 = u 0 , u H 1 = u 1 for all H ∈ H * .Then f has at least a third critical point v, which is invariant by symmetrization in V, namely v = v * in V.
Proof.We prove this theorem in two steps.
Step 1. First, we prove the existence of a sequence {u n } ∈ X \ {u 0 , The proof is the same as in Theorem 2.17.
In Case 1, we use the inequality (2.2) and the assumption X = S. Then we can replace Lemma 2.16 by its symmetric version [11,Theorem 3.10].Thus for all n > c−a 2 and δ = 3 n , there exists where C Θ and K are some constants.Now, the assertion follows at once.
Step 2. Now, we apply the assertion of Step 1 for n sufficiently large.
In Case 1, the obtained sequence {u n } is a (CPS) sequence, so it possesses a subsequence which will be denoted also by {u n }, which is convergent to some v ∈ X, with f (v) = c > a, |D f (v)| = 0 and v = v * as seen in the proof of Lemma 2.18.
In Case 2, the constructed (CPS) sequence admits a subsequence converging to some v ∈ In both cases v is a critical point of f , different from u 0 and u 1 , and it is invariant by symmetrization in V.

Main result
Let Ω be a ball, Let (X, V, * , H * , S = X) satisfy the Main assumptions, where X is a reflexive, real Banach space, which verifies the following embedding condition: (EC) there exists p ∈]1, N[, such that the embedding X → L q (Ω) is continuous for q ∈ [1, p * ] and compact if q ∈ [1, p * [.We denote the best embedding constant by C q > 0, i.e. u q ≤ C q u , for all u ∈ X and q ∈ [1, p * ].
Further we assume, that J : X → R is a convex functional such that the following properties hold (J 2 ) J is continuous and weakly lower semicontinuous; (J 3 ) J(u H ) ≤ J(u) for all u ∈ X and H ∈ H * , where (X, V, * , H * , S = X) satisfies the Main assumptions, with V = L p (Ω) and 0 ∈ H for all H ∈ H * .
Let f : Ω × R → R be a Carathéodory function satisfying the following assumptions x ∈ Ω and all t ∈ R; ( f 2 ) F(x, t) = F(y, t) for a.e.x, y ∈ Ω, with |x| = |y|, and all t ∈ R; ( f 3 ) F(x, t) ≤ F(x, −t) for a.e.x ∈ Ω and all t ∈ R − ; We say that u ∈ X is a (weak) solution of the equation if be the energy functional associated to the problem (3.1).The critical points of the energy functional E λ are exactly the (weak) solutions of (3.1).
We use the following notation i) For every r ∈ F (X) \ sup u∈X F (u), the F −1 ([r, ∞)) ∩ X H is a non-empty, weakly closed subset of X and also F −1 (I r ) ∩ X H is non-empty, where I r = (r, ∞).Moreover, is well-defined.
ii) If furthermore ( f 3 ) holds, then for all r ∈ F (X) \ inf u∈X F (u) also the set F −1 ((−∞, r]) ∩ X H is a non-empty, weakly closed subset of X and F −1 (I r ) ∩ X H is non-empty, where now I r = (−∞, r).Furthermore, is well-defined.
The next two lemmas provide us two different symmetric local minima of the energy functional E λ .Lemma 3.3.Let (J 1 )-(J 3 ), ( f 1 )-( f 4 ) hold and let f be non constant.Assume that E λ is coercive, bounded below and there exists a real number r, with r ∈ F (X) \ sup u∈X F (u). Then the infimum of Moreover, u 0 is a local minimizer of E λ in X, u H 0 = u 0 for every H ∈ H , and F (u 0 ) > r.If ( f 3 ) is replaced by the stronger condition ( f 3 ) , then the result continues to hold for all λ ∈ R, with λ > ϕ 1 (r).Lemma 3.4.Let (J 1 )-(J 3 ), ( f 1 )-( f 3 ) and ( f 3 ) hold, and let f be non constant.Assume that E λ is coercive, bounded below and there exists a real number r, with r ∈ F (X) \ inf u∈X F (u). Then the infimum of E λ = J − λF in F −1 ((−∞, r]) ∩ X H is attained at some point u 1 , provided that λ ∈ R satisfies the inequality λ < ϕ 2 (r).Moreover, u 1 is a local minimizer of E λ in X, u H 1 = u 1 for every H ∈ H , and F (u 1 ) < r.Now, we can state the main result of this section, which extends the Theorem 3.12 of [6].Theorem 3.5.Let the functionals J, F and E λ satisfy the (J 1 )-(J 3 ) and ( f 1 )-( f 4 ) conditions.We assume in addition that (E 1 ) F (u H ) ≥ F (u), for every u ∈ X and H ∈ H * ; (E 2 ) E λ = J − λF is coercive in X, for all λ ∈ I; (E 3 ) E λ satisfy the (CPS) c condition, for every λ ∈ R.
Assume also that there exists r ∈ R such that Then E λ has at least three critical points in X, for every λ ∈ (ϕ 1 (r), ϕ 2 (r)), which are symmetric invariant in V.
Proof.By Lemmas 3.3 and 3.4 we have two different local minima u 0 and u 1 in X for the energy functional E λ for every λ ∈ (ϕ 1 (r), ϕ 2 (r)) and these minima are also in X H * .From the assumptions (E 2 ) and (E 3 ) we have that E λ is coercive and satisfies the (CPS) c condition for all λ ∈ R. Furthermore (E 1 ) implies that E λ (u H ) ≤ E λ (u), for all u ∈ X, H ∈ H * and λ ∈ (ϕ 1 (r), ϕ 2 (r)).So, we can apply Theorem 2.19, which ensures the existence of a third invariant critical point u 2 for E λ , with λ ∈ (ϕ 1 (r), ϕ 2 (r)).

Application
Let Ω be a ball in R N , X = W 1,2 0 (Ω) and let E = C ∞ 0 (Ω), which is dense in X.We consider the problem (1.1), namely where a ij satisfy the conditions (a 1 )-(a 3 ).We recall the corresponding energy functional E λ defined in (1.2), as where J(u) = 1 2 Ω ∑ N i,j=1 a ij (u)D i uD j udx and F (u) = Ω F(x, u)dx, F(x, t) = t 0 f (x, s)ds.By the conditions (a 1 ) − (a 3 ), the functional E λ is continuous, E-differentiable and We say that u ∈ W 1,2 0 (Ω) is a weak solution of the quasilinear problem (1.1) if u is a critical point of E λ .
We assume that f : Ω × R → R satisfies the conditions ( f 1 )-( f 4 ) with p = 2 and in addition the following assumptions hold ( f 5 ) there exists c > 0 such that | f (x, t)| ≤ c|t| for a.e.x ∈ Ω and all t ∈ R; ( f 6 ) there exist q ∈ (2, 2 * ) and a positive constant M > 0, such that for a.e.x ∈ Ω and all t ∈ R, |F(x, t)| ≤ M|t| q .b) The embedding condition (EC) also remains true for p = 2. Now, we give a concrete example for the function f , which satisfies the conditions ( f 1 )-( f 6 ).We have to prove that {u n } contains a strongly convergent subsequence.Since E λ is coercive, we have that the sequence {u n } is bounded.

Remark 4. 1 .
a) Condition ( f 6 ) is stronger than ( f 5 ) for t ∈ [−1, 1] while the condition ( f 5 ) is stronger when t is outside of the interval [−1, 1].We need both conditions in the proof of our main result (Theorem 4.1).

Proof of Theorem 4 . 1 .
By Lemma 4.4, Lemma 4.5, Lemma 4.7, the conditions (E 1 )-(E 3 ) of Theorem 3.5 are satisfied.In what follows we verify the conditions (i)-(ii).Let λ 1 be the first eigenvalue of the problem− u = λu, in H 1 0 (Ω), that is λ 1 is defined by the Rayleigh quotient