MULTIPLE SYMMETRIC INVARIANT NON TRIVIAL SOLUTIONS FOR A CLASS OF QUASILINEAR ELLIPTIC VARIATIONAL SYSTEMS CSABA FARKAS AND CSABA VARGA

Abstract. In the present paper we prove a multiplicity result for a model quasilinear elliptic system, coupled with the homogeneous Dirichlet boundary condition (Sλ ) on the unit ball, depending on a positive parameter λ. By variational methods, we prove that for large values of λ, the problem (Sλ ) has at least two non-zero symmetric invariant weak solutions.

Keywords: Elliptic system, Symmetrization, Ekeland’s variational principle, Spherical cap symmetrizations. MSC 2010:35B06, 35J50,35J65 1. Introduction Consider the following quasi-linear, elliptic differential system coupled with the homogeneous Dirichlet boundary condition,   −∆p u = λFu (x, u(x), v(x)) in Ω, −∆q v = λFv (x, u(x), v(x)) in Ω, (Sλ )  u=v=0 on ∂Ω, where λ is a positive parameter and N > p, q > 1, Ω = B(0, 1) ⊂ RN is the unit ball, F ∈ C 1 (Ω × R2 , R), Fz denotes the partial derivative of F with respect to z, ∆α is the α-Laplacian operator, i.e., ∆α = div(|∇u|α−2 ∇u). Systems of the type (Sλ ) have been the object of intensive investigations on bounded domains; We refer to the works of Boccardo and de Figueiredo [1], de Figueiredo [3], de N´apoli, Mariani [4] and Krist´aly, Rˇadulescu, Varga [6]. From the articles dealing with systems, we would like to highlight the paper of A. Krist´aly and I. Mezei, see [5], which studies a gradient-type system defined on a strip like domain, depending on two parameters, and proving a Ricceri-type three critical point result. While keeping some conditions from [5], we also aim to give a multiplicity theorem for our problem, three solutions, which are invariant under symmetrization. As we alredy have pointed out, our aim is to examine the above problem in the point of view of symmetrizations, namely to prove a result which ensures the existence of symmetrically invariant solutions. Despite of the fact, that symmetrizations don’t really occur in modelling real situations of the everyday life, they are very useful and highly applied topic in the 1

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CS.FARKAS AND CS. VARGA

theory of partial differential equations. Many mathematicians worked and work in the study of symmetrizations, trying to describe some new phenomena. Here we would like to mention the work of Brock and Solynin, [2], J. Van Schaftingen [8], M. Squassina [7] who have proven many results among symmetrizations in the past few years, here we thinking of the symmetric minimax principle, Ekeland-, Borvein Preiss variational principles etc., which have opened many new ways to applications of this topic. In [2], Brock and Solynin proved that the Steiner symmetrization of a function can be approximated in Lp (Rn ) by a sequence of very simple rearrangements which are called polarizations. Moreover, they introduced the concept of rearrangement and investigated some general properties. The aforementioned problem is interesting not only from a mathematical point of view but also from its applicability in mathematical physics. The problem (Sλ ) is a generalization of the equation of the spring pendulum. A spring pendulum is a physical system where a piece of mass is connected to a spring so that the resulting motion contains elements of a simple pendulum motion as well as a spring motion. The equation of spring pendulum is the following:     l 2 0  x(t) = ω0 x(t) 1 − √ 2  −¨ x(t) +y(t)2    l 2 0  y (t) = ω0 y(t) 1 − √ 2 ,  −¨ 2

(S)

x(t) +y(t)

q g where ω0 = and l0 is the length of the spring at rest. A simple simulation l0 shows how our numerical solutions be represented, and that the orbit of this kind of pendulum has a fractal-like shape. Such phenomena are often studied in chaos theory. The problem (S) can be treated as a variational problem, if we choose F (x, y) =

p ω02 (x2 + y 2 ) − ω02 · l0 x2 + y 2 , 2

then the energy functional associated to problem S is defined by Z Z 0 2 0 2 E(x, y) = (x ) + (y ) dt − F (x, y)dt, I

I

where I ⊂ R+ . The main objective of our paper is to ensure the existence of symmetric invariant non-trivial solutions for the problem (Sλ ) where the natural functional framework is the Sobolev space W01,p,q (Ω) = W01,p (Ω) × W01,q (Ω). In order to present our main result, we first recall that (u, v) ∈ W01,p × W01,q is a weak solution to problem (Sλ ) if

MULTIPLE SYMMETRIC SOLUTIONS

(1.1)

3

Z  Z p−2   |∇u| ∇u∇w1 dx − λ Fu (x, u(x), v(x))w1 (x)dx = 0   Ω Ω Z Z     |∇v|q−2 ∇v∇w2 dx − λ Fv (x, u(x), v(x))w2 (x)dx = 0, Ω

Ω

W01,p

W01,q .

for every (w1 , w2 ) ∈ × In the sequel, we outline our approach and state the main result. We assume that the following hypotheses hold: (F1 ) F : Ω × R2 → R is a continuous function, (s, t) 7→ F (x, s, t) is of C 1 and F (x, 0, 0) = F (x, s, 0) = F (x, 0, t) = 0 and Fs (x, s, t) · s− + Ft (x, s, t) · t− ≤ 0 for all x, s, t, where τ− = min{0, τ }; F (x, s, t) (F2 ) lim = 0, uniformly for every x ∈ Ω; (s,t)→(0,0) |s|p + |t|q F (x, s, t) (F3 ) lim = 0, uniformly for every x ∈ Ω; |s|+|t|→+∞ |s|p + |t|q (F4 ) There exists,(u0 , v0 ) ∈ W01,p (Ω) × W01,q (Ω) such that Z F (x, u0 (x), v0 (x))dx > 0; Ω

(F5 ) For F (x, s, t) = F (y, s, t) for each x, y ∈ Ω with |x| = |y| and s, t ∈ R and for x ∈ Ω and a ≤ b and c ≤ d F (x, a, c) + F (x, b, d) ≥ F (x, a, d) + F (x, b, c); (F6 ) For all x, s, t one has F (x, s, t) ≤ F (x, |s|, |t|). Our main result reads as follows: Theorem 1.1. Assume that p, q > 1, and let Ω ⊂ RN be the unit ball. Let F ∈ C 1 (Ω × R2 , R) be a function which satisfies (F1 ) − (F6 ). There exists a λ0 such that, for every λ > λ0 the problem (Sλ ) has at least two weak solutions in W01,p,q (Ω), invariant by spherical cap symmetrization. Remark 1.1. Let p = q = 2, then the function F : Ω × R × R → R defined by F (x, s, t) = kxk ln(1 + s2+ · t2+ ) fulfills the hypotheses (F1 ) − (F6 ), where τ+ = max{0, τ }. Remark 1.2. From (F1 ) and (F5 ) one can conclude the following inequality: F (x, 0, 0) + F (x, s, t) ≥ F (x, 0, t) + F (x, s, 0), for t, s ≥ 0, therefore F (x, s, t) ≥ 0 for s, t ≥ 0.

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CS.FARKAS AND CS. VARGA

Remark 1.3. If SF = Cmax

|sFs (x, s, t) + tFt (x, s, t)| < ∞, |s|p + |t|q (s,t)6=(0,0) sup

then there exists a λF such that, for every 0 < λ ≤ λF the problem (Sλ ) has only the trivial solution. Indeed, a solution of (Sλ ) is a pair (u, v) ∈ W01,p (Ω) × W01,q (Ω) such that Z  Z p−2   |∇u| ∇u∇w1 dx − λ Fu (x, u(x), v(x))w1 (x))dx = 0   Ω Ω Z Z    q−2  |∇v| ∇v∇w2 dx − λ Fv (x, u(x), v(x))w2 (x)dx = 0, Ω

Ω

W01,p (Ω)

W01,q (Ω).

for all w1 ∈ and w2 ∈ Choosing w1 = u and w2 = v, we obtain that Z Z SF p q kuk1,p + kvk1,q = λ (Fu (x, u, v)u + Fv (x, u, v)v)dx ≤ λ |u|p + |v|q ≤ C max Ω Ω ≤λ

SF (Cpp kukp1,p + Cqq kvkq1,q ) ≤ λSF (kukp1,p + kvkq1,q ), Cmax

where, Cmax = max{Cpp , Cqq }, therefore if λ < S1F then we necessarily have that (u, v) = (0, 0), which concludes the proof of this remark. Remark 1.4. Note that if (u, v) is a weak solution Z  Z p−2   |∇u| ∇u∇u− dx − λ Fu (x, u(x), v(x))u− (x)dx = 0   Ω Ω Z Z     |∇v|q−2 ∇v∇v− dx − λ Fv (x, u(x), v(x))v− (x)dx = 0, Ω

Ω

follows that u− = v− = 0. The proof of Theorem 1.1 is based on the symmetric version of the general minimax theorem and on a symmetric version of Ekeland’s variational principles. Therefore, in the next section we recall besides of the aforementioned results, some properties from the critical point theory(Palas-Smale sequence, Mountain pass theorem), while in Section 3, we prove our main result. As we already pointed out, the proof of this theorem is based on variational arguments. To see this, we consider the space W01,α (Ω) endowed with the norm Z 1/α α kuk1,α = |∇u| α ∈ {p, q}, Ω

MULTIPLE SYMMETRIC SOLUTIONS

5

and for β ∈ [α, α∗ ] we have the Sobolev embeddings W01,α (Ω) ,→ Lβ (Ω). The product space W01,p (Ω) × W01,q (Ω) is endowed with the norm k(u, v)k1,p,q = kuk1,p + kvk1,q . We define the function F : W01,p (Ω) × W01,q (Ω) → R by Z F(u, v) = F (x, u, v)dx Ω

W01,p (Ω),

for u ∈ defined by (1.2)

v∈

W01,q (Ω).

The energy functional associated to problem (Sλ ) is

1 1 Aλ (u, v) = kukp1,p + kvkq1,q − λ p q

Z F (x, u, v)dx. Ω

2. Preliminaries This section is devoted to present some preparatory results. First, we recall the definition of the Palais-Smale condition, the Mountain Pass theorem, and we recall the definition of the spherical cap symmetrization and polarization, and some abstract results from symmetrization theory. Definition 2.1. (Palais-Smale condition,[10], [6]) (a) A function ϕ ∈ C 1 (X, R) satisfies the Palais-Smale condition at level c ∈R (shortly, (P S)c -condition) if every sequence {un } ⊂ X such that lim ϕ(un ) = c and lim kϕ0 (un )k = 0,

n→∞

n→∞

possesses a convergent subsequence. (b) A function ϕ ∈ C 1 (X, R) satisfies the Palais-Smale condition (shortly, (P S)condition) if it satisfies the Palais-Smale condition at every level c ∈ R. Theorem 2.1. (Mountain Pass Theorem, [10]) Let X be a Banach space, ϕ ∈ C 1 (X, R), e ∈ X and r > 0 be such that ||e|| > r and b := inf ϕ(u) > ϕ(0) ≥ ϕ(e). ||u||=r

Then for each ε > 0 there exists u ∈ X such that (a) c − 2ε ≤ ϕ(u) ≤ c + 2ε (b) ||ϕ0 (u)|| < 2ε where c := inf max ϕ(γ(t)) γ∈Γ t∈[0,1]

and Γ := {γ ∈ C([0, 1]) : γ(0) = 0, γ(1) = e}. Moreover, if ϕ satisfies the Palais-Smale condition at level c , then c is a critical value of ϕ.

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CS.FARKAS AND CS. VARGA

Definition 2.2 (Spherical cap symmetrization). Let P ∈ ∂B(0, 1)∩RN . The spherical cap symmetrization of the set A with respect to P is the unique set A∗ such that A∗ ∩ {0} = A ∩ {0} and for any r ≥ 0, A∗ ∩ ∂B(0, r) = Bg (rP, ρ) ∩ ∂B(0, r) for someρ ≥ 0, HN −1 (A∗ ∩ ∂B(0, r)) = HN −1 (A ∩ ∂B(0, r)), where HN −1 is the outer Hausdorff (N − 1)-dimensional measure and Bg (rP, ρ) denotes the geodeisc ball on the sphere ∂B(0, r) of center rP and radius ρ. By definition Bg (rP, 0) = ∅. Definition 2.3. The spherical cap symmetrization of a function u : Ω → R is the unique function u∗ : Ω∗ → R such that, for all c ∈ R, {u∗ > c} = {u > c}∗ . Definition 2.4 (Polarization). A subset H of RN is called a polarizer if it is a closed affine half-space of RN , namely the set of points x which satisfy α · x ≤ β for some α ∈ RN and β ∈ R with |α| = 1. Given x in RN and a polarizer H, the reflection of x with respect to the boundary of H is denoted by xH . The polarization of a function u : RN → R+ by a polarizer H is the function uH : RN → R+ defined by ( max{u(x), u(xH )}, if x ∈ H H (2.1) u (x) = min{u(x), u(xH )}, if x ∈ RN \ H. The polarization C H ⊂ RN of a set C ⊂ RN is defined as the unique set which satisfies χC H = (χC )H , where χ denotes the characteristic function. The polarization uH of a positive function u defined on C ⊂ RN is the restriction to C H of the polarization of the extension u˜ : RN → R+ of u by zero outside C. The polarization of a function which may change sign is defined by uH := |u|H , for any given polarizer H. 2.1. Abstract framework of symmetrizations. Based on [8], consider the following abstract framework: Let X and V be two real Banach spaces, with X ⊂ V and let S ⊂ X. For the better understanding of the reader we present some crucial abstract symmetrization and polarization results of Van Schaftingen [8] and of Squassina [7]. Let us first introduce the following main assumption. Definition 2.5. Let H? be a pathconnected topological space and denote by h : S × H? → S, (u, H) 7→ uH , the polarization map. Let ? : S → V, u 7→ u? , be any symmetrization map. Assume that the following properties hold. 1) The embeddings X ,→ V and V ,→ W are continuous; 2) h is continuous; 3) (u? )H = (uH )? = u? and (uH )H = uH for all u ∈ S and H ∈ H? ;

MULTIPLE SYMMETRIC SOLUTIONS

7

4) for all u ∈ S there exists a sequence (Hm )m ⊂ H? such that uH1 ...Hm → u? in V ; 5) kuH − v H kV ≤ ku − vkV for all u, v ∈ S and H ∈ H? . Since there exists a map Θ : (X, k · kV ) → (S, k · kV ) which is Lipschitz continuous, with Lipschitz constant CΘ > 0, and such that Θ|S = Id|S , both maps h : S × H? → S and ? : S → V 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 (2.2)

kuH − v H kV ≤ CΘ ku − vkV ,

ku? − v ? kV ≤ CΘ ku − vkV

for all u, v ∈ X and for all H ∈ H? . We are now able to state the uniform approximation of symmetrization established by Van Schaftingen in [8]. Proposition 2.1 (Corollary 3.4 of [8]). For all ε > 0 there exists a continuous map Tε : S → S such that Tε u is built via iterated polarization and kTε u − u? kV < ε for all u ∈ S. Example 2.1. (Spherical cap symmetrization with Dirichlet boundary condition) ? Let Ω be now a ball or an annulus of RN . Put X = W01,p (Ω), V = Lp (Ω) ∩ Lp (Ω), Np with p? = . Denote by ? the spherical cap symmetrization and let H? be N −p defined as above. Again the assumptions stated in Definition 2.5 are satisfied by Proposition 2.19, Theorem 2.20 and Proposition 2.20 of [8]. Example 2.2. (Schwarz symmetrization) Let again X = W01,p (B), V = Lp (B) ∩ Np ? Lp (B), with p? = , S = W01,p (B), u? = |u|? , where ? denotes the Schwarz N −p symmetrization and H? is defined as above for the Schwarz symmetrization, but uH = |u|H . Then the assumptions stated in Definition 2.5 are satisfied again by Proposition 2.19, Theorem 2.20 and Proposition 2.20 of [8]. We recall three results which are crucial in our further investigations. Proposition 2.2. Let H ∈ H. Suppose Ω = ΩH ⊂ RN , u, v : Ω → R are measurable and nonnegative. If G : Ω × R × R → R+ is a Borel measurable function such that G(x, s, t) = G(xH , s, t) and if x ∈ Ω and a ≤ b and c ≤ d, G(x, a, c) + G(x, b, d) ≥ G(x, a, d) + G(x, b, c) then Z Z G(x, u, v)dx ≤ G(x, uH , v H ). Ω

Ω

We recall a symmetric version of Ekeland’s variational principle, which appears in the paper [7] of Squassina. Theorem 2.2 (Theorem 2.8 of [7]). Let (X, V, ?, H? , S) satisfy the assumptions given in Definition 2.5. Denote by κ > 0 any constant with the property kukV ≤ κkuk for all u ∈ X.

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CS.FARKAS AND CS. VARGA

Assume that Φ : X → R ∪ {∞} is a proper lower semi–continuous functional bounded from below such that Φ(uH ) ≤ Φ(u) for all u ∈ S and H ∈ H? .

(2.3)

Moreover, assume that for all u ∈ dom(Φ) there exists ξ ∈ S such that Φ(ξ) ≤ Φ(u). Then for all ε > 0 and σ > 0 there exists v ∈ X such that a) kv − v ? kV < (κ(CΘ + 1) + 1)ε; b) Φ(w) ≥ Φ(v) − σkw − vk for all w ∈ X. In addition, Φ(v) ≤ Φ(u) and kv − uk ≤ ε + kTε u − uk, where u ∈ S is some element which satisfies Φ(u) ≤ inf Φ + εσ and Tε is the continuous map given in X Proposition 2.1. Now, we recall a symmetric version of Minimax Theorem, due to Van Schaftigen in [8]. In what follows X denotes a real Banach space, e a fixed element of X \ {0}, Γ = {γ ∈ C([0, 1], X) : γ(0) = 0,

(2.4)

γ(1) = e},

and Φ a functional of class C 1 (X) such that (2.5)

c = inf max Φ(γ(t)) > a = max{Φ(0), Φ(e)}. γ∈Γ t∈[0,1]

Theorem 2.3 (Theorem 3.5 of [8]). Let (X, V, ?, H? , S) satisfy the assumptions of Definition 2.5. Denote by κ > 0 any constant with the property kukV ≤ κkuk for all u ∈ X. Let Φ ∈ C 1 (X) verify (2.5) and assume that Φ(uH ) ≤ Φ(u) for all u ∈ S and H ∈ H? . For every ε ∈ (0, (c − a)/2), δ > 0 and γ ∈ Γ, with the properties i) sup Φ(γ(t)) ≤ c + ε; t∈[0,1]

ii) γ([0, 1]) ⊂ S; iii) {γ(0), γ(1)}H0 = {γ(0), γ(1)} for some H0 ∈ H? , then there exists uε ∈ X such that a) c − 2ε ≤ Φ(uε ) ≤ c + 2ε; b) kuε − u?ε kV ≤ 2(2κ + 1)δ; c) kΦ0 (uε )kX ? ≤ 8ε/δ. Remark 2.1. In our case Γ = {γ ∈ C([0, 1], W01,p,q ) : γ(0) = (0, 0),

γ(1) = (eu , ev )},

where (eu , ev ) ∈ W01,p,q is a fixed element different from (0, 0). 3. Proof of Theorem 1.1 Before proving our main result, we prove that our functional Aλ is coercive and satisfies the Palais- Smaile condition on W01,p,q (Ω) = W01,p (Ω) × W01,q (Ω). Lemma 3.1. The functional Aλ : W01,p,q (Ω) → R is coercive for every λ ≥ 0.

MULTIPLE SYMMETRIC SOLUTIONS

9

Proof. Let us fix a λ ≥ 0. Due to (F2 ) and (F3 ) one has that for every ε > 0, there exists δ1 = δ1 (ε) > 0 and δ2 = δ2 (ε) such that F (x, s, t) ≤ ε (|s|p + |t|q ) , whenever |s| + |t| > δ1 ,

(3.1) and

F (x, s, t) ≤ ε (|s|p + |t|q ) , whenever |s| + |t| < δ2 .

(3.2)

From the definition of the function F , there exists Mε > 0 such that |F (x, s, t)| ≤ Mε whenever |s| + |t| ∈ [δ2 , δ1 ]. Therefore, we obtain Z F(u, v) =

F (x, u, v)dx = Z F (x, u, v)dx + Ω

Z = {kuk1,p +kvk1,q >δ1 }

F (x, u, v)dx+

{kuk1,p +kvk1,q <δ2 }

Z F (x, u, v)dx ≤

+

where Cα is the

{kuk1,p +kvk1,q ∈[δ2 ,δ1 ]} ≤ 2εCpp kukp1,p + 2εCqq kvkq1,q + Mε ωn , embedding constant in W01,α (Ω) ,→ Lα (Ω), α







∈ {p, q}. Therefore



1 1 − 2ελCpp · kukp1,p + − 2ελCqq kvkq1,q − λMε ωn ≥ p q   1 − 2ελCmax (kukp1,p + kvkq1,q ) − λMε ωn , ≥ p+q where Cmax = max{Cpp , Cqq }. In particular if 0 < ε < (2(p + q)λCmax )−1 , then Aλ is coercive, which concludes our proof.  Aλ (u, v) ≥

Lemma 3.2. One has, Aλ (uH , v H ) ≤ Aλ (u, v). Proof. First of all, we mention that, from F6 one can conclude the following inequality Z Z F (x, u(x), v(x))dx ≤ Ω

F (x, |u|(x), |v|(x))dx. Ω

Based on the above inequality and on Remark 1.2 we can apply Proposition 2.2 for (u, v) ∈ W01,p,q (Ω)+ . Therefore, one has that k∇uH kLp = k∇ukLp , k∇v H kLq = k∇vkLq and kuH kLp ≤ kukLP , kv H kLq ≤ kukLq . On the other hand, due to F (x, s, t) = F (xH , s, t) one has, Z Z F (x, u(x), v(x))dx ≤ F (x, uH (x), v H (x))dx, Ω

Ω

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CS.FARKAS AND CS. VARGA

therefore Aλ (uH , v H ) ≤ Aλ (u, v).  Remark 3.1. Using the Sobolev embeddings, (F1 ) and (F2 ) and (F4 ), one can prove in a standard way that F is of class C 1 , its differential being Z F(u, v)(w, y) = [Fu (x, u, v)w + Fv (x, u, v)y], Ω

for every u, w ∈

W01,p (Ω)

and v, y ∈ W01,q (Ω).

Lemma 3.3. Let λ ≥ 0 be fixed and let {(un , vn )} be a bounded sequence in W01,p,q (Ω) such that kAλ0 (un , vn )k? → 0 as n → ∞. Then {(un , vn )} contains a strongly convergent subsequence in W01,p,q (Ω). Proof. Because W01,p,q (Ω) is a reflexive Banach space and {(un , vn )} is a bounded sequence, we can assume that (un , vn ) → (u, v) weakly in W01,p,q ; (un , vn ) → (u, v) strongly in Lp (Ω) × Lq (Ω).

(3.3) (3.4)

On the other hand, we have Aλ0 (un , vn )(u

Z

|∇un |p−2 ∇un (∇u − ∇un )

− un , v − vn ) = Ω

Z

|∇vn |q−2 ∇vn (∇v − ∇vn ) − λF 0 (un , vn )(u − un , v − vn )

+ Ω

and Aλ0 (u, v)(un

Z − u, vn − v) =

|∇u|p−2 ∇u(∇un − ∇u)

Ω

Z

|∇v|q−2 ∇v(∇vn − ∇v) − λF 0 (u, v)(un − u, vn − v).

+ Ω

Adding these two relations, one has Z not. (|∇un |p−2 ∇un − |∇u|p−2 ∇u)(∇un − ∇u) an = Ω Z + (|∇vn |q−2 ∇vn − |∇v|q−2 ∇v)(∇vn − ∇v) Ω

=

−Aλ0 (un , vn )(u − un , v − vn ) − Aλ0 (u, v)(un − u, vn − v) −λF 0 (un , vn )(u − un , v − vn ) − λF 0 (u, v)(un − u, vn − v).

We can easily see that the last two terms tends to 0 as n → ∞. Due to (3.3), the second terms tends to 0, while the inequality |Aλ0 (un , vn )(u − un , v − vn )| ≤ kAλ0 (un , vn )k? k(u − un , v − vn )k1,p,q

MULTIPLE SYMMETRIC SOLUTIONS

11

and the assumption implies that the first term tends to 0 too. Thus, (3.5)

lim an = 0.

n→∞

From the well-known inequality  (|t|α−2 t − |s|α−2 s)(t − s), if α ≥ 2, α |t − s| ≤ α−2 α−2 α/2 α α (2−α)/2 ((|t| t − |s| s)(t − s)) (|t| + |s| ) , if 1 < α < 2, for all t, s ∈ RN , and (3.5), we conclude that Z (|∇un − ∇u|p + |∇vn − ∇v|q ) = 0, lim n→∞

Ω

hence, the sequence {(un , vn )} converges strongly to (u, v) in W01,p,q .  Proof of Theorem 1.1 The proof is divided into two steps. In the first one we prove that the critical point obtained by the symmetric version of Ekeland principle is symmetric invariant, while in the second step we prove that the critical point obtained by the symmetric version of the minimax theorem is invariant under symmetrization. (Step 1.) For the minimizing sequence (un , vn ) we consider the following sequence  Aλ (un , vn ) − d, if Aλ (un , vn ) − d > 0 εn = 1 , if Aλ (un , vn ) − d = 0. n where d = inf Aλ . We have that, Aλ (un , vn ) ≤ d + εn and εn → 0 as n → ∞. Applying Theorem 2.2, yields that there exists a sequence {(an , bn )} ⊂ W01,p,q (Ω) such that: (a) Aλ (an , bn ) ≤ Aλ (un , vn ); (b) k(an , bn ) − (a?n , b∗n )kLp ×Lq → 0; Since Aλ satisfies the (P S) condition, up to a subsequence (an , bn ) → (a, b), (a, b) ∈ W01,p,q (Ω). From the fact that (an , bn ) → (a, b) and (a∗n , b∗n ) → (a∗ , b∗ ) we can conclude that k(a, b) − (a∗ , b∗ )kLp ×Lq ≤ ≤ k(an , bn ) − (a, b)kLp ×Lq + k(an , bn ) − (a∗n , b∗n )kLp ×Lq + k(a∗n , b∗n ) − (a∗n , b∗n )kLp ×Lq . Therefore from the above outcomes one has that (a, b) = (a∗ , b∗ ). Which conclude the proof of the first step. (Step 2.) Z 1 1 p q Aλ ((u0 , v0 )) = ku0 k1,p + kv0 k1,q − λ F (x, u0 (x), v0 (x))dx = A − λB, p q Ω R q p where A = ku0 k1,p +kv0 k1,q > 0, and B = Ω F (x, u0 (x), v0 (x))dx > 0. Consequently, there exists λ0 > 0 such that for every λ > λ0 , we have that h(λ) = A − λB < 0,

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therefore 1 1 Aλ ((u0 , v0 )) = ku0 kp1,p + kv0 kq1,q − λ p q

Z F (x, u0 (x), v0 (x))dx < 0. Ω

In fact, we may choose, ) (1 p q 1 kuk + kvk 1,p 1,q p q λ0 = inf : u ∈ W01,p (Ω), v ∈ W01,q (Ω), F(u, v) > 0 . F(u, v) Now, fix λ > λ0 . First, relations (F2 ) and (F3 ) imply that for every ε > 0 there exists δ1 = δ1 (ε), δ2 = δ2 (ε) such that for every (s, t) ∈ R2 with |s| + |t| ∈ (0, δ1 ) ∪ (δ2 , +∞), one has F (x, s, t) 0≤ p < ε. |s| + |t|q   ∗ ∗ is Fix γ ∈ 1, min{ pp , qq } . Note that the continuous function (s, t) 7→ |s|Fγp(x,s,t) +|t|γq bounded on the set {(s, t) ∈ R2 : |s| + |t| ∈ [δ1 , δ2 ]}. Therefore there exists mε > 0, such that mε ε p q F (x, s, t) ≤ (|s|pγ +|t|γq ), for all (s, t) ∈ R2 . p q (|s| +|t| )+ max{pCp , qCq } (max{p, q})γ Therefore, for each (u, v) ∈ W01,p (Ω) × W01,q (Ω) we get Z F (x, u, v) ≤ F(u, v) = Ω

Z

Z mε ε p q |u| + |v| dx + (|u|pγ + |v|γq ) ≤ ≤ max{pCpp , qCqq } Ω (max{p, q})γ Ω ε mε p q ≤ (kukpLpγ + kvkqLγq ) ≤ p q (kukLp + kvkLq ) + γ max{pCp , qCq } (max{p, q}) mε ε p q p q p (C p kukp1,p + Cγq kvkq1,q ) ≤ ≤ p q (Cp kuk1,p + Cq kvk1,q ) + max{pCp , qCq } (max{p, q})γ γp   γ  γ   1 1 1 1 p q p q ≤ε kuk1,p + kvk1,q + mε Cγ kuk1,p + kvk1,q ≤ p q p q    γ 1 1 1 1 p q p q ≤ε kuk1,p + kvk1,q + mε Cγ kuk1,p + kvk1,q , p q p q p q where Cγ = max{Cγp , Cγq } and in the last inequality we used the fact that the function 1 (s, t) 7→ (sγ + tγ ) γ , s, t ≥ 0, is decreasing. Consequently Aλ (u, v) ≥    γ 1 1 1 1 p q p q kuk1,p + kvk1,q − λmε Cγ kuk1,p + kvk1,q ≥ (1 − ε) p q p q

MULTIPLE SYMMETRIC SOLUTIONS

13

 γ
γ 1−ε 1 1 p q ≥ (kuk1,p + kvk1,q ) − λmε Cγ max , kukp1,p + kvkq1,q = p+q p q    γ
γ−1 1−ε 1 1 q p (kukp1,p + kvkq1,q ). − λmε Cγ max , kuk1,p + kvk1,q p+q p q Now let 0 < ρ < 1, and k(u, v)k = kuk1,p + kvk1,q . Then we have  ρ max{p,q} ≤ kukp1,p + kvkq1,q ≤ ρ. 2 Therefore for ρ small enough Aλ (u, v) > 0. Since inf Aλ (u, v) > 0 = Aλ (0, 0) > Aλ (u0 , v0 ) k(u,v)k=ρ

and Aλ satisfies the Palais-Smale condition, we are in the position to apply the mountain pass theorem with Γ = {γ ∈ C([0, 1], W01,p,q ) : γ(0) = (0, 0),

γ(1) = (eu , ev )}.,

which means that c = inf γ∈Γ supt∈[0,1] Aλ (γ(t)) is a critical value of Aλ , therefore there exists a critical point (u, v) such that Aλ (u, v) = c. From the definition of c, 1 sup Aλ (γ(t)) ≤ c + 2 . n t∈[0,1] From the above inequality and from the first step, Aλ has a global minimum which is invariant by spherical symmetrization, and (0, 0) is also invariant by spherical symmetrization( therefore the assumption iii) in Theorem 2.3 is fulfilled) we are in the position to apply Theorem 2.3 with Γ = {γ ∈ C([0, 1], W01,p,q ) : γ(0) = (0, 0),

γ(1) = (eu , ev )},

where (eu , ev ) is the global minimum of Aλ . Choosing ε = n12 , and δ = n1 , yields that there exist (un , vn ) ∈ W01,p,q (Ω) such that (a) |Aλ (un , vn ) − c| ≤ n22 ; (b) k(un , vn ) − (u∗n , vn∗ )kLp ×Lq ≤ 2(2K + 1) n1 ; (c) kAλ0 (un , vn )k∗ ≤ n4 ; Since Aλ satisfies the (P S) condition by Lemma 3.3 there exist a subsequence (unk , vnk )k of (un , vn ) and (u, v) ∈ W01,p,q (Ω) such that (unk , vnk ) → (u, v) as k → ∞. Hence the inequalities (a), (b), (c) imply that Aλ (u, v) = c and Aλ0 (u, v) = 0 and (u, v) = (u∗ , v ∗ ) since k(u, v)−(u∗ , v ∗ )kLP ×Lq ≤ k(u, v)−(unk , vnk )kLP ×Lq +k(unk , vnk )−(u∗nk , vn∗ k )kLP ×Lq + +k(u∗nk , vn∗ k ) − (u∗ , v ∗ )kLP ×Lq → 0, which concludes our proof.



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Acknowledgements. Both authors were supported by the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, project No. PN-II-ID-PCE-20113-0241 and partialy by Domus Hungarica DSZ/66/2012 of the Hungarian Academy of Sciences. References [1] L. Boccardo and D. G. de Figueiredo, Some remarks on a system of quasilinea r elliptic equations, Nonlin. Diff. Eqns Applic. 9 (2002), 309–323. [2] F. BROCK and A. YU. SOLYNIN An approach to symmetrization via polarization TRAN. AMS Volume 352, No. 4, Pages 1759–1796. [3] D. G. de Figueiredo, Semilinear elliptic systems , in Nonlinear Functional Analysis and Applications to Differential Equations, Trieste, 1997, pp. 122–152 (World Scientific, River Edge, NJ, 1998). [4] P. L. de Napoli and M. C. Mariani, Quasilinear elliptic systems of resonant type and nonlinear eigenvalue problems, Abstr. Appl. Analysis 7 (2002) 155–167. ´ ly and I. Mezei Multiple solutions for a perturbed system on strip-like domains, [5] A. Krista Discrete Cont Dyn S. SER. S 5 (2012), no. 4, 789–796. ´ ly, V. Ra ˇ dulescu, Cs. Varga, Variational principles in mathematical physisc, [6] A. Krista geometry and economics, Cambridge University Press, 2011. [7] M. Squassina, Symmetry in variational principles and applications, J. London Math. Soc. 85 (2012), 323–348. [8] J. Van Schaftingen, Symmetrization and minimax principles, Comm. Contemp. Math. 7 (2005), 463–481. [9] D. Smets and M. Willem, Partial symmetry and asymptotic behavior for some elliptic problems, Calc. Var. Partial Differential Equations 18 (2003), no. 1, 57–75. [10] M. Willem, Minimax theorems, Progress in Nonlinear Differential Equations and their Applications 24, Birkh¨ auser, Boston, 1996. Faculty of Mathematics and Computer Science, Babes¸–Bolyai University, 400084 Cluj–Napoca, Romania E-mail address: [email protected] Faculty of Mathematics and Computer Science, Babes¸–Bolyai University, 400084 Cluj–Napoca, Romania E-mail address: [email protected]