Another four critical points theorem BIAGIO RICCERI Dedicated to the memory of Ky Fan, with my immense esteem and admiration 2 1 Abstract: In this paper, making use of Theorem 2 of [5], we establish a new four 0 critical points theorem which can be regarded as a companion to Theorem 1 of [4]. We also 2 present an application to the Dirichlet problem for a class of quasilinear elliptic equations. n a J 8 Key words: Criticalpoint, globalminimum, multiplicity,minimaxinequality,Dirich- 2 let problem. ] P A 2010 Mathematics Subject Classification: 47J10, 47J30, 58E05, 49J35, 35J92. . h t a m [ 2 The aim of this paper is to establish a new four critical points theorem (Theorem 1 v below) that can be regarded as a companion to Theorem 1 of [4]. 4 5 Asin[4], ourkeytoolisthemultiplicityresultonglobalminimaestablishedin[3]. The 6 1 use of such a result requires the validity of a strict minimax inequality which is explicitly 1. assumed in [4]. 0 A novelty of Theorem 1 is that no minimax inequality appears among the hypotheses. 2 1 This is possible thanks to the use of Theorem 2 of the very recent [5] which just highlights : v a rather general situation where the strict minimax inequality occurs. i X In other words, Theorem 1 should be regarded as the fruit that one obtains by com- r bining the underlying ideas of [4] with Theorem 2 of [5]. a For the reader convenience, we start just recalling Theorem 2 of [5]. First, we introduce the following notations. If X is a non-empty set and Γ,Ψ,Φ : X → R are three given functions, for each µ > 0 and r ∈]inf Φ,sup Φ[, we put X X µΓ(x)+Ψ(x)−infΦ−1(]−∞,r])(µΓ+Ψ) α(µΓ+Ψ,Φ,r) = inf x∈Φ−1(]−∞,r[) r −Φ(x) and µΓ(x)+Ψ(x)−infΦ−1(]−∞,r])(µΓ+Ψ) β(µΓ+Ψ,Φ,r) = sup . r −Φ(x) x∈Φ−1(]r,+∞[) 1 When Ψ+Φ is bounded below, for each r ∈]inf Φ,sup Φ[ such that X X inf Γ(x) < inf Γ(x) , x∈Φ−1(]−∞,r]) x∈Φ−1(r) we put Ψ(x)−γ +r µ∗(Γ,Ψ,Φ,r) = inf : x ∈ X,Φ(x) < r,Γ(x) < η , r η −Γ(x) r (cid:26) (cid:27) where γ = inf (Ψ(x)+Φ(x)) x∈X and η = inf Γ(x) . r x∈Φ−1(r) THEOREM A ([5], Theorem 2). - Let X be a topological space and Γ,Ψ,Φ : X → R three sequentially lower semicontinuous functions, with Γ also sequentially inf-compact, satisfying the following conditions: (a) inf (µΓ(x)+Ψ(x)) = −∞ for all µ > 0 ; x∈X (b) inf (Ψ(x)+Φ(x)) > −∞ ; x∈X (c) there exists r ∈]inf Φ,sup Φ[ such that X X inf Γ(x) < inf Γ(x) . x∈Φ−1(]−∞,r]) x∈Φ−1(r) Under such hypotheses, for each µ > max{0,µ∗(Γ,Ψ,Φ,r)}, one has α(µΓ+Ψ,Φ,r) = 0 and β(µΓ+Ψ,Φ,r) > 0 . As we said, the key tool in the proof of Theorem 1 is provided by the following THEOREM B ([3], Theorem 1). - Let X be a topological space, A ⊆ R an open interval and P : X ×A → R a function satisfying the following conditions: (a ) for each x ∈ X, the function P(x,·) is quasi-concave and continuous ; 1 (a ) for each λ ∈ A, the function P(·,λ) is lower semicontinuous and inf-compact ; 2 (a ) one has 3 sup inf P(x,λ) < inf supP(x,λ) . λ∈Ax∈X x∈Xλ∈A Under such hypotheses, there exists λ∗ ∈ A such that the function P(·,λ∗) has at least two global minima. Here is our main result: 2 THEOREM 1. - Let X be a reflexive real Banach space; I : X → R a sequentially weakly lower semicontiunuous and coercive C1 functional whose derivative admits a con- tinuous inverse on X∗; J,Ψ,Φ : X → R three C1 functionals with compact derivative satisfying the following conditions: J(x) J(x) liminf ≥ 0 , limsup < +∞ , (1) kxk→+∞ I(x) kxk→+∞ I(x) Ψ(x) liminf = −∞ , (2) kxk→+∞ I(x) inf (Ψ(x)+λΦ(x)) > −∞ (3) x∈X for all λ > 0. Moreover, assume that there exist a strict local minimum x of I, with 0 I(x ) = J(x ) = Ψ(x ) = Φ(x ) = 0, and another point x ∈X such that 0 0 0 0 1 max{J(x ),Ψ(x ),Φ(x )} < 0 , (4) 1 1 1 J(x) Φ(x) min liminf ,liminf ≥ 0 (5) x→x0 I(x) x→x0 I(x) (cid:26) (cid:27) and Ψ(x) liminf > −∞ . x→x0 I(x) Under such hypotheses, for each ν,µ satisfying I(x ) 1 ν > max 0,− (6) J(x ) (cid:26) 1 (cid:27) and Ψ(x) µ > max 0,−liminf , inf µ∗(I +νJ,Ψ,Φ,r) , (7) (cid:26) x→x0 I(x) r>supMν Φ (cid:27) where M is the set of all global minima of I + νJ, there exists λ∗ > 0 such that the ν functional µ(I +νJ) +Ψ+λ∗Φ has at least four critical points. Precisely, among them, one is x as a strict local, not global minimum and two are global minima. 0 PROOF. First of all, observe that, since X is reflexive, the functionals J,Ψ,Φ are sequentially weakly continuous, being with compact derivative ([6], Corollary 41.9). Fix ν as in (6). For x ∈ X \I−1(0), we have J(x) I(x)+νJ(x) = I(x) 1+ν I(x) (cid:18) (cid:19) and so, since I is coercive, in view of (1), it follows that lim (I(x)+νJ(x)) = +∞ . (8) kxk→+∞ 3 By the reflexivity of X again, this implies that the set M is non-empty and bounded. As ν a consequence, Φ is bounded in M . Also, by (2) and (3), we have sup Φ = +∞. Now, ν X fix µ as in (7). Let r > sup Φ be such that µ > µ∗((I +νJ),Ψ,Φ,r). Since Φ−1(r) is M ν non-empty and sequentially weakly closed, there exists x¯ ∈ Φ−1(r) such that I(x¯)+νJ(x¯) = inf (I(x)+νJ(x)) . x∈Φ−1(r) The choice of r implies that x¯ 6∈ M . So, we infer that ν inf (I(x)+νJ(x)) < inf (I(x)+νJ(x)) . x∈Φ−1(]−∞,r]) x∈Φ−1(r) Moreover, by (2), there exists a sequence {x } in X such that n Ψ(x ) n lim kx k = +∞ , lim = −∞ . (9) n n→∞ n→∞ I(xn) For any ρ ∈ R and for n large enough, we have Ψ(xn) ρ(I(x )+νJ(x ))+Ψ(x ) = (I(x )+νJ(x )) ρ+ I(xn) . (10) n n n n n  1+νJ(xn) I(xn)   Clearly, from (1), (8), (9) and (10), it follows that lim (µ(I(x )+νJ(x ))+Ψ(x )) = −∞ . n n n n→∞ So, if we consider X endowed with the weak topology, all the assumptions of Theorem A (with Γ = I +νJ) are satisfied, and so we have α(µ(I +νJ)+Ψ,Φ,r) < β(µ(I +νJ)+Ψ,Φ,r) . But, by Theorem 1 of [1], this inequality is equivalent to sup inf ((µ(I(x)+νJ(x))+Ψ(x)+λ(Φ(x)−r)) < inf sup((µ(I(x)+νJ(x))+Ψ(x)+λ(Φ(x)−r)). λ≥0x∈X x∈Xλ≥0 At this point, after observing that, in view of (3) and (8), one has lim (µ(I(x)+νJ(x))+Ψ(x)+λΦ(x)) = +∞ (11) kxk→+∞ for all λ > 0, we realize that we can apply Theorem B, with A =]0,+∞[, considering X with the weak topology again and taking P(x,λ) = µ(I(x)+νJ(x))+Ψ(x)+λ(Φ(x)−r)) . 4 Therefore, there exists λ∗ > 0 such that the functional µ(I +νJ)+Ψ+λ∗Φ has at least two gobal minima. Now, choose ǫ,σ > 0 so that Ψ(x) liminf > −µ+ǫ x→x0 I(x) and ǫ σ < . µν +λ∗ In view of (5) and recalling that x is a strict local minimum of I (with I(x ) = 0), we can 0 0 find a neighbourhood V of x such that, for each x ∈ V \{x }, one has 0 0 I(x) > 0 , Ψ(x) > −µ+ǫ , I(x) J(x) > −σ , I(x) and Φ(x) > −σ . I(x) Consequently, for each x ∈ V \{x }, we have 0 J(x) Ψ(x) Φ(x) µ(I(x)+νJ(x))+Ψ(x)+λ∗Φ(x) = I(x) µ+µν + +λ∗ > I(x) I(x) I(x) (cid:18) (cid:19) > I(x)(ǫ−σ(µν +λ∗)) > 0 . Hence, since µ(I(x )+νJ(x ))+Ψ(x )+λ∗Φ(x ) = 0, it follows that x is a strict local 0 0 0 0 0 minimum of the functional µ(I +νJ)+Ψ+λ∗Φ. On the other hand, in view of (4) and (6), we have µ(I(x )+νJ(x ))+Ψ(x )+λ∗Φ(x ) < 0 , 1 1 1 1 and hence x is not a global minimum of the functional µ(I +νJ) +Ψ+ λ∗Φ. Now, we 0 remark that this functional, due to (11) and to our assumptions on I,J,Φ,Ψ turns out to satisfythePalais-Smalecondition([6],Example38.25). Summarizing: thefunctional µ(I+ νJ)+Ψ+λ∗Φ is C1, satisfies the Palais-Smale condition, has at least two global minima and admits x as a local, not global minimum. At this point, we can invoke Theorem 0 (1.ter) of [2] to ensure the existence of a fourth critical point for the same functional, and the proof is complete. △ Now, we are going to present an application of Theorem 1 to quasilinear elliptic equations. 5 So, let Ω ⊂ Rn be a bounded domain with smooth boundary and let p > 1. On the Sobolev space W1,p(Ω), we consider the norm 0 1 p kuk = |∇u(x)|pdx . (cid:18)ZΩ (cid:19) If n ≥ p, we denote by A the class of all Carath´eodory functions f : Ω×R → R such that |f(x,ξ)| sup < +∞ , 1+|ξ|q (x,ξ)∈Ω×R where 0 < q < pn−n+p if p < n and 0 < q < +∞ if p = n. While, when n < p, we denote n−p by A the class of all Carath´eodory functions f : Ω×R → R such that, for each r > 0, the function x → sup |f(x,ξ)| belongs to L1(Ω). |ξ|≤r Given f ∈ A, consider the following Dirichlet problem −div(|∇u|p−2∇u) = f(x,u) in Ω (P ) f ( u = 0 on ∂Ω . Let us recall that a weak solution of (P ) is any u ∈ W1,p(Ω) such that f 0 |∇u(x)|p−2∇u(x)∇v(x)dx− f(x,u(x))v(x)dx = 0 ZΩ ZΩ for all v ∈ W1,p(Ω). 0 The functionals T,J : W1,p(Ω) → R defined by f 0 1 T(u) = kukp p J (u) = F(x,u(x))dx , f ZΩ where ξ F(x,ξ) = f(x,t)dt , Z0 are C1 with derivatives given by T′(u)(v) = |∇u(x)|p−2∇u(x)∇v(x)dx ZΩ J′(u)(v) = f(x,u(x))v(x)dx f ZΩ 6 for all u,v ∈ W1,p(Ω). Consequently, the weak solutions of problem (P ) are exactly the 0 f critical points in W1,p(Ω) of the functional T −J which is called the energy functional of 0 f problem (P ). Moreover, J′ is compact, while T′ is a homeomorphism between W1,p(Ω) f f 0 and its dual. The announced application of Theorem 1 is as follows: THEOREM 2. - Let q > p, with q < pn when n > p, and let f,g,h: Ω×R → R be n−p three functions belonging to A and satisfying the following conditions: inf F(x,ξ) sup F(x,ξ) lim x∈Ω = +∞ , limsup x∈Ω < +∞ , (12) ξ→+∞ ξp |ξ|→+∞ |ξ|q inf G(x,ξ) x∈Ω lim = +∞ , (13) |ξ|→+∞ |ξ|q sup H(x,ξ) inf H(x,ξ) limsup x∈Ω ≤ 0 , liminf x∈Ω > −∞ , (14) |ξ|→+∞ |ξ|p |ξ|→+∞ |ξ|p sup F(x,ξ) limsup x∈Ω < +∞ , (15) |ξ|p ξ→0 inf G(x,ξ) x∈Ω liminf ≥ 0 , (16) ξ→0 |ξ|p sup H(x,ξ) limsup x∈Ω ≤ 0 . (17) |ξ|p ξ→0 Finally, assume that there exist a measurable set B ⊂ Ω, with meas(B) > 0, and ξ ∈ R 1 such that max{−F(x,ξ ),G(x,ξ ),−H(x,ξ )} < 0 1 1 1 for all x ∈ B. Under such hypotheses, for each ν > 0 large enough, there exists ǫ > 0 with the ν following property: for each ǫ ∈]0,ǫ [ there exists λ∗ > 0 such that the problem ν −div(|∇u|p−2∇u) = ǫf(x,u)−λ∗g(x,u)+νh(x,u) in Ω ( u = 0 on ∂Ω has at least three non-zero weak solutions, two of which are global minima in W1,p(Ω) of 0 the corresponding energy functional. PROOF. First, observe that from the first assumption in (12) it follows J (u) f limsup = +∞ . kukp kuk→+∞ This is proved in the proof of Theorem 4 of [5], and so we do not repeat the argument here. Moreover, from the second assumption in (12) and from (13), it clearly follows that, 7 for each λ > 0, the function λG − F is bounded below in R (see [5] again), and so the functional λJ −J is bounded below in W1,p(Ω). Moreover, by (14), there is c > 0 and, g f 0 for each ǫ > 0, another c > 0, such that ǫ −c(|ξ|p +1) ≤ H(x,ξ) ≤ ǫ|ξ|p +c ǫ for all (x,ξ) ∈ Ω×R. This clearly implies that J (u) h limsup ≤ 0 kukp kuk→+∞ and J (u) h liminf > −∞ . kuk→+∞ kukp Furthermore, by (15) and by the second assumption in (12), there is a constant c > 0 1 such that F(x,ξ) ≤ c (|ξ|p+|ξ|q) 1 for all (x,ξ) ∈ Ω×R. Since q > p, this implies that J (u) f limsup < +∞ . kukp u→0 Now, suppose n ≥ p. By (16) and (17), taking into account that g,h ∈ A, for some r > p and for each ǫ > 0 there is d > 0 such that ǫ G(x,ξ) ≥ −ǫ|ξ|p −d |ξ|r ǫ and H(x,ξ) ≤ ǫ|ξ|p +d |ξ|r ǫ for all (x,ξ) ∈ Ω×R. From this, we get J (u) g liminf ≥ 0 u→0 kukp and J (u) h limsup ≤ 0 . kukp u→0 In the case n < p, we get again these two inequalities thanks to (16) and (17) and to the continuous embedding of W1,p(Ω) into C0(Ω). Now, let ω ∈ L1(Ω) be such that 0 max{|F(x,ξ)|,|G(x,ξ)|,|H(x,ξ)|}≤ ω(x) for all (x,ξ) ∈ Ω×[−|ξ |,|ξ |]. Next, choose a closed set C ⊂ B, an open set D ⊂ Ω, with 1 1 C ⊂ D, and θ ∈ R in such a way that η := max − F(x,ξ )dx, G(x,ξ )dx,− H(x,ξ )dx < θ < 0 1 1 1 (cid:26) ZC ZC ZC (cid:27) 8 and ω(x)dx < θ−η . ZD\C Finally, let v : Ω → [−|ξ |,|ξ |] be a function belonging to W1,p(Ω) such that v (x) = ξ 1 1 1 0 1 1 for all x ∈ C and v (x) = 0 for all x ∈ Ω\D. Clearly, we have 1 max − F(x,v (x))dx, F(x,v (x))dx,− H(x,v (x))dx < η +(θ−η) < 0 . 1 1 1 (cid:26) ZΩ ZΩ ZΩ (cid:27) At this point, the conclusion comes directly from that of Theorem 1 applied taking I(u) = 1kukp, J(u) = −J (u), Ψ(u) = −J (u), Φ(u) = J (u) for all u ∈ W1,p(Ω). △ p h f g 0 References [1] G. CORDARO, On a minimax problem of Ricceri, J. Inequal. Appl., 6 (2001), 261-285. [2] N. GHOUSSOUB and D. PREISS, A general mountain pass principle for locating and classifying critical points, Ann. Inst. H. Poincar´e Anal. Non Lin´eaire, 6 (1989), 321-330. [3] B. RICCERI, Multiplicity of global minima for parametrized functions, Rend. Lincei Mat. Appl., 21 (2010), 47-57. [4] B.RICCERI, A class of nonlinear eigenvalue problems with four solutions, J. Nonlinear Convex Anal., 11 (2010), 503-511. [5] B. RICCERI, A further refinement of a three critical points theorem, Nonlinear Anal., 74 (2011), 7446-7454. [6] E. ZEIDLER, Nonlinear functional analysis and its applications, vol. III, Springer- Verlag, 1985. Department of Mathematics University of Catania Viale A. Doria 6 95125 Catania Italy e-mail address: [email protected] 9