A class of functionals possessing multiple global minima
aa r X i v : . [ m a t h . A P ] F e b A class of functionals possessing multiple global minima
BIAGIO RICCERI
To Professor Gheorghe Morosanu, with friendship, on his 70th birthday
Abstract.
We get a new multiplicity result for gradient systems. Here is a very particular corollary:Let Ω ⊂ R n ( n ≥
2) be a smooth bounded domain and let Φ : R → R be a C function, with Φ(0 ,
0) = 0,such that sup ( u,v ) ∈ R | Φ u ( u, v ) | + | Φ v ( u, v ) | | u | p + | v | p < + ∞ where p >
0, with p = n − when n > S ⊆ L ∞ (Ω) × L ∞ (Ω) dense in L (Ω) × L (Ω), there exists ( α, β ) ∈ S suchthat the problem − ∆ u = ( α ( x ) cos(Φ( u, v )) − β ( x ) sin(Φ( u, v )))Φ u ( u, v ) in Ω − ∆ v = ( α ( x ) cos(Φ( u, v )) − β ( x ) sin(Φ( u, v )))Φ v ( u, v ) in Ω u = v = 0 on ∂ Ωhas at least three weak solutions, two of which are global minima in H (Ω) × H (Ω) of the functional( u, v ) → (cid:18)Z Ω |∇ u ( x ) | dx + Z Ω |∇ v ( x ) | dx (cid:19) − Z Ω ( α ( x ) sin(Φ( u ( x ) , v ( x ))) + β ( x ) cos(Φ( u ( x ) , v ( x )))) dx . Mathematics Subject Classification (2010):
Keywords: minimax; multiple global minima; variational methods; semilinear elliptic systems.
1. Introduction
Let S be a topological space. A function g : S → R is said to be inf-compact if, for each r ∈ R , the set g − (] − ∞ , r ]) is compact.If Y is a real interval and f : S × Y → R is a function inf-compact and lower semicontinuous in S , andconcave in Y , the occurrence of the strict minimax inequalitysup Y inf S f < inf S sup Y f implies that the interior of the set A of all y ∈ Y for which f ( · , y ) has at least two local minima is non-empty.This fact was essentially shown in [4], giving then raise to an enormous number of subsequent applications1o the multiplicity of solutions for nonlinear equations of variational nature (see [7] for an account up to2010).In [6] (see also [5]), we realized that, under the same assumptions as above, the occurrence of the strictminimax inequality also implies the existence of ˜ y ∈ Y such that the function f ( · , ˜ y ) has at least two globalminima. It may happen that ˜ y is unique and does not belong to the closure of A (see Example 7 of [1]).In [8] and [12], we extended the result of [6] to the case where Y is an arbitrary convex set in a vectorspace. We also stress that such an extension is not possible for the result of [4]. We then started to build anetwork of applications of the results of [8] and [12] which touches several different topics: uniquely remotalsets in normed spaces ([8]); non-expansive operators ([9]); singular points ([10]); Kirchhoff-type problems([11]); Lagrangian systems of relativistic oscillators ([13]); integral functional of the Calculus of Variations([14]); non-cooperative gradient systems ([15]); variational inequalities ([16]).The aim of this short paper is to establish a further application within that network.
2. Results
The main abstract result is as follows:THEOREM 2.1. -
Let X be a topological space, ( Y, h· , · , i ) a real Hilbert space, T ⊆ Y a convex set densein Y and I : X → R , ϕ : X → Y two functions such that, for each y ∈ T , the function x → I ( x )+ h ϕ ( x ) , y i islower semicontinuous and inf-compact. Moreover, assume that there exists a point x ∈ X , with ϕ ( x ) = 0 ,such that ( a ) x is a global minimum of both functions I and k ϕ ( · ) k ; ( b ) inf x ∈ X h ϕ ( x ) , ϕ ( x ) i < k ϕ ( x ) k .Then, for each convex set S ⊆ T dense in Y , there exists ˜ y ∈ S such that the function x → I ( x )+ h ϕ ( x ) , ˜ y i has at least two global minima in X . PROOF. In view of ( b ), we can find ˜ x ∈ X and r > I (˜ x ) + r k ϕ ( x ) k h ϕ (˜ x ) , ϕ ( x ) i < I ( x ) + r k ϕ ( x ) k . (1)Thanks to ( a ), we have I ( x ) + r k ϕ ( x ) k = inf x ∈ X ( I ( x ) + r k ϕ ( x ) k ) . (2)The function y → inf x ∈ X ( I ( x ) + h ϕ ( x ) , y i ) is weakly upper semicontinuous, and so there exists ˜ y ∈ B r suchthat inf x ∈ X ( I ( x ) + h ϕ ( x ) , ˜ y i ) = sup y ∈ B r inf x ∈ X ( I ( x ) + h ϕ ( x ) , y i ) , (3) B r being the closed ball in X , centered at 0, of radius r . We distinguish two cases. First, assume that˜ y = rϕ ( x ) k ϕ ( x k . As a consequence, taking into account that r k ϕ ( x ) k is the maximum of the restriction to B r ofthe continuous linear functional h ϕ ( x ) , ·i (attained at the point rϕ ( x ) k ϕ ( x ) k only), we haveinf x ∈ X ( I ( x ) + h ϕ ( x ) , ˜ y i ) ≤ I ( x ) + h ϕ ( x ) , ˜ y i < I ( x ) + r k ϕ ( x ) k . (4)Now, assume that ˜ y = rϕ ( x ) k ϕ ( x k . In this case, due to (1), we haveinf x ∈ X ( I ( x ) + h ϕ ( x ) , ˜ y i ) ≤ I (˜ x ) + h ϕ (˜ x ) , ˜ y i = I (˜ x ) + r k ϕ ( x ) k h ϕ (˜ x ) , ϕ ( x ) i < I ( x ) + r k ϕ ( x ) k . (5)Therefore, from (2), (3), (4) and (5), it follows thatsup y ∈ B r inf x ∈ X ( I ( x ) + h ϕ ( x ) , y i ) < inf x ∈ X sup y ∈ B r ( I ( x ) + h ϕ ( x ) , y i ) . (6)2ow, let S ⊆ T be a convex set dense in Y . By continuity, we clearly havesup y ∈ B r ∩ S h ϕ ( x ) , y i = sup y ∈ B r h ϕ ( x ) , y i for all x ∈ X . Therefore, in view of (6), we havesup y ∈ B r ∩ S inf x ∈ X ( I ( x )+ h ϕ ( x ) , y i ) ≤ sup y ∈ B r inf x ∈ X ( I ( x )+ h ϕ ( x ) , y i ) < inf x ∈ X sup y ∈ B r ( I ( x )+ h ϕ ( x ) , y i ) = inf x ∈ X sup y ∈ B r ∩ S ( I ( x )+ h ϕ ( x ) , y i ) . At this point, the conclusion follows directly applying Theorem 1.1 of [12] to the restriction of the function( x, y ) → I ( x ) + h ϕ ( x ) , y i to X × ( B r ∩ S ). △ We now present an application of Theorem 2.1 to elliptic systems.In the sequel, Ω ⊆ R n ( n ≥
2) is a bounded domain with smooth boundary.We denote by A the class of all functions H : Ω × R → R which are measurable in Ω, C in R andsatisfy sup ( x,u,v ) ∈ Ω × R | H u ( x, u, v ) | + | H v ( x, u, v ) | | u | p + | v | p < + ∞ where p >
0, with p < n +2 n − when n > H ∈ A , we are interested in the problem − ∆ u = H u ( x, u, v ) in Ω − ∆ v = H v ( x, u, v ) in Ω u = v = 0 on ∂ Ω , ( P H ) H u (resp. H v ) denoting the derivative of H with respect to u (resp. v ).As usual, a weak solution of ( P H ) is any ( u, v ) ∈ H (Ω) × H (Ω) such that Z Ω ∇ u ( x ) ∇ ϕ ( x ) dx = Z Ω H u ( x, u ( x ) , v ( x )) ϕ ( x ) dx , Z Ω ∇ v ( x ) ∇ ψ ( x ) dx = Z Ω H v ( x, u ( x ) , v ( x )) ψ ( x ) dx for all ϕ, ψ ∈ H (Ω).Define the functional I H : H (Ω) × H (Ω) → R by I H ( u, v ) = 12 (cid:18)Z Ω |∇ u ( x ) | dx + Z Ω |∇ v ( x ) | dx (cid:19) − Z Ω H ( x, u ( x ) , v ( x )) dx for all ( u, v ) ∈ H (Ω) × H (Ω).Since H ∈ A , the functional I H is C in H (Ω) × H (Ω) and its critical points are precisely the weak solu-tions of ( P H ). Moreover, due to the Sobolev embedding theorem, the functional ( u, v ) → R Ω H ( x, u ( x ) , v ( x ))has a compact derivative and, as a consequence, it is sequentially weakly continuous in H (Ω) × H (Ω).Also, we denote by λ the first eigenvalue of the Dirichlet problem ( − ∆ u = λu in Ω u = 0 on ∂ Ω .Our result is as follows: 3HEOREM 2.2. -
Let
F, G ∈ A , with p = n − when n > , and let K ∈ A , with K ( x, ,
0) = 0 for all x ∈ Ω , satisfy the following conditions: ( a ) one has lim s + t → + ∞ sup x ∈ Ω ( | F ( x, s, t ) | + | G ( x, s, t ) | ) s + t = 0 ;( a ) there is η ∈ (cid:3) , λ (cid:2) such that K ( x, s, t ) ≤ η ( s + t ) for all x ∈ Ω , s, t ∈ R ; ( a ) one has meas( { x ∈ Ω : 0 < | F ( x, , | + | G ( x, , | } ) > and | F ( x, , | + | G ( x, , | ≤ | F ( x, s, t ) | + | G ( x, s, t ) | (8) for all x ∈ Ω , s, t ∈ R ; ( a ) one has meas (cid:18)(cid:26) x ∈ Ω : inf ( s,t ) ∈ R ( F ( x, , F ( x, s, t ) + G ( x, , G ( x, s, t )) < | F ( x, , | + | G ( x, , | (cid:27)(cid:19) > . Then, for every convex set S ⊆ L ∞ (Ω) × L ∞ (Ω) dense in L (Ω) × L (Ω) , there exists ( α, β ) ∈ S such thatthe problem − ∆ u = α ( x ) F u ( x, u, v ) + β ( x ) G u ( x, u, v ) + K u ( x, u, v ) in Ω − ∆ v = α ( x ) F v ( x, u, v ) + β ( x ) G v ( x, u, v ) + K v ( x, u, v ) in Ω u = v = 0 on ∂ Ω has at least three weak solutions, two of which are global minima in H (Ω) × H (Ω) of the functional ( u, v ) → (cid:18)Z Ω |∇ u ( x ) | dx + Z Ω |∇ v ( x ) | dx (cid:19) − Z Ω ( α ( x ) F ( x, u ( x ) , v ( x ))+ β ( x ) G ( x, u ( x ) , v ( x ))+ K ( x, u ( x ) , v ( x ))) dx . PROOF. We are going to apply Theorem 2.1, with the following choices: X is the space H (Ω) × H (Ω)endowed with the weak topology induced by the scalar product h ( u, v ) , ( w, ω ) i X = Z Ω ( ∇ u ( x ) ∇ w ( x ) + ∇ v ( x ) ∇ ω ( x )) dx ; Y is the space L (Ω) × L (Ω) with the scalar product h ( f, g ) , ( h, k ) i Y = Z Ω ( f ( x ) h ( x ) + g ( x ) k ( x )) dx ; T is L ∞ (Ω) × L ∞ (Ω); I is the function defined by I ( u, v ) = 12 (cid:18)Z Ω |∇ u ( x ) | dx + Z Ω |∇ v ( x ) | dx (cid:19) − Z Ω K ( x, u ( x ) , v ( x )) dx for all ( u, v ) ∈ X ; ϕ is the function defined by ϕ ( u, v ) = ( F ( · , u ( · ) , v ( · )) , G ( · , u ( · ) , v ( · )))for all ( u, v ) ∈ X ; x is the zero of X . Let us show that the assumptions of Theorem 2.1 are satisfied. First,from (7) and (8) it clearly follows, respectively, that k ϕ (0 , k Y = Z Ω ( | F ( x, , | + | G ( x, , | ) dx > k ϕ (0 , k Y ≤ k ϕ ( u, v ) k Y for all ( u, v ) ∈ X . Moreover, from ( a ), thanks to the Poincar´e inequality, we get Z Ω K ( x, u ( x ) , v ( x )) dx ≤ η Z Ω ( | u ( x ) | + | v ( x ) | ) dx ≤ ηλ Z Ω ( |∇ u ( x ) | + |∇ v ( x ) | ) dx (9)for all ( u, v ) ∈ X . In particular, since K ( x, ,
0) = 0 in Ω and ηλ < , from (9) we infer that (0 ,
0) is a globalminimum of I in X . So, condition ( a ) is satisfied. Now, let us verify condition ( b ). To this end, set P ( x, s, t ) = F ( x, , F ( x, s, t ) + G ( x, , G ( x, s, t ) − | F ( x, , | − | G ( x, , | for all ( x, s, t ) ∈ Ω × R and D = (cid:26) x ∈ Ω : inf ( s,t ) ∈ R P ( x, s, t ) < (cid:27) . By ( a ), D has a positive measure. In view of the Scorza-Dragoni theorem, there exists a compact set C ⊂ D , with positive measure, such that the restriction of P to C × R is continuous. Fix a point ˜ x ∈ C such that the intersection of C and any ball centered at ˜ x has a positive measure. Choose ˜ s, ˜ t ∈ R \ { } sothat P (˜ x, ˜ s, ˜ t ) <
0. By continuity, there is r > P ( x, ˜ s, ˜ t ) < x ∈ C ∩ B (˜ x, r ). Set γ = sup ( x,s,t ) ∈ Ω × [ −| ˜ s | , | ˜ s | ] × [ −| ˜ t | , | ˜ t | ] | P ( x, t, s ) | . Since
F, G ∈ A , γ is finite. Now, choose an open set A such that C ∩ B (˜ x, r ) ⊂ A ⊂ Ωand meas( A \ ( C ∩ B (˜ x, r ))) < − R C ∩ B (˜ x,r ) P ( x, ˜ s, ˜ t ) dxγ . (10)Finally, choose two functions ˜ u, ˜ v ∈ H (Ω) such that˜ u ( x ) = ˜ s , ˜ v ( x ) = ˜ t for all x ∈ C ∩ B (˜ x, r ) , ˜ u ( x ) = ˜ v ( x ) = 0for all x ∈ Ω \ A and | ˜ u ( x ) | ≤ | ˜ s | , | ˜ v ( x ) | ≤ | ˜ t | for all x ∈ Ω. Then, taking (10) into account, we have h ϕ (˜ u, ˜ v ) , ϕ (0 , i Y −k ϕ (0 , k Y = Z Ω P ( x, ˜ u ( x ) , ˜ v ( x )) dx = Z C ∩ B (˜ x,r ) P ( x, ˜ s, ˜ t ) dx + Z A \ ( C ∩ B (˜ x,r )) P ( x, ˜ u ( x ) , ˜ v ( x )) dx< Z C ∩ B (˜ x,r ) P ( x, ˜ s, ˜ t ) dx + γ meas( A \ ( C ∩ B (˜ x, r )) < . This shows that ( b ) is satisfied. Finally, fix α, β ∈ L ∞ (Ω). Clearly, the function( x, s, t ) → α ( x ) F ( x, s, t ) + β ( x ) F ( x, s, t ) + K ( x, s, t )5elongs to A , and so the functional ( u, v ) → I ( u, v ) + h ϕ ( u, v ) , ( α, β ) i Y is sequentially weakly lower semicontinuous in X . Let us show that it is coercive. Set θ = max (cid:8) k α k L ∞ (Ω) , k β k L ∞ (Ω) (cid:9) and fix ǫ > ǫ < θ (cid:18) λ − η (cid:19) . (11)By ( a ), there is c ǫ > | F ( x, s, t ) | + | G ( x, s, t ) | ≤ ǫ ( | s | + | t | ) + c ǫ for all ( x, s, t ) ∈ Ω × R . Then, for each u, v ∈ H (Ω), recalling (9), we have I ( u, v )+ h ϕ ( u, v ) , ( α, β ) i Y ≥ (cid:18) − ηλ (cid:19) Z Ω ( |∇ u ( x ) | + |∇ v ( x ) | ) dx − Z Ω | α ( x ) F ( x, u ( x ) , v ( x ))+ β ( x ) G ( x, u ( x ) , v ( x )) | dx ≥ (cid:18) − ηλ (cid:19) Z Ω ( |∇ u ( x ) | + |∇ v ( x ) | ) dx − θǫ Z Ω ( | u ( x ) | + | v ( x ) | ) dx − θc ǫ meas(Ω) ≥ (cid:18) − ηλ − θǫλ (cid:19) Z Ω ( |∇ u ( x ) | + |∇ v ( x ) | ) dx − θc ǫ meas(Ω) . Notice that, in view of (11), we have − ηλ − θǫλ >
0, and solim k ( u,v ) k X → + ∞ ( I ( u, v ) + h ϕ ( u, v ) , ( α, β ) i Y ) = + ∞ , as claimed. In particular, this also implies that the functional ( u, v ) → I ( u, v ) + h ϕ ( u, v ) , ( α, β ) i Y is weaklylower semicontinuous, by the Eberlein-Smulyan theorem. Thus, the assumptions of Theorem 2.1 are satisfied.Therefore, for each convex set S ⊆ L ∞ (Ω) × L ∞ (Ω) dense in H (Ω) × H (Ω), there exists ( α, β ) ∈ S , suchthat the functional( u, v ) → (cid:18)Z Ω |∇ u ( x ) | dx + Z Ω |∇ v ( x ) | dx (cid:19) − Z Ω ( α ( x ) F ( x, u ( x ) , v ( x ))+ β ( x ) G ( x, u ( x ) , v ( x ))+ K ( x, u ( x ) , v ( x ))) dx has at least two global minima in H (Ω) × H (Ω). Finally, by Example 38.25 of [17], the same functionalsatisfies the Palais-Smale condition, and so it admits at least three critical points, in view of Corollary 1 of[3]. The proof is complete. △ REMARK 2.1. - We are not aware of known results close enough to Theorem 2.2 in order to do a propercomparison. This sentence also applies to the case of single equations, that is to say when
F, G, K dependon x and s only. For an account on elliptic systems, we refer to [2].Among the various corollaries of Theorem 2.2, we wish to stress the following ones:COROLLARY 2.1. - Let K ∈ A , with K ( x, ,
0) = 0 for all x ∈ Ω , satisfy condition ( a ) . Moreover, let Φ : R → R be a non-constant C function, with Φ(0 ,
0) = 0 , belonging to A , with p = n − when n > .Then, for every convex set S ⊆ L ∞ (Ω) × L ∞ (Ω) dense in L (Ω) × L (Ω) , there exists ( α, β ) ∈ S suchthat the problem − ∆ u = ( α ( x ) cos(Φ( u, v )) − β ( x ) sin(Φ( u, v )))Φ u ( u, v ) + K u ( x, u, v ) in Ω − ∆ v = ( α ( x ) cos(Φ( u, v )) − β ( x ) sin(Φ( u, v )))Φ v ( u, v ) + K v ( x, u, v ) in Ω u = v = 0 on ∂ Ω6 as at least three weak solutions, two of which are global minima in H (Ω) × H (Ω) of the functional ( u, v ) → (cid:18)Z Ω |∇ u ( x ) | dx + Z Ω |∇ v ( x ) | dx (cid:19) − Z Ω ( α ( x ) sin(Φ( u ( x ) , v ( x ))) + β ( x ) cos(Φ( u ( x ) , v ( x ))) + K ( x, u ( x ) , v ( x ))) dx . PROOF. It suffices to apply Theorem 2.2 to the functions
F, G : R → R defined by F ( s, t ) = sin(Φ( s, t )) ,G ( s, t ) = cos(Φ( s, t ))for all ( s, t ) ∈ R . △ COROLLARY 2.2. -
Let
F, G : R → R belong to A , with p = n − when n > . Moreover, assume that F, G are twice differentiable at and that < | F (0) | + | G (0) | = inf s ∈ R ( | F ( s ) | + | G ( s ) | ) ,F ′′ (0) F (0) + G ′′ (0) G (0) < . (12) Then, for every convex set S ⊆ L ∞ (Ω) × L ∞ (Ω) dense in L (Ω) × L (Ω) , there exists ( α, β ) ∈ S such thatthe problem ( − ∆ u = α ( x ) F ′ ( u ) + β ( x ) G ′ ( u ) in Ω u = 0 on ∂ Ω has at least three weak solutions, two of which are global minima in H (Ω) of the functional u → Z Ω |∇ u ( x ) | dx − Z Ω ( α ( x ) F ( u ( x )) + β ( x ) G ( u ( x ))) dx . PROOF. We apply Theorem 2.2 taking K = 0. Since 0 is a global minimum of the function | F ( · ) | + | G ( · ) | , we have F ′ (0) F (0) + G ′ (0) G (0) = 0and so, in view of (12), 0 is a strict local maximum for the function F ( · ) F (0) + G ( · ) G (0). Hence, ( a ) issatisfied and Theorem 2.2 gives the conclusion. △ eferences [1] A. CABADA and A. IANNIZZOTTO, A note on a question of Ricceri , Appl. Math. Lett., (2012),215-219.[2] D. G. de FIGUEIREDO, Semilinear elliptic systems: existence, multiplicity, symmetry of solutions .Handbook of differential equations: stationary partial differential equations. Vol. V, 1-48, Handb. Differ.Equ., Elsevier/North-Holland, Amsterdam, 2008.[3] P. PUCCI and J. SERRIN,
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