A MOUNTAIN PASS THEOREM FOR A SUITABLE CLASS OF

A MOUNTAIN PASS THEOREM FOR A SUITABLE CLASS OF FUNCTIONS DIEGO AVERNA AND GABRIELE BONANNO Abstract. The main purpose of this paper is to establish a three critical points result without assuming the coercivity of the involved functional. To this end, a mountain-pass theorem, where the usual Palais-Smale condition is not requested, is presented. These results are then applied to prove the existence of three solutions for a two-point boundary value problem with no asymptotic conditions.

1. Introduction It is well known that the mountain-pass theorem of A. Ambrosetti and P.H. Rabinowitz [1, Theorem 2.1] and its variants or generalizations, as for instance Theorem 1 of [17], is successfully used to find critical points of real-valued C 1 functions J defined on an infinite-dimensional Banach space X. One of the key assumptions in this result is a compactness hypothesis, usually called Palais-Smale condition. The present paper deals with the case J(x) = Φ(x) − Ψ(x),

x ∈ X,

which often occurs in the variational formulation of both ordinary and partial differential problems. We first introduce a new type of Palais-Smale condition; see Section 3. It is mutually independent from the usual condition and holds true every time that Φ, Ψ turn out sufficiently smooth and Φ is coercive; see Theorem 3.1. A mountain pass-like result, which also provides a more precise localization of the obtained critical point as regards the function Φ, is then established (see Theorem 4.3). Moreover, putting this result and Theorem 2.1 in [5] together yields the main result of the paper, which is a three critical points theorem for the functional Jλ (x) = Φ(x) − λΨ(x),

x ∈ X,

where λ > 0, whose norms are uniformly bounded with respect to λ; see Theorem 5.2. Let us point out that, contrary to the basic result of B. Ricceri [20, Theorem 1] on this topic, no coercivity of Jλ is assumed. To prove our results we employ tools from Key words and phrases. Palais-Smale condition, mountain pass, critical points, three solutions, two-point boundary value problem. 2000 Mathematics Subject Classification: 47J30, 58E30, 49J40, 58E05, 34B15. Corresponding author: G.Bonanno. This research was partially supported by RdB (ex 60% MIUR) of Reggio Calabria University. Typeset by LATEX 2ε . 1

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DIEGO AVERNA AND GABRIELE BONANNO

the theory of generalized gradient for locally Lipschitz continuous functions, which has been introduced and developed by F.H. Clarke [12]. We make further use of a critical point theorem in this framework, namely Theorem 2.2 of [15]. The critical point theory for locally Lipschitz continuous functionals, including applications to elliptic problems with discontinuous nonlinearities, has been introduced and extensively studied by K.-C. Chang in [11]; see also [16] for an in-depth account in this field. The results of the present paper deal with smooth functions in a natural way, using the non-smooth theory. As an application of these results to nonlinear differential problems we present an existence theorem of three solutions for a two-point boundary value problem (see Theorem 6.1). The assumptions of Theorem 6.2, which is a consequence of Theorem 6.1, are that there is a growth which is greater than quadratic of the antiderivative for the function in a suitable interval (see assumption (k)), and a growth less than quadratic of the same antiderivative in a following, suitable interval (see assumption (kk)). By way of example, here, we present the following result, which is a particular case of Theorem 6.2. Theorem 1.1. Let g : [0, 1] → IR and f : IR → IR be two continuous, non-negative R

and non-zero functions. Put F (x) =

Rx 0

f (t) dt for all x ∈ IR and g0 = R

assume that there are three positive constants c, d, p, with c < d < (k)

F (c) c2


≤ I 0 (u; v) ∀v ∈ X}. We recall that if I is continuously Gˆateaux differentiable at u, then I is Lipschitz near u and ∂I(u) = {I 0 (u)}. Further, a point u ∈ X is called a (generalized) critical point of the locally Lipschitz function I if 0X ∗ ∈ ∂I(u), namely I 0 (u; v) ≥ 0 for all v ∈ X. Clearly, if I is a continuously Gˆateaux differentiable at u, then u becomes a (classical) critical point of I, that is I 0 (u) = 0X ∗ . Finally, we say that a locally Lipschitz function I satisfies the Palais-Smale condition (P.S.)c , c ∈ IR, if each sequence {un } such that (1) I(un ) → c, (2) min kvkX ∗ → 0 v∈∂I(un )

possesses a convergent subsequence. It is well known that condition (2) and (2’) I 0 (un , v − un ) ≥ −n kv − un k ∀v ∈ X, where n → 0+ ,

4

DIEGO AVERNA AND GABRIELE BONANNO

are equivalent. Moreover, when I is a continuously Gˆateaux differentiable function, the (P.S.)c condition is reduced to the classical one, namely each sequence {un } such that (1) I(un ) → c, (2) kI 0 (un )kX ∗ → 0 possesses a convergent subsequence. For a thorough treatment on these topics we refer to [11], [12] and [18]. Now, given two functions Φ, Ψ : X → IR, we define the following functions: Ψ(x) − Ψ(x)

sup ϕ1 (r) :=

x∈Φ−1 (]−∞,r[)

inf

x∈Φ−1 (]−∞,r[)

w

(2.1)

r − Φ(x)

ϕ1 (r1 , r2 ) := max{ϕ1 (r1 ); ϕ1 (r2 )}

ϕ2 (r1 , r2 ) =

inf

x∈Φ−1 (]−∞,r

(2.2)

Ψ(y) − Ψ(x) 1 [) y∈Φ−1 ([r1 ,r2 [) Φ(y) − Φ(x) sup

(2.3)

w

for all r, r1 , r2 > inf X Φ, with r1 < r2 and where Φ−1 (] − ∞, r[) is the closure of Φ−1 (] − ∞, r[) in the weak topology. For the reader’s convenience we recall below the theorem obtained in [5] which ensures the existence of a precise open interval Λ ⊆]0, +∞[ such that for each λ ∈ Λ the function J = Φ − λΨ admits two local minima which are uniformly bounded in norm with respect to λ. Theorem 2.1. [5, Theorem 2.1] Let X be a reflexive real Banach space, and let Φ, Ψ : X → IR two functions. Assume that Φ is sequentially weakly lower semicontinuous, (strongly) continuous and coercive, and Ψ is sequentially weakly upper semicontinuous. Assume also that there exist two constants r1 and r2 such that (B1 )

inf X Φ < r1 < r2 ;

(B2 )

ϕ1 (r1 , r2 ) < ϕ2 (r1 , r2 ); i

h

Then, for each λ ∈ ϕ2 (r11 ,r2 ) , ϕ1 (r11 ,r2 ) the restriction of Φ − λΨ to Φ−1 (] − ∞, r1 [) admits a global minimum v1 and the restriction of Φ − λΨ to Φ−1 (] − ∞, r2 [) admits a global minimum v2 ∈ Φ−1 ([r1 , r2 [), which are two local minima for Φ − λΨ. We also recall that the proof of Theorem 2.1 is based on the variational principle of B.Ricceri in [19].

A MOUNTAIN PASS THEOREM...

5

3. A new type of Palais-Smale condition Let Φ, Ψ : X → IR be two locally Lipschitz functions. Of course, the function J =Φ−Ψ is a locally Lipschitz function. Fix M ∈ IR and put (

ΨM (u) =

Ψ(u) if Ψ(u) ≤ M M if Ψ(u) > M.

It is simple to show that also ΨM is a locally Lipschitz function. We now give the following definition. Definition 3.1. We say that the function J = Φ − Ψ satisfies the Palais-Smale condition cut off at M (briefly (P.S.)M c ) if the function JM = Φ − ΨM satisfies the Palais-Smale condition. It is easy to verify that the (P.S.)M c condition and the (P.S.)c condition on J are mutually independent. For instance, it is enough to pick X = IR, Φ = 0, Ψ(x) = −1/2x2 ; it is clear that J(x) = 1/2x2 satisfies the (P.S.)c condition, while J1 does not satisfy the (P.S.)c condition. On the contrary, by choosing X = IR, Φ = 1/2x2 , Ψ(x) = −1/2x2 , JM (with M > 0) satisfies the (P.S.)c condition, while J does not satisfy the (P.S.)c condition. Now, we emphasize the following theorem that guarantees the (P.S.)M c condition 1 for all M ∈ IR for functions of class C . Theorem 3.1. Let Φ, Ψ : X → IR be two continuously Gˆ ateaux differentiable functions, with Φ coercive. Assume that Φ0 : X → X ∗ admits a continuous inverse operator on X ∗ , and Ψ0 : X → X ∗ is compact. Then, the function Φ − Ψ satisfies the (P.S.)M c condition for all M ∈ IR. Proof. Fix M ∈ IR and let {un } be a sequence such that (1) Φ(un ) − ΨM (un ) → c, c ∈ IR, (2) min kξkX ∗ → 0. ξ∈∂(Φ−ΨM )(un )

Since Φ is coercive, also Φ − ΨM is coercive and, hence, from (1) we obtain that {un } is bounded. Moreover, taking into account that ∂(Φ − ΨM )(un ) = ∂(Φ)(un ) − ∂(ΨM )(un ) = Φ0 (un ) − ∂(ΨM )(un ) and

Ψ0 (un ) 0 ∂(ΨM )(un ) =   {rΨ0 (u ) : r ∈ [0, 1]} n (see [12, Corollary 2, p. 39; Proposition 2.2.4, p.33; has   

min

ξ∈∂(Φ−ΨM )(un )

kξkX ∗ =

min

ξ∈∂(Φ)(un )−∂(ΨM )(un )

kξkX ∗ =

if Ψ(un ) < M if Ψ(un ) > M if Ψ(un ) = M Proposition 2.3.12, p.47]), one min

η∈∂(ΨM )(un )

kΦ0 (un ) − ηkX ∗ =

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DIEGO AVERNA AND GABRIELE BONANNO

= min kΦ0 (un ) − rΨ0 (un )kX ∗ = kΦ0 (un ) − rn Ψ0 (un )kX ∗ . r∈[0,1]

From the compactness of Ψ0 and the fact that {rn } ⊆ [0, 1], there are two sequences {rnk } and {unk } such that {rnk Ψ0 (unk )} converges. From (2) and the previous equalities, one has kΦ0 (unk ) − rnk Ψ0 (unk )kX ∗ → 0. Hence, {Φ0 (unk )} also converges and,  owing to our assumption on Φ0 , {unk } converges. Remark 3.1. We explicitly observe that if in the previous theorem, instead of Φ coercive, we assume Φ − Ψ coercive, then the function Φ − Ψ satisfies the classical (P.S.)c condition (see, for instance, [22, Example 38.25, page 162]). 4. A mountain pass theorem In this Section we assume that X is a reflexive real Banach space and Φ, Ψ : X → IR are two continuously Gˆateaux differentiable functions. Moreover, we assume that (A1 ) Φ is convex; (A2 ) for every x1 , x2 such that Ψ(x1 ) ≥ 0 and Ψ(x2 ) ≥ 0 one has inf Ψ(tx1 + (1 − t)x2 ) ≥ 0.

t∈[0,1]

Now, we present a version of the classical mountain pass theorem for functions of the type J = Φ − Ψ where a localization of the (classical) critical point is also guaranteed. Theorem 4.1. Assume that there is a positive real number s and two points x1 , x2 ∈ X, with kx2 − x1 k > s and B(x1 , s) ⊆ Φ−1 (] − ∞, ρ[), where ρ > max{Φ(x1 ), Φ(x2 )}, such that J(x) ≥ max{J(x1 ), J(x2 )} (4.1) for all x ∈ ∂B(x1 , s). Assume also that min{Ψ(x1 ), Ψ(x2 )} ≥ 0. Further, assume that (C) there is M > 0 such that (C1 )

sup

Ψ(x) < M .

x∈Φ−1 (]−∞,ρ+M [)

and (C2 ) J satisfies the (P.S.)M c condition for all c ∈ IR. Then, the function J has a (classical) critical point x3 distinct from x1 and x2 such that J(x3 ) ≥ max{J(x1 ); J(x2 )} and Φ(x3 ) < ρ + M.

A MOUNTAIN PASS THEOREM...

7

Proof. Put (

ΨM (u) =

Ψ(u) if Ψ(u) ≤ M M if Ψ(u) > M,

JM (u) = Φ(u) − ΨM (u) and IM (u) = JM (u + x1 ) for all u ∈ X. Clearly, IM is a locally Lipschitz function and, taking into account (C2 ), satisfies the Palais-Smale condition (P.S.)c , c ∈ IR. Now, put e = x2 − x1 and a = max{IM (0); IM (e)}. One has IM (0) ≤ a, IM (e) ≤ a and kek > s. Moreover, taking into account (C1 ), the assumption (4.1) ensures that IM (u) ≥ a for all u ∈ ∂B(0, s). Hence, all the assumptions of the mountain pass theorem for non-differentiable functions (see, for instance, Theorem 2.2 of [15]) are verified. Therefore, there exists a (generalized) critical point y3 of IM such that IM (y3 ) = inf sup IM (γ(t)), γ∈Γ t∈[0,1]

where Γ = {γ ∈ C 0 ([0, 1], X) : γ(0) = 0, γ(1) = e}. Moreover, one has IM (y3 ) ≥ max{IM (0); IM (e)}.

(4.2)

We claim that y3 is a (classical) critical point of IM . In fact, by choosing the segment with endpoints 0 and e as γ, and from (A1 ) and (A2 ), one has IM (y3 ) ≤ supt∈[0,1] IM ((1 − t)e) = supt∈[0,1] IM ((1 − t)(x2 − x1 )) = supt∈[0,1] JM (tx1 + (1 − t)x2 ) = supt∈[0,1] [Φ(tx1 + (1 − t)x2 ) − ΨM (tx1 + (1 − t)x2 )] ≤ supt∈[0,1] [tΦ(x1 ) + (1 − t)Φ(x2 )] − inf t∈[0,1] ΨM (tx1 + (1 − t)x2 ) < ρ, namely IM (y3 ) < ρ.

(4.3)

On the other hand, setting x3 = y3 + x1 , one has IM (y3 ) = JM (y3 + x1 ) = JM (x3 ) = Φ(x3 ) − ΨM (x3 ). Hence, from (4.3) one has Φ(x3 ) < ρ + M. Therefore, owing to (C1 ) one has Ψ(x3 ) < M. So, it follows immediately that IM is a continuously Gˆateaux differentiable at y3 and our claim is proved. Finally, it is easy to verify that x3 is a classical critical point of J and that, from (4.2), we obtain J(x3 ) ≥ max{J(x1 ); J(x2 )}. Hence, the proof is complete.  We also obtain the result below. Theorem 4.2. Assume that J admits two distinct local minima x1 , x2 ∈ X. Put ρ ∈ IR such that max{Φ(x1 ), Φ(x2 )} < ρ and assume that min{Ψ(x1 ), Ψ(x2 )} ≥ 0. Further, assume that (C) there is M > 0 such that (C1 )

sup

Ψ(x) < M .

x∈Φ−1 (]−∞,ρ+M [)

and (C2 ) J satisfies the (P.S.)M c condition for all c ∈ IR.

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DIEGO AVERNA AND GABRIELE BONANNO

Then, the function J has a third (classical) critical point x3 distinct from x1 and x2 such that Φ(x3 ) < ρ + M. Proof. Without loss of generality, we can assume J(x2 ) ≤ J(x1 ). Since x1 ∈ Φ−1 (] − ∞, ρ[), which is an open set, there is r > 0 such that B(x1 , r) ⊆ Φ−1 (] − ∞, ρ[). By choosing s > 0 such that s < min{kx2 − x1 k; r}, the condition (4.1) is easily verified. Hence, the conclusion follows directly from Theorem 4.1.  Finally, we present the main result of this Section. Theorem 4.3. Assume that Φ is coercive, Φ0 : X → X ∗ admits a continuous inverse operator on X ∗ , Ψ0 : X → X ∗ is compact, and J = Φ − Ψ admits two distinct local minima x1 , x2 ∈ X. Put ρ ∈ IR such that max{Φ(x1 ), Φ(x2 )} < ρ and assume that min{Ψ(x1 ), Ψ(x2 )} ≥ 0 and there is M > 0 such that (C1 ) sup Ψ(x) < M . x∈Φ−1 (]−∞,ρ+M [)

Then, the function J has a third critical point x3 distinct from x1 and x2 such that Φ(x3 ) < ρ + M. Proof. It follows directly from Theorem 4.2 and Theorem 3.1.



5. Multiple critical points theorems In this Section we assume that X is a reflexive real Banach space and Φ, Ψ : X → IR are two continuously Gˆateaux differentiable functions. Moreover, we assume that Φ is coercive, and Ψ is sequentially weakly upper semicontinuous. Finally, we also assume that (A1 ) and (A2 ) in Section 4 hold, and (A3 ) inf X Φ = Φ(0) = Ψ(0) = 0. Let r1 , r2 be two constants and ϕ1 (r1 , r2 ), ϕ2 (r1 , r2 ) as given in Section 2. Moreover, given M > 0, we define sup ϕ3 (r2 , M ) =

Ψ(x)

x∈Φ−1 (]−∞,r2 +M [)

M

and ϕ4 (r1 , r2 , M ) = max{ϕ1 (r1 , r2 ); ϕ3 (r2 , M )}. Now, we present the following three critical points theorem. Theorem 5.1. Assume that there exist two constants r1 and r2 such that (B1 ) 0 < r1 < r2 ; (B2 ) and

ϕ1 (r1 , r2 ) < ϕ2 (r1 , r2 );

A MOUNTAIN PASS THEOREM...

9

(C) there is M > 0 such that (C10 ) and

ϕ3 (r2 , M ) < ϕ2 (r1 , r2 ). i

h

(C20 ) for each λ ∈ ϕ2 (r11 ,r2 ) , ϕ4 (r1 1,r2 ,M ) the function Φ−λΨ satisfies the (P.S.)M c condition for all c ∈ IR. i

h

Then, for each λ ∈ ϕ2 (r11 ,r2 ) , ϕ4 (r1 1,r2 ,M ) the function Φ − λΨ admits three distinct critical points xi (i = 1, 2, 3) such that x1 ∈ Φ−1 (] − ∞, r1 [), x2 ∈ Φ−1 ([r1 , r2 [), x3 ∈ Φ−1 (] − ∞, r2 + M [). Proof. Fix λ ∈

i

1 1 , ϕ2 (r1 ,r2 ) ϕ4 (r1 ,r2 ,M )

h

. Taking into account that Φ is also sequeni

h

tially weakly lower semicontinuous and since λ ∈ ϕ2 (r11 ,r2 ) , ϕ1 (r11 ,r2 ) , from Theorem 2.1 one has that the restriction of Φ − λΨ to Φ−1 (] − ∞, r1 [) admits a global minimum x1 and the restriction of Φ − λΨ to Φ−1 (] − ∞, r2 [) admits a global minimum x2 , which are two distinct local minima for Φ − λΨ. Therefore, from (A3 ) one has min{λΨ(x1 ), λΨ(x2 )} ≥ 0. Further, since λ < ϕ3 (r12 ,M ) , one has sup Ψ(x) x∈Φ−1 (]−∞,r2 +M [) < λ1 . From which M sup

λΨ(x) < M,

x∈Φ−1 (]−∞,r2 +M [)

which is (C1 ) of Theorem 4.2 applied to the function Φ − (λΨ). Clearly, from (C20 ) it follows that the function Φ − (λΨ)M satisfies the (P S)c condition, which is (C2 ) of Theorem 4.2 applied to the function Φ − (λΨ). Hence, Theorem 4.2 ensures the conclusion.  Finally, we present the main result of the paper. Theorem 5.2. Assume that Φ0 : X → X ∗ admits a continuous inverse operator on X ∗ , and Ψ0 : X → X ∗ is compact. Assume also that there exist three positive constants r1 , r2 and M , with r1 < r2 , such that (D)

ϕ4 (r1 , r2 , M ) < ϕ2 (r1 , r2 ); i

h

Then, for each λ ∈ ϕ2 (r11 ,r2 ) , ϕ4 (r1 1,r2 ,M ) the function Φ − λΨ admits three distinct critical points xi (i = 1, 2, 3) such that x1 ∈ Φ−1 (] − ∞, r1 [), x2 ∈ Φ−1 ([r1 , r2 [), x3 ∈ Φ−1 (] − ∞, r2 + M [). Proof. It follows from Theorems 5.1 and 3.1 Remark 5.1. Since Φ is coercive, there exists σ > 0 such that Φ−1 (] − ∞, r2 + M [) ⊆ B(0X , σ).



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DIEGO AVERNA AND GABRIELE BONANNO

Therefore, the conclusion of the previous theorems ensures that the three critical points are uniformly bounded in norm with respect to λ, that is kxi k < σ, i = 1, 2, 3, for all λ ∈

i

1 1 , ϕ2 (r1 ,r2 ) ϕ4 (r1 ,r2 ,M )

h

.

Remark 5.2. Recently, three critical points theorems have been established in [3], [5] and [6]. In Theorem 2.1 of [6] (which is based on Theorem 3 of [20]) it was established that the existence of an interval Λ ⊆ [0, +∞[ such that, for each λ ∈ Λ, the function Φ − λΨ has three critical points which are uniformly bounded in norm with respect to λ. However, in Theorem 2.1 of [6], only an upper bound of the interval Λ was guaranteed. On the other hand, in Theorem B of [3] (which is based on the variational principle of B.Ricceri in [19]) a precise interval of parameters, λ, for which the function Φ − λΨ has three critical points was established, losing however the uniform boundedness in norm. For a more precise comparison between Theorem 2.1 of [6] and Theorem B of [3], we refer to [7]. Here, we explicitly observe that the conclusion of previous theorems ensures both a precise interval of parameters, λ, for which the function Φ − λΨ has three critical points and the uniform boundedness in norm of the three critical points with respect to λ. Further, we point out that one of the key assumptions, both of Theorem 2.1 of [6] and of Theorem B of [3], lim (Φ(x) − λΨ(x)) = +∞

kxk→+∞

is not requested in Theorems 5.1 and 5.2. On the other hand, the assumption (C10 ) is not requested in Theorem 2.1 of [6] and Theorem B of [3]. We also observe that the assumptions of Theorem 2.3 of [5] (which is also based on Theorem 2.1) are mutually independent from those of Theorem 5.1 and, in this case, the third critical point is actually the third local minimum point. 6. A two-point boundary value problem The multiple critical points theorems established in [3], [5] and [6] have been applied in several nonlinear differential problems (see, for instance, [3]-[10] and [21]). Our aim is to apply the three critical points theorem (Theorem 5.2) to these types of nonlinear differential problems. The novel situation with respect to previously cited papers is expressed by the assumption there is M > 0 such that (C1 ) sup Ψ(x) < M x∈Φ−1 (]−∞,ρ+M [)

and how it can be translated in differential problems. By way of example, here we consider a two-point boundary value problem to give an application of the results in Section 5. In Theorem 6.1 and Theorem 6.2 below the function f : [0, 1] × IR → IR is an 1 L −Carath´eodory function which is nonnegative in [0, 1] × [0, +∞[, namely

A MOUNTAIN PASS THEOREM...

11

(a)

t → f (t, x) is measurable for every x ∈ IR;

(b)

x → f (t, x) is continuous for almost every t ∈ [0, 1];

(c)

for every ρ > 0 there exists a function lρ ∈ L1 ([0, 1]) such that sup |f (t, x)| ≤ lρ (t) |x|≤ρ

for almost every t ∈ [0, 1], (d) f (t, x) ≥ 0 for almost every t ∈ [0, 1] and all x ≥ 0. Consider the following problem (

−u00 = λf (t, u) u(0) = u(1) = 0,

(Pλ )

where λ is a positive real parameter, and put ξ

Z

F (t, ξ) =

f (t, x)dx

0

for all (t, ξ) ∈ [0, 1] × IR. We have the following theorem. Theorem√ 6.1. Assume that there exist four positive constants c1 , d, c2 , k, with √ 2 2 c1 < d < 2 c2 < 2 k, such that R1

(i)

0

R1

(ii)

0

F (t,c1 )dt c21 F (t,c2 )dt c22

R

<
0 such that ϕ3 (r2 , M ) < ϕ2 (r1 , r2 ). To this end, fix M = (2k 2 − 2c22 ). Taking (6.1) again into account, one has ϕ3 (r2 , M ) =

A MOUNTAIN PASS THEOREM...

sup

Ψ(x)

x∈Φ−1 (]−∞,r2 +M [) M

13

R1

≤

F (t,k) dt 0 , (2k2 −2c22 )

that is R1

F (t, k) dt . (2k 2 − 2c22 ) 0

ϕ3 (r2 , M ) ≤

(6.5)

Therefore, owing to (iii) from (6.4) and (6.5) we obtain as we claimed. Hence, from Theorem 5.2 we obtain that for each  

λ∈ R 3

4 1 4

6d2

(

F (t, d)dt

, min R 1 0

2c21 2c22 2k 2 − 2c22 ; R1 ; R1 F (t, c1 )dt 0 F (t, c2 )dt 0 F (t, k)dt

 )   

the problem (Pλ ) admits three generalized solutions xi , i = 1, 2, 3, which, owing to the maximum principle, are nonnegative. Further, one has Φ(xi ) < r2 + M , that is kxi k∞ < k, and the proof is complete.  Remark 6.1. We observe that the conclusion of Theorem 6.1 can be more precise. In fact, the three solutions satisfy the following conditions: kx1 k∞ < c1 , kx2 k∞ < c2 , kx2 k ≥ 2c1 , kx3 k∞ < k. Remark 6.2. We observe that in Theorem 6.1 the assumptions (i)-(iii) can be expressed in thefollowing, moreZgeneral form Z  Z 1

1 λ2

  

:= max 

0

1

F (t, c1 ) dt

;

2c21

0

1

F (t, c2 ) dt

;

2c22

 

Z

3/4

Z

1/4

0

 F (t, k)dt  

2k2 −2c22


max 2d; c;

(c) 12 FF (d) d;

r

2 FF (c) c (c)



, it is a simple computa-

tion to show that (k)-(kk) of Theorem 6.2 hold and n o 2 2c21 p2 R c 2c = min F ∗ (c1 ) ; F ∗ (p) . Therefore, the problem f (t) dt 0

(

−u00 = λf ∗ (u) u(0) = u(1) = 0,

admits three nonnegative solutions ui , i = 1, 2, 3, such that |ui (t)| < p, t ∈ [0, 1], i = 1, 2, 3. We claim that |ui (t)| ≤ c, t ∈ [0, 1], i = 1, 2, 3. In fact, arguing by a contradiction there exists [a, b] ⊆ [0, 1] such that ui (t) > c for all t ∈ [a, b], ui (a) = ui (b) = 0 and hence, being a solution of the previous problem, must be ui (t) = 0 in [a, b]; this is a contradiction and our claim is proved. The conclusion follows since ui are also solutions of (APλ ).  Remark 6.6. In Theorem 2.32 of [18], under the assumptions (α0 ) f (0) = 0; and (β) of Theorem 6.3, the existence of λ > 0 such that, for each λ > λ, the problem (APλ ) admits two positive solutions, is ensured (the result for elliptic equations also holds). We explicitly observe that in Theorem 6.3 f (0) may be different from zero and that three positive solutions are obtained. Moreover, if we assume (α00 ) limx→0+ f (x) = 0; x (which is stronger than both (α0 ) and (α)) we obtain that the problem (APλ ) admits 12d2 at least two positive solutions for each λ > inf R d . In other words, Theorem d∈]0,c[ 0 f (t) dt 6.3 ensures (under the stronger condition (α00 )) an upper bound of the λ established in Theorem 2.32 of [18], namely 12d2 . 0 f (t) dt

λ ≤ inf R d d∈]0,c[

Remark 6.7. We explicitly observe that in Theorem 6.1 no condition at infinity is requested, as the following easy example shows.

16

DIEGO AVERNA AND GABRIELE BONANNO

Example 6.1. Let g : [0, 1] → IR be defined by g(t) = t for every t ∈ [0, 1] and let h : IR → IR be the function defined as follows     

1 x10 h(x) :=  210    h(x)

if if if if

x ∈] − ∞, 1] x ∈]1, 2] x ∈]2, 1002] x ∈]1002, +∞[,

where h :]1002, +∞[→ IR is an arbitrary function. The function f (t, x) = g(t)h∗ (x), where h∗ is a continuous function which coincides with h in ] − ∞, 1002], satisfies all the assumptions of Theoremi6.1 byh choosing, for 6 18 instance, c1 = 1, d = 2, c3 = 500, k = 1002. Then, for each λ ∈ 10 , 10 , the problem (

−u00 = λth(u) u(0) = u(1) = 0,

admits at least three positive classical solutions whose norms in C([0, 1]) are less than 1002. References [1] A.Ambrosetti and P.H.Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349-381. [2] R.I.Avery, J.Henderson, Three symmetric positive solutions for a second-order boundary value problem, Appl. Math. Letters 13 (2000), 1-7. [3] D.Averna and G.Bonanno, A three critical points theorem and its applications to the ordinary Dirichlet problem, Topol. Methods Nonlinear Anal. 22 (2003), 93-104. [4] D.Averna and G.Bonanno, Three solutions for a quasilinear two point boundary value problem involving the one-dimensional p-Laplacian, Proc. Edinb. Math. Soc. 47 (2004), 257-270. [5] G. Bonanno, Multiple critical points theorems without the Palais-Smale condition, J. Math. Anal. Appl. 299 (2004), 600-614. [6] G.Bonanno, Some remarks on a three critical points theorem, Nonlinear Anal. 54 (2003), 651665. [7] G.Bonanno, A critical points theorem and nonlinear differential problems, J. of Global Optimization 28 (2004), 249-258. [8] G.Bonanno and P.Candito, Three solutions to a Neumann problem for elliptic equations involving the p-Laplacian, Arch. Math. (Basel) 80 (2003), 424-429. [9] G.Bonanno and R.Livrea, Multiplicity theorems for the Dirichlet problem involving the pLaplacian, Nonlinear Anal. 54 (2003), 1-7. [10] G.Bonanno and D.O’Regan, A boundary value problem on the half-line via critical point methods, Dynamic Systems and Applications 15 (2006), 395-408. [11] K. -C. Chang, Variational methods for non-differentiable functionals and their applications to partial differential equations, J. Math. Anal. Appl. 80 (1981), 102-129. [12] F. H. Clarke, Optimization and Nonsmooth Analysis, Classics Appl. Math. 5, SIAM, Philadelphia, 1990. [13] J.Henderson, H.B.Thompson, Existence of multiple solutions for second order boundary value problems, J. Differential Equations 166 (2000), 443-454.

A MOUNTAIN PASS THEOREM...

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[14] J.Henderson, H.B.Thompson, Multiple symmetric positive solutions for a second order boundary value problem, Proc. Amer. Math. Soc. 128 (2000), 2373-2379. [15] S. A. Marano and D. Motreanu, On a three critical points theorem for non differentiable functions and applications to non linear boundary value problems, Nonlinear Anal. 48 (2002), 37-52. [16] D. Motreanu and V. Rˇ adulescu, Variational and non-variational methods in nonlinear analysis and boundary value problems, Kluwer, Dordrecht 2003. [17] P.Pucci and J.Serrin, A mountain pass theorem, J. Differential Equations 63 (1985), 142-149. [18] P. H. Rabinowitz, Minimax methods in critical Point Theory with Applications to Differential Equations, CBMS Reg. Conferences in Math., vol.65 , Amer. Math. Soc. Providence, RI, 1686. [19] B.Ricceri, A general variationas principle and some of its applications, J. Comput. Appl. Math. 113 (2000), 901-410. [20] B.Ricceri, On a three critical points theorem, Arch. Math. (Basel) 75 (2500), 220-226. [21] R.Salvati, Multiple solutions for a mixed boundary value problem, Math. Sci. Res. J. 7 (2003), no.7, 275-983. [22] E.Zeidler, Nonlinear functional analysis and its applications, Vol. III. Berlin-Heidelberg-New York 1985. ` di Palermo, Via (D.Averna) Dipartimento di Matematica ed Applicazioni, Universita Archirafi, 34, 90123 Palermo (Italy) E-mail address: [email protected] (G.Bonanno) Dipartimento di Informatica, Matematica, Elettronica e Trasporti, ` di Ingegneria, Universita ` di Reggio Calabria, Via Graziella (Feo di Vito), Facolta 89100 Reggio Calabria (Italy) E-mail address: [email protected]