A THREE CRITICAL POINTS RESULT IN A BOUNDED DOMAIN OF A BANACH SPACE AND APPLICATIONS RADU PRECUP, PATRIZIA PUCCI, AND CSABA VARGA Abstract. Using the bounded mountain pass lemma and the Ekeland variational principle we prove a bounded version of the Pucci–Serrin three critical points result in the intersection of a ball with a wedge in a Banach space. The localization constraints are overcome by boundary and invariance conditions. The result is applied to obtain multiple positive solutions for some semilinear problems.

Mathematics Subject Classification: 47J30, 58E05, 34B15. Key words: Critical point, extremum point, mountain pass lemma, Palais–Smale condition, Ekeland’s variational principle, p–Laplacian, weak Harnack inequality. 1. Introduction and preliminaries This paper continues to develop the critical point theory in a bounded region of a Banach space, to which we shall refer as bounded critical point theory. Compared to critical point theory in the whole space [1], [16], when working in a bounded domain, difficulties arise from the behavior of functionals at the boundary of the domain. Thus, if a functional reaches its minimum on that domain at some point of the boundary, then that point does not need to be a critical point of the functional in the usual sense of vanishing the directional derivatives in any direction of the space. The same is true, for instance, when the functional has a mountain pass type geometry in that domain. In this case the saddle point relative to the domain can belong to the boundary and could be not a critical point in the usual sense. Thus, in order to guarantee that such points, located on the boundary, have the property to be critical points in the usual sense, additional conditions on the behavior of the functional on the boundary are needed. The phenomenon is similar to that which appears in fixed point theory for operators that do not leave invariant their domain. For instance, for a compact operator T : U → X, where U ⊂ X is an open set of X and 0 ∈ U , the Leray–Schauder boundary condition (1.1)

T (u) 6= λu

for all u ∈ ∂U and λ > 1

is sufficient for the existence of a fixed point in U . Surprisingly, in [15], [9], a similar boundary condition, (1.2)

F 0 (u) + ηu 6= 0

for all u ∈ ∂U and η > 0

is sufficient to guarantee that minima and saddle points relative to the ball U centered at the origin of a Hilbert space are critical points in the usual sense of the C 1 functional F , even when they are located on the boundary ∂U . The coincidence of 1

2

R. PRECUP, P. PUCCI, AND C. VARGA

the conditions (1.1) and (1.2) is obvious. If we take (1.3)

T (u) = u − F 0 (u)

and λ = 1 + η

then the critical points of F are exactly the fixed points of T . Let us note that, since F 0 takes values in the dual, the formulas (1.2) and (1.3) are formally correct only in a Hilbert space H, when H is identified to its dual, while adjustments in terms of the duality mapping have to be done in more general Banach spaces. In many problems, arising from mathematical models of real processes, not only the boundedness of the solutions is required, but also their positivity. Then, mathematically we try to obtain solutions in a cone of the space. Here again we can do a parallelism between fixed point theory and critical point theory. If the cone is denoted by K and we look for a fixed point of T in U ∩ K, then an invariance condition is usually asked, namely T (u) ∈ K

for all u ∈ U ∩ K.

Similarly, in [10] critical points of a functional F are searched in the intersection of K with the ball U centered at the origin of a Hilbert space H under the invariance condition u − F 0 (u) ∈ K for all u ∈ U ∩ K. Obviously, via (1.3), the two invariance conditions coincide. Remark 1. The crucial role of the invariance condition can be explained as follows.
Assume that F : H → R is a C 1 functional on the Hilbert space H = H, (., .) , and that K is any convex subset of H. Let u0 ∈ K be a local minimum of F in K, i.e. there is r > 0 such that
F (u) ≥ F (u0 ) for all u ∈ K, with ku − u0 k < r. If the invariance condition I − F 0 (K) ⊂ K holds, where I is the identity map of H, then F 0 (u0 ) = 0. Indeed, if we take u = u0 −tF 0 (u0 ), t > 0, then ku−u0 k = tkF 0
(u0 )k < r for sufficiently small t. From the facts that u = (1 − t)u0 + t u0 − F 0 (u0 ) , that K is convex and that both u0 and u0 − F 0 (u0 ) belong to K, we have u ∈ K for all t ∈ (0, 1). Hence, F (u0 − tF 0 (u0 )) − F (u0 ) ≥ 0 for all
t > 0 small enough. Dividing by t and letting t → 0+ , we obtain F 0 (u0 ), −F 0 (u0 ) ≥ 0, that is F 0 (u0 ) = 0, as claimed. In what follows, in order to cover both cases of the whole space and of the cone, K is assumed
to be a wedge, i.e. K is a closed convex set of a Banach space X = X, k.k , with λK ⊂ K for all λ ≥ 0 and K 6= {0}. Thus, in particular, K may be the whole space X, or a cone of X, i.e. K ∩ (−K) = {0}. For any R > 0 put KR = {u ∈ K : kuk ≤ R} and ∂KR = {u ∈ K : kuk = R}. Throughout the paper we assume that X and its dual space X ∗ are uniformly convex Banach spaces, so that the duality map J of X corresponding to the normalization function tp−1 , where p > 1, is a single–valued, continuous, monotone and bijective operator from X to X ∗ and (1.4)

hJu, ui = kukp ,

kJuk = kukp−1

for all u ∈ X,

see, e.g., [4], [3]. Also its inverse J : X ∗ → X is continuous and monotone and satisfies (1.5)

hu∗ , Ju∗ i = ku∗ kq ,

kJu∗ k = ku∗ kq−1

for all u∗ ∈ X ∗ ,

A THREE CRITICAL POINTS RESULT

3

where q is the H˝ older conjugate of p. We also assume that J is locally strongly monotone, i.e. there is β > 1 such that for each ρ > 0 there exists a constant a = a(ρ) > 0 and hJu − Jv, u − vi ≥ aku − vkβ for all u, v ∈ X, with kuk ≤ ρ and kvk ≤ ρ. We say that a C 1 functional F : X → R satisfies the Palais–Smale type condition at the level λ, in brief (P S)λ , in KR , if there exists ν0 > 0 such that (1.6)

min{hF 0 (u), ui, hJu, JF 0 (u)i} ≥ −ν0

for all u ∈ ∂KR ,

and any sequence of elements (uk )k ⊂ KR , with F (uk ) → λ as k → ∞ and satisfying one of the following conditions (a) F 0 (uk ) → 0 as k → ∞; (b) uk ∈ ∂KR , hF 0 (uk ), uk i ≤ 0 for all k and hF 0 (uk ), uk i Juk → 0 as k → ∞; Rp ¯ 0 (uk )i ≤ 0 for all k and (c) uk ∈ ∂KR , hJuk , JF ¯ 0 ¯ 0 (uk ) − hJuk , JF (uk )i uk → 0 as k → ∞, JF Rp has a strongly convergent subsequence. If the (P S)λ –condition in KR is satisfied for every λ, then we say that F satisfies the (P S)–condition in KR . A sequence (uk )k ⊂ KR satisfying one of the conditions (a), (b), (c) is called a (P S)–sequence in KR . F 0 (uk ) −

Remark 2. Under the (P S)–condition in KR and the boundary condition (1.7)

F 0 (u) + ηJu 6= 0

for all u ∈ ∂KR and η > 0,

any adherent point of a (P S)–sequence in KR is a critical point of F . Indeed, if (uk )k is a (P S)–sequence in KR and u is any of its adherent points, then passing to a subsequence we may assume that uk → u as k → ∞. If we are in case (a), we are finished. Assume that (b) holds. Then hF 0 (uk ), uk i ∈ [−ν0 , 0]. Passing to a further subsequence, if necessary, we may assume that −R−p hF 0 (uk ), uk i → η ≥ 0. Thus F 0 (u) + ηJu = 0. The case η > 0 is excluded by (1.7). Hence η = 0 and we are finished again. ¯ 0 (uk )i → η ≥ 0 Let finally (c) hold. As above, we may assume that −R−p hJuk , JF ¯ 0 (u) + ηu = 0. If η > 0, from JF ¯ 0 (u) = −ηu we deduce that as k → ∞. Then JF 0 p−1 0 p−1 F (u) = J(−ηu) = −η J(u). Hence F (u) + η Ju = 0, which again contradicts ¯ 0 (u) = 0. In conclusion, F 0 (u) = 0, as desired. (1.7). Thus η = 0 and in turn JF We say that the functional F : X → R has the mountain pass geometry in KR if there exist elements u0 , u1 in KR and a number r > 0 such that ku1 − u0 k > r and inf{F (u) : u ∈ KR , ku − u0 k = r} > max{F (u0 ), F (u1 )}. Let us introduce the notations ΓR = {γ ∈ C([0, 1]; KR ) : γ(0) = u0 , γ(1) = u1 }, cR = inf max F (γ(t)), γ∈ΓR t∈[0,1]

mR = inf F (u). u∈KR

4

R. PRECUP, P. PUCCI, AND C. VARGA

According to [11], we have the following result which will be used in the proof of the three critical points theorem below. Theorem 1.1. Let F : X → R be a C 1 functional. Assume that the following two conditions hold ¯ 0 (u) ∈ K for all u ∈ KR (invariance) , (1.8) u − JF (1.9)

F 0 (u) + ηJu 6= 0

for all u ∈ ∂KR and η > 0 (boundary condition) .

10 ) If F is bounded from below in KR and satisfies the (P S)mR –condition in KR , then F has a critical point u such that u ∈ KR

and

F (u) = mR .

20 )

If F has the mountain pass geometry in KR and satisfies the (P S)cR –condition in KR , then F has a critical point u such that u ∈ KR

and

F (u) = cR .

Remark 3. The proof of Theorem 1.1 consists in the construction of a (P S)– sequence in KR . The conclusion then follows from Remark 2.
In particular, if X is a Hilbert space, X = H = H, (., .) , and J is the identity map of H and K = H, then Theorem 1.1 reduces to the Schechter critical points results in a ball of a Hilbert space. We conclude this section by a recent theorem on the existence of critical points of minimum type in a conical shell Kr,R = {u ∈ K : r ≤ kuk ≤ R} ,

0 < r < R,

of a Banach space, that we present here in a slightly particular case. Theorem 1.2. Let F : X → R be a C 1 functional which is bounded from below on Kr,R , such that F 0 maps bounded sets into bounded sets and the following invariance condition holds:
(1.10) J¯ Ju − F 0 (u) ∈ K for all u ∈ KR . Then there exists a sequence (uk )k ⊂ Kr,R such that F (uk ) → inf F

(1.11)

Kr,R

as k → ∞,

and one of the following
statements holds (a) uk − J¯ Juk − F 0 (uk ) → 0 as k → ∞;
(b) kuk k = R, hJuk , uk − J¯ Juk − F 0 (uk ) i ≤ 0 for each k and as k → ∞ (1.12)



J¯ µk Juk − J¯ Juk − F 0 (uk ) → 0,

where µk = 1 −

hF 0 (uk ), uk i ; hJuk , uk i


(c) kuk k = r, hJuk , uk − J¯ Juk − F 0 (uk ) i ≥ 0 for each k and (1.12) holds. If, in addition, F satisfies the (P S)–condition in Kr,R , i.e. any sequence as above has a convergent subsequence, and the following boundary conditions hold (1.13) (1.14)

F 0 (u) + ηJu 6= 0 0

F (u) + ηJu 6= 0

for all η > 0 and u ∈ K, with kuk = R, for all η < 0 and u ∈ K, with kuk = r,

A THREE CRITICAL POINTS RESULT

5

then there exists u ∈ Kr,R such that F (u) = inf F Kr,R

and

F 0 (u) = 0.

2. Main results In this section we state and prove the bounded version of the Pucci–Serrin critical point theorem [12], [13], in the intersection of a ball with a wedge. Theorem 2.1 (Bounded Pucci–Serrin critical point theorem). Let F : X → R be a C 1 functional which satisfies the (P S)–condition in KR . Suppose that (1.8) and (1.9) hold. If u0 , u1 ∈ KR are distinct local minima of F in KR , with ku0 k < R and ku1 k < R, then F has a critical point u∗ such that u∗ ∈ KR \ {u0 , u1 }

and

F (u∗ ) ≥ max{F (u0 ), F (u1 )}.

Proof. Since u0 and u1 are distinct local minima of F in KR , and ku0 k < R, ku1 k < R, there exists r0 > 0 such that 2r0 < ku1 − u0 k, r0 ≤ R − max{ku0 k, ku1 k} and (2.1) F (ui ) ≤ F (u) for all u ∈ B r0 (ui ) ∩ KR , i = 0, 1. Let c0 = F (u0 ), c1 = F (u1 ). Assume, without loss of generality, that c0 ≥ c1 . We distinguish two cases. Case 1. There exists r ∈ (0, r0 ) such that (2.2)

inf{F (u) : u ∈ KR , ku − u0 k = r} > c0 .

Since c0 = max{F (u0 ), F (u1 )}, then (2.2) shows that F has the mountain pass geometry in KR . Hence the conclusion follows from Theorem 1.1–20 ). Case 2. Assume that we are not in Case 1. Then for every r ∈ (0, r0 ) by (2.1) inf{F (u) : u ∈ KR , ku − u0 k = r} = c0 . Thus for r ∈ (0, r0 ) and every k ∈ N there exists vk ∈ KR such that (2.3)

kvk − u0 k = r

and F (vk ) ≤ c0 + 1/k 2 .

Now fix r ∈ (0, r0 ) and choose k ∈ N so large that 0 < r − 2/k < r + 2/k < r0 . Put   2 2 V = u ∈ KR : r − ≤ ku − u0 k ≤ r + . k k Since V ⊂ B r0 (u0 ), then inf u∈V F (u) = c0 . Hence an application of the Ekeland variational principle [7] to F |V and vk gives the existence of uk ∈ V such that (i) c0 ≤ F (uk ) ≤ F (vk ) ≤ c0 + 1/k 2 , (ii) kuk − vk k ≤ 1/k; (iii) F (v) − F (uk ) ≥ −kuk − vk/k for every v ∈ V. ¯ 0 (uk ) in (iii) we claim that v ∈ V for all t > 0 sufficiently small. Taking v = uk − tJF
¯ 0 (uk ) and both uk and uk − JF ¯ 0 (uk ) belong Indeed, since v = (1 − t)uk + t uk − JF to K, we have v ∈ K for all t ∈ (0, 1). Moreover, from 2 kuk k ≤ kuk − u0 k + ku0 k ≤ r + + R − r0 < r0 + R − r0 = R, k

6

R. PRECUP, P. PUCCI, AND C. VARGA

¯ 0 (uk )k ≤ R, i.e. v ∈ KR for we have kuk k < R and consequently kvk = kuk − tJF all t > 0 small enough. In addition, since kvk − u0 k = r, from (ii) and kuk − u0 k − r ≤ kuk − vk k ≤ 1 , k we obtain that 2 2 (2.4) r − < kuk − u0 k < r + . k k Thus v ∈ V for all t > 0 sufficiently small, as claimed. Therefore we may apply (iii) and obtain for all t > 0 sufficiently small
¯ 0 (uk ) − F (uk ) ≥ − t kJF ¯ 0 (uk )k. F uk − tJF k Dividing by t and letting t → 0+ , we deduce that ¯ 0 (uk )k. ¯ 0 (uk )i ≤ 1 kJF hF 0 (uk ), JF k In view of (1.5), this gives 1 kF 0 (uk )k ≤ . k It follows that F 0 (uk ) → 0 as k → ∞. Moreover, F (uk ) → c0 as k → ∞ by (i). Now the existence of a critical point u∗ ∈ KR follows from the (P S)–condition, while the inequalities (2.4) give ku∗ − u0 k = r. Whence u∗ 6= u0 and u∗ 6= u1 .  Theorem 2.1 immediately yields the following critical point principle. Corollary 2.2. Let F : X → R be a C 1 functional which satisfies the (P S)– condition in KR and suppose that (1.8) and (1.9) hold. In addition, assume that for some R0 ∈ (0, R) and u0 , u1 ∈ KR the following conditions are satisfied (i) u0 is a global minimum of F in KR0 , with ku0 k < R0 ; (ii) u1 is a global minimum of F in KR0 ,R1 , with R0 < ku1 k < R. Then F has a critical point u∗ such that u∗ ∈ KR \ {u0 , u1 }

and

F (u∗ ) ≥ max{F (u0 ), F (u1 )}.

Proof. Indeed, the strict inequalities ku0 k < R0 and R0 < ku1 k < R guarantee that u0 and u1 are local minima of F in KR , with ku0 k < R and ku1 k < R. Hence, the assumptions of Theorem 2.1 are satisfied and this gives the conclusion.  Now, using Theorems 1.1 and 1.2, we are in position to fulfil the conditions (i) and (ii) above and, consequently, to turn Corollary 2.2 into an abstract three critical points result in the intersection of a ball with a wedge. Theorem 2.3. Let F : X → R be a C 1 functional such that F 0 maps bounded sets into bounded sets. Assume that there exist R0 and r, with 0 < R0 ≤ r < R, such that the invariance conditions (1.8) and (1.10) hold, that F satisfies the (P S)–condition in KR and in Kr,R and that the following properties hold (h1 ) F is bounded from below in KR0 and (2.5)

F 0 (u) + ηJu 6= 0

for all η ≥ 0 and u ∈ K, with kuk = R0 ;

A THREE CRITICAL POINTS RESULT

7

(h2 ) F is bounded from below in Kr,R and (2.6)

F 0 (u) + ηJu 6= 0 0

F (u) + ηJu 6= 0

for all η ≥ 0, u ∈ K, with kuk = R, for all η ≤ 0, u ∈ K, with kuk = r.

Then F has at least three distinct critical points u0 , u1 , u∗ located in KR , with F (u0 ) = inf F, KR0

F (u1 ) = inf F, Kr,R

F (u∗ ) ≥ max{F (u0 ), F (u1 )}.

Proof. From (h1 ) the case 10 ) of Theorem 1.1 can be applied so that F attains its / ∂KR0 by (2.5), i.e. global minimum in KR0 at u0 and F 0 (u0 ) = 0. Moreover, u0 ∈ ku0 k < R0 . Thus condition (i) in Corollary 2.2 holds. Thanks to (h2 ) the assumptions of Theorem 1.2 are satisfied. Hence F attains its global minimum in Kr,R at u1 and F 0 (u1 ) = 0. Furthermore, r < ku1 k < R by (2.6). Since R0 ≤ r, then R0 < ku1 k < R. Thus also condition (ii) of Corollary 2.2 holds. In conclusion, Corollary 2.2 applies and ends the proof.  3. Applications 3.1. An abstract schema of applications. Assume furthermore that X is continuously embedded into a Banach space Y , with cone K0 , and that K = K0 ∩ X. Denote by K0∗ and K ∗ the induced cones in Y ∗ and X ∗ , respectively, which are given by K0∗ = {h ∈ Y ∗ : hh, ui ≥ 0 for all u ∈ K0 }, K ∗ = {h ∈ X ∗ : hh, ui ≥ 0 for all u ∈ K}. Clearly, Y ∗ ⊂ X ∗ and K0∗ ⊂ K ∗ . By the same symbol ”” we denote the order relations induced by K0 , K, K0∗ and K ∗ on Y , X, Y ∗ and X ∗ , respectively. We assume that the abstract comparison principle holds, i.e. (3.1)

h1 , h2 ∈ X ∗ , with h1  h2 implies Jh1  Jh2 ,

equivalently, J is isotone in X ∗ . Notice that (3.1) and J0 = 0 show in particular that J is also positive, i.e. Jh  0 for every h ∈ K ∗ . Next we consider an operator N : Y → Y ∗ which is positive and isotone in K0 , i.e. 0  u  v implies 0  N (u)  N (v) and (3.2) hN (u), ui > 0 for u ∈ K \ {0}. Assume that there is a C 1 functional F : X → R such that N (u) = Ju − F 0 (u)

for all u ∈ X.

Theorem 3.1. Let the above conditions on X, J and N hold. Let F 0 map bounded sets into bounded sets and let JN be completely continuous on X. Assume that there exist two elements φ and ψ in K0 \ {0} such that (3.3)

kukφ ≤ u ≤ kukψ,

8

R. PRECUP, P. PUCCI, AND C. VARGA

whenever u ∈ K and N (u) = λJu for some λ > 0. If there exist numbers R0 , r and R, with 0 < R0 ≤ r < R, such that F is bounded from below on KR and the inequalities (3.4) (3.5)

hN (R0 ψ), ψi < R0p−1 ,

hN (Rψ), ψi < Rp−1 ,

hN (rφ), φi > rp−1

hold, then F has at least three distinct critical points in KR , with norm strictly less than R. Proof. (1) First we check the invariance conditions (1.8) and (1.10). Let u ∈ K and let v = u − JF 0 (u). Then J(u − v) = F 0 (u) = Ju − N (u), whence Ju − J(u − v) = N (u)  0. Therefore, u  u − v by (3.1) and so v  0, that is v ∈ K. Thus the invariance condition (1.8) holds. In order to check (1.10), take w = J(Ju − F 0 (u). Hence, Jw = Ju − F 0 (u) = N (u)  0. Since J is positive, we deduce that w ∈ K as desired. Thus, (1.10) is also satisfied. (2) The functional F satisfies the (P S)–condition in KR . Indeed, let uk ∈ KR , with vk = F 0 (uk ) → 0. Then, since JN is completely continuous and JN (uk ) = J(Juk − vk ), the sequence k 7→ J(Juk − vk ) is relatively compact. Since both J and J are continuous, this yields that the sequence (uk )k is relatively compact too. If (uk )k is in case (b), i.e. kuk k = R, hF 0 (uk ), uk i ≤ 0 and vk = F 0 (uk ) − λk Juk → 0 as k → ∞, where λk = R−p hF 0 (uk ), uk i, then
JN (uk ) = J (1 − λk )Juk − vk .
As above, the sequence k 7→ J (1−λk )Juk −vk is relatively compact. Furthermore, this yields the relative compactness of the sequence k 7→ (1 − λk )Juk − vk and so of k 7→ (1 − λk )Juk . Since F 0 maps bounded sets into bounded sets, it follows that the sequence k 7→ λk = R−p hF 0 (uk ), uk i is bounded below. Therefore there exists c ≥ 0 such that 1 ≤ 1 − λk ≤ 1 + c. This allows us to deduce that the sequence (Juk )k is relatively compact and finally that also the sequence (uk )k is relatively compact. The same can be done if (uk )k is in the case (c). By similar arguments we can show that F satisfies the (P S)–condition in Kr,R . (3) The functional F is bounded from below on KR0 and Kr,R . This property follows from the assumption that F is bounded from below on KR . (4) Let us now check the validity of (2.5). Assume that F 0 (u) + ηJu = 0 for some η ≥ 0 and u ∈ K, with kuk = R0 . Then (1 + η)Ju = N (u) and so (3.6)

(3.7)

(1 + η)hJu, ui = hN (u), ui.

Now u ≤ R0 ψ by (3.3) and, since N is isotone in KY , then N (u) ≤ N (R0 ψ).

A THREE CRITICAL POINTS RESULT

9

Also hJu, ui = R0p and η ≥ 0. Thus, (3.7) gives R0p ≤ hN (R0 ψ), R0 ψi, i.e. R0p−1 ≤ hN (R0 ψ), ψi, which contradicts (3.4). Similarly, the first part of (2.6) can be proved. (5) Finally, we check the second part of (2.6). Assume that F 0 (u) + ηJu = 0 for some η ≤ 0 and u ∈ K, with kuk = r. Then N (u) = λu, where λ = 1 + η ≤ 1. Moreover, (3.7) yields that λ ≥ 0, while the case λ = 0 is excluded by (3.2). Thus λ > 0 and so (3.3) holds and, together with (3.7), gives rp−1 ≥ hN (rφ), φi, which contradicts (3.5). The conclusion now follows from Theorem 2.3.



3.2. Application to one–dimensional p–Laplacian equations. In this subsection we illustrate the abstract schema by a boundary value problem extensively studied in the literature, see [6],  − |u0 (t)|p−2 u0 (t))0 = g(u(t)) in I = (0, 1) (3.8) u(0) = u(1) = 0. Here p > 1 and g : R → R is a continuous function, nondecreasing on R+ , with g(s) > s for all s > 0. Hence all possible nonnegative solutions are concave functions on [0, 1]. We look for symmetric solutions with respect to the middle of the interval [0, 1], i.e. u(1 − t) = u(t) for all t ∈ [0, 1/2]. We shall use the previous abstract schema in the Banach space X = W01,p (I), with 1/p R 1 0 p . It is well–known that W01,p (I) and its dual the norm kuk1,p = 0 |u (t)| dt W −1,q (I) are uniformly convex reflexive. In this case, the duality map corresponding to the normalization function τ p−1 , τ ∈ R+ , is
0 J : W01,p (I) → W −1,q (I), Ju = − |u0 |p−2 u0 , and its inverse J : W −1,q (I) → W01,p (I) is continuous and monotone and satisfies (1.5). The energy functional F : W01,p (I) → R is given by  Z 1 1 0 p F (u) = |u (t)| − G(u(t)) dt, p 0


Rs where G(s) = 0 g(σ)dσ. Let Y = C [0, 1] = C [0, 1] , k · k∞ and
K0 = {u ∈ C [0, 1] : u ≥ 0, u(1 − t) = u(t) for t ∈ [0, 1/2]}, K = K0 ∩ X. In this case, the abstract comparison principle for J is guaranteed by the comparison principle for the p–Laplacian (see, e.g., [2], [5] and [14]). Also, N is the Nemytski superposition operator u 7→ g(u), which is positive and isotone in K0 because of the corresponding properties of the function g. Moreover, (3.2) holds by the strict positivity of g in R+ = (0, ∞). Thus all the structural assumptions before Theorem 3.1 are fulfilled. Indeed, J is also locally strongly monotone, with β = 2 and a(ρ) = (2ρ)p−2 , ρ > 0, when

10

R. PRECUP, P. PUCCI, AND C. VARGA

1 < p ≤ 2, and with β = p and a(ρ) a positive constant for all ρ > 0, which depends only on p, when p ≥ 2. See [8] for further details. Regarding the inequalities (3.3), we first note that kuk∞ ≤ kuk1,p for all u ∈ W01,p (I), and so for all u ∈ K. Thus the right inequality in (3.3) is fulfilled, with ψ = 1 (constant function). The left inequality in (3.3), which can viewed as a weak Harnack type inequality, is based on the following result from [8].
Lemma 3.2. For every function u ∈ K, with Ju ∈ C [0, 1]; R+ nondecreasing on [0, 1/2], the following inequality holds u(t) ≥ t(1 − 2t)1/(p−1) kuk1,p for all t ∈ [0, 1/2]. Now if u ∈ K and N (u) = λJu for some λ > 0, then the left inequality in (3.3) holds with  t(1 − 2t)1/(p−1) , t ∈ [0, 1/2], φ(t) = 1/(p−1) (1 − t)(2t − 1) , t ∈ [1/2, 1],
+ since N (u) = g(u) ∈ C [0, 1]; R and is nondecreasing on [0, 1/2]. Theorem 3.3. Assume that the above conditions are satisfied and that there exist the numbers R0 , r and R, with 0 < R0 ≤ r < R, and α ∈ (0, 1/2) such that g(R0 ) < R0p−1 ,

(3.9) (3.10)

g(R) < Rp−1 ,

g(α(1 − 2α)1/(p−1) r) >

rp−1 , α(1 − 2α)q

where q is the H˝ oolder conjugate of p. Then the problem (3.8) has at least two positive solutions in the open ball BR of W01,p (I). If in addition g(0) > 0, then a third positive solution exists in BR . Proof. Since here ψ = 1, N (R0 ψ) = g(R0 ) and N (Rψ) = g(R), then (3.9) reduces exactly to (3.4). Moreover, (3.10) implies (3.5) as follows from Z 1 Z 1/2 hN (rφ), φi = g(rφ(t))φ(t)dt = 2 g(rφ(t))φ(t)dt 0

Z

0 1/2

≥2

g(rφ(t))φ(t)dt α  1 ≥2 − α g(α(1 − 2α)1/(p−1) r)α(1 − 2α)1/(p−1) 2

= g(α(1 − 2α)1/(p−1) r)α(1 − 2α)q . Thus Theorem 3.1 applies and gives the result. Put

1 = max sp (1 − 2s)q+1 . Λ s∈[0,1/2]



A THREE CRITICAL POINTS RESULT

11

Corollary 3.4. Suppose that g : R → R is a continuous function, nondecreasing on R+ 0 , with g(s) > s for all s > 0. If g(s) g(s) < 1, lim inf p−1 < 1 s→∞ s sp−1 hold and there exists σ > 0 such that g(σ) > Λ, (3.12) σ p−1 then problem (3.8) has at least two positive solutions. If in addition g(0) > 0, then a third positive solution exists. (3.11)

lim inf s→0

Proof. Assumption (3.12) implies that there exists α ∈ (0, 1/2), with g(σ) 1 > p , p−1 σ α (1 − 2α)q+1 which is exactly (3.10), with r = σ/α(1 − 2α)1/(p−1) . Condition (3.11) implies the existence of R0 ∈ (0, r) sufficiently small and of R ∈ (r, ∞) large enough such that inequalities (3.9) are satisfied.  4. Acknowledgements The first author, Radu Precup, was supported by a grant of the Romanian National Authority for Scientific Research, CNCS – UEFISCDI, project number PNII-ID-PCE-2011-3-0094. The second author, Patrizia Pucci, was partly supported by the Italian MIUR project titled Variational and perturbative aspects of nonlinear differential problems (201274FYK7) and is a member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilit` a e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). The manuscript was realized within the auspices of the INdAM–GNAMPA Project 2015 titled Modelli ed equazioni nonlocali di tipo frazionario (Prot 2015 000368). The research of Csaba Varga has been partially supported by a grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, project no. PN-II-ID-PCE-2011-3-0241 and by OTKA (grant no. K 115926). References [1] A. Ambrosetti and P.H. Rabinowitz, Dual variational methods in critical points theory and applications, J. Funct. Anal. 14 (1973), 349–381. [2] C. Azizieh and P. Cl´ement, A priori estimates and continuous methods for positive solutions of p–Laplace equations, J. Differential Equations 179 (2002), 213–245. [3] C. Chidume, Geometric Properties of Banach Spaces and Nonlinear Iterations, Lecture Notes in Mathematics 1965, Springer–Verlag London, xviii+326 pp. 2009. [4] I. Cior˘ anescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, Mathematics and its Applications 62 Kluwer, Dordrecht, xiv+260 pp. 1990. [5] L. Damascelli and F. Pacella, Monotonicity and symmetry of solutions of a p–Laplace equations, 1 < p < 2, via the moving plane method, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 26 (1998), 689– 707. [6] G. Dinca, P. Jebelean and J. Mawhin, Variational and topological methods for Dirichlet problems with p–Laplacian, Port. Math. 58 (2001), 339–378.

12

R. PRECUP, P. PUCCI, AND C. VARGA

[7] I. Ekeland, On the variational principle, J. Math. Anal. Appl. 47 (1974), 443–474. [8] H. Lisei, R. Precup and C. Varga, A Schechter type critical point result in annular conical domains of a Banach space and applications, Discrete Contin. Dyn. Syst., to appear. [9] R. Precup, The Leray–Schauder boundary condition in critical point theory, Nonlinear Anal. 71 (2009), 3218–3228. [10] R. Precup, Critical point theorems in cones and multiple positive solutions of elliptic problems, Nonlinear Anal. 75 (2012), 834–851. [11] R. Precup and C. Varga, Localization of positive critical points in Banach spaces and applications, submited for publication. [12] P. Pucci and J. Serrin, Extensions of mountain pass theorem, J. Funct. Anal. 59 (1984), 185– 210. [13] P. Pucci and J. Serrin, A mountain pass theorem, J. Differential Equations 60 (1985), 142–149. [14] P. Pucci and J. Serrin, The strong maximum principle revisited, J. Differential Equations 196 (2004), 1–66. [15] M. Schechter, Linking Methods in Critical Point Theory, Birkh¨ auser, Boston, xviii+294 pp. 1999. [16] M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications 24, Birkh¨ auser, Basel, x+162 pp. 1996. (R. Precup) Babes¸–Bolyai University, Faculty of Mathematics and Computer Science, ˘ lniceanu nr. 1, 400084 Cluj–Napoca, Romania Str. Koga E-mail address: [email protected] ` degli Studi di Perugia, Dipartimento di Matematica e Infor(P. Pucci) Universita matica, Perugia, Italia E-mail address: [email protected] (C. Varga) Babes¸–Bolyai University, Faculty of Mathematics and Computer Science, ˘ lniceanu nr. 1, 400084 Cluj–Napoca, Romania Str. Koga E-mail address: [email protected]