a nonlinear eigenvalue problem for the periodic scalar $p$-laplacian

COMMUNICATIONS ON doi:10.3934/cpaa.2014.13.1075PURE AND APPLIED ANALYSISVolume 13, Number 3, May 2014 pp. 1075–1086

A NONLINEAR EIGENVALUE PROBLEM FOR THE PERIODIC

SCALAR p-LAPLACIAN

Giuseppina Barletta

Department DICEAM, University of Reggio Calabria, Reggio Calabria, 89100, Italy

Roberto Livrea and Nikolaos S. Papageorgiou

Department DICEAM, University of Reggio Calabria, Reggio Calabria, 89100, Italy

National Technical University, Department of Mathematics, Zografou Campus,Athens, 15780, Greece

(Communicated by Wei Feng)

Abstract. We study a parametric nonlinear periodic problem driven by the

scalar p-Laplacian. We show that if λ1 > 0 is the first eigenvalue of the periodic

scalar p-Laplacian and λ > λ1, then the problem has at least three nontrivial

solutions one positive, one negative and the third nodal. Our approach isvariational together with suitable truncation, perturbation and comparison

techniques.

1. Introduction. In this paper, we study the following nonlinear periodic eigen-value problem:{

−(|u′(t)|p−2u′(t)

)′= λ|u(t)|p−2u(t)− f(t, u(t)) a. e. on T = [0, b],

u(0) = u(b), u′(0) = u′(b).

}(Eλ)

Here, 1 < p < +∞, λ > 0 is the parameter and f(t, x) is a Caratheodoryperturbation – i.e., for all x ∈ IR, t 7→ f(t, x) is measurable and for a.a. t ∈ T ,x 7→ f(t, x) is continuous – which exhibits (p − 1)-superlinear growth in the x-

variable near ±∞. Let λ1 denote the first nonzero eigenvalue of the periodic scalar

p-Laplacian. We show that, if λ > λ1, the problem (Eλ) has at least three nontrivialsolutions providing precise sign information for all the solutions.

Multiplicity theorems for nonlinear periodic problems were proved by Aizicovici-Papageorgiou-Staicu [1, 3, 4], Del Pino-Manasevich-Murua [5], Gasinski [8], Gasinski-Papageorgiou [10], Papageorgiou-Papageorgiou [11] and Yang [13]. Parametric pe-riodic problems (i.e., nonlinear eigenvalue problems) were studied in [3, 4]. In [3]the authors treat equations with concave terms plus a (p − 1)-linear pertubationand they show that for all λ > 0 small the problem has at least three nontrivialsolutions but do not provide sign information for all of them. In [4] the probleminvolves competing nonlinearities (concave-convex nonlinearities) and the authorslook for positive solutions. Their main result is a bifurcation-type theorem for smallλ > 0.

2010 Mathematics Subject Classification. Primary: 34B15, 34B18; Secondary: 34C25.Key words and phrases. Constant sign and nodal solutions, parametric equation, second de-

formation theorem, extremal solutions.

1075

http://dx.doi.org/10.3934/cpaa.2014.13.1075

1076 G. BARLETTA, R. LIVREA AND N. S. PAPAGEORGIOU

Our approach is variational, based on the critical point theory, coupled with suit-able truncation and comparison techniques. In the next section, for easy reference,we recall the main mathematical tools which will be used in this paper.

2. Mathematical background. Let X be a Banach space and X∗ its topologicaldual. By 〈·, ·〉 we denote the duality brackets for the pair (X∗, X). Let ϕ ∈ C1(X).We say that ϕ satisfies the ‘Palais-Smale condition’ (the ‘PS-condition’, for short),if the following is true:

(PS) ‘Every sequence {xn}n≥1 ⊆ X such that {ϕ(xn)}n≥1 ⊆ IR is bounded and

ϕ′(xn)→ 0 in X∗ as n→∞

admits a strongly convergent subsequence’.

This compactness-type condition compensates for the lack of local compactnessin the ambient space X which in general is an infinite dimensional Banach space.It leads to a deformation theorem, which in turn produces the minimax theoryof certain critical values of ϕ. One such result is the well-known ‘mountain passtheorem’.

Theorem 2.1. If ϕ ∈ C1(X) satisfies the PS-condition, x0, x1 ∈ X, ‖x1 − x0‖ >ρ > 0,

max{ϕ(x0), ϕ(x1)} < inf‖x−x0‖=ρ

ϕ(x) = ηρ

and

c = infγ∈Γ

max0≤t≤1

ϕ(γ(t)) where Γ = {γ ∈ C([0, 1], X) : γ(0) = x0, γ(1) = x1} ,

then c ≥ ηρ and c is a critical value of ϕ.

Recall that a map A : X → X∗ is said to be of type (S)+, if the followingimplication is true

xn ⇀ x in X and lim supn→∞

〈A(xn), xn − x〉 ≤ 0⇒ xn → x in X.

In the study of problem (Eλ), we will use the Sobolev space

W 1,pper(0, b) := {u ∈W 1,p(0, b) : u(0) = u(b)}.

Recall that W 1,pper(0, b), endowed with the norm

‖u‖ =[‖u‖pp + ‖u′‖pp

]1/p,

for all u ∈ W 1,pper(0, b), is compactly embedded in C(T ) (by the Sobolev embedding

theorem) and so in the above definition the evaluations of u at t = 0 and t = bmake sense. In what follows, for notational economy, we set

W = W 1,pper(0, b).

In addition to W , we will also use the Banach space

C1(T ) := {u ∈ C1(T ) : u(0) = u(b)}.

This is an ordered Banach space with positive cone C+ = {u ∈ C1(T ) : u(t) ≥0 for all t ∈ T}. This cone has nonempty interior given by

intC+ = {u ∈ C+ : u(t) > 0 for all t ∈ T}.

A NONLINEAR EIGENVALUE PROBLEM 1077

Let A : W →W ∗ be the nonlinear map defined by

〈A(u), y〉 =

∫ b

0

|u′(t)|p−2u′(t)y′(t) dt for all u, y ∈W. (1)

The next proposition summarizes the properties of A (see, for example, [1]).

Proposition 1. If A : W → W ∗ is the nonlinear map defined by (1), then A iscontinuous, bounded (maps bounded sets to bounded sets), strictly monotone (hencemaximal monotone too) and of type (S)+.

Let f0 : T × IR→ IR be a Caratheodory function such that

|f0(t, x)| ≤ a(t)(1 + |x|r−1) for a.a. t ∈ T, all x ∈ IR,

with a ∈ L1(T )+ and 1 < r <∞. We set F0(t, x) =∫ x

0f0(t, s) ds and consider the

C1-functional ψ0 : W → IR defined by

ψ0(u) =1

p‖u′‖pp −

∫ b

0

F0(t, u(t)) dt for all u ∈W.

From [1] we have the following result relating the C1(T ) and local W -minimizerfor the functional ψ0.

Proposition 2. If u0 ∈ W is a local C1(T )-minimizer of ψ0, i.e., there existsρ0 > 0 such that

ψ0(u0) ≤ ψ0(u0 + h) for all h ∈ C1(T ), ‖h‖C1(T ) ≤ ρ0,

then u0 ∈ C1(T ) and it is also a local W -minimizer of ψ0, i.e., there exists ρ1 > 0such that

ψ0(u0) ≤ ψ0(u0 + h) for all h ∈W, ‖h‖ ≤ ρ1.

Next let us recall some basic facts about the spectrum of the periodic scalarp-Laplacian. So, we consider the following nonlinear eigenvalue problem{

−(|u′(t)|p−2u′(t)

)′= λ|u(t)|p−2u(t) a. e. on T = [0, b],

u(0) = u(b), u′(0) = u′(b).

}(2)

We say that λ is an eigenvalue, if problem (2) has a nontrivial solution u, whichis a corresponding eigenfunction. Evidently, a necessary condition for λ ∈ IR to bean eigenvalue is that λ ≥ 0. Note that λ0 = 0 is an eigenvalue with eigenspaceIR (i.e., the space of constant functions). The eigenvalue λ0 is the only havingeigenfunctions of constant sign. Every other eigenvalue λ > 0 has eigenfunctionswhich are nodal (i.e., sign changing).

Let

πp :=2π(p− 1)1/p

p sin(πp

) .

Then the sequence {λn =(

2nπpb

)p}n≥0 is the complete set of eigenvalues of (2). If

p = 2 (linear eigenvalue problem), then πp = π and we recover the well-known se-

quence of eigenvalues of the periodic scalar Laplacian, given by{λn =

(2nπb

)2}n≥0

.

Every eigenfunction u ∈ C1(T ) of (2) satisfies u(t) 6= 0 a.e. on T and in factu has a finite number of zeros (see [6] and [9]). Let M = W ∩ ∂BLp1 , where


∂BLp

1 := {u ∈ Lp(T ) : ‖u‖p = 1}. The eigenvalues{λn

}n≥1

admit minimax char-

acterizations provided by the Ljusternik-Schnirelmann theory (see [6, 9]). For

λ1 > 0, namely, the first nonzero eigenvalue, we have the following alternative min-imax characterization (see [3]). Here u0(t) = 1

b1/pfor all t ∈ T is the Lp-normalized

principal eigenfunction.

Proposition 3. The following variational characterization holds

λ1 = infγ∈Γ

max−1≤t≤1

∥∥∥∥ d

dtγ(t)

∥∥∥∥pp

,

where Γ = {γ ∈ C([−1, 1],M) : γ(−1) = −u0, γ(1) = u0}.

Finally, let us fix our notation. For x ∈ IR, we set x± = max{±x, 0} and foru ∈W , we set u±(·) = u(·)±.

We know that

u± ∈W, |u| = u+ + u−, u = u+ − u−.By |·| we denote the Lebesgue measure on IR . Also, if h : T×IR→ IR is a measurablefunction, we set Nh(u)(·) = h(·, u(·)) for all u ∈ W . Finally, if ϕ ∈ C1(X) andc ∈ IR, we set Kϕ = {x ∈ X : ϕ′(x) = 0} and ϕc = {x ∈ X : ϕ ≤ c}.

3. Solutions of Constant Sign. In this section we produce solutions of constantsign for problem (Eλ). The hypotheses on the perturbation f(t, x) are the following:

(H) f : T × IR → IR is a Caratheodory function such that f(t, 0) = 0 a.e. on Tand(i) for every ρ > 0 there exists aρ ∈ L1(T ) such that

|f(t, x)| ≤ aρ(t)for a.a. t ∈ T and all |x| ≤ ρ;

(ii) limx→±∞

f(t, x)

|x|p−2x= +∞ uniformly for a.a t ∈ T ;

(iii) limx→±0

f(t, x)

|x|p−2x= 0 uniformly for a.a t ∈ T ;

(iv) for every ρ > 0 there exists θρ > 0 such that for a.a. t ∈ T x 7→θρ|x|p−2x− f(t, x) is nondecreasing on [−ρ, ρ].

Remark 1. Hypothesis H(ii) implies that for a.a. t ∈ T , f(t, ·) is (p−1)-superlinearnear ±∞. A similar polynomial growth is assumed near zero by virtue of H(iii).

Example 1. The following functions satisfy hypotheses H (for the sake of simplicity,we drop the t-dependence):

f1(x) = |x|r−2x,

for all x ∈ IR, with p < r <∞;

f2(x) = |x|p−2x ln(1 + |x|).for all x ∈ IR.

By virtue of hypothesis H(ii), given any η > 0, we can find M1 = M1(η) > 0such that

f(t, ξ) ≥ ηξp−1 for a.a. t ∈ T, all ξ ≥M1

f(t, ξ) ≤ η|ξ|p−2ξ for a.a. t ∈ T, all ξ ≤ −M1.(3)


Let η = λ and set u(t) = ξ, u(t) = −ξ for all t ∈ T and with ξ ≥M1.Then, see ([3]),

0 = A(u) ≥ λup−1 −Nf (u) and 0 = A(u) ≤ λ|u|p−2u−Nf (u) (4)

in W ∗. In what follows we consider the following two ordered intervals

[0, u] = {u ∈W : 0 ≤ u(t) ≤ u(t) for all t ∈ T},

[u, 0] = {u ∈W : u(t) ≤ u(t) ≤ 0 for all t ∈ T}.Next we show that for all parameters λ > 0, problem (Eλ) has at least two nontrivialconstant sign solutions, a positive one in [0, u] and a negative one in [u, 0]. To thisend, we introduce the following truncations-perturbations of the reaction in (Eλ):

g+λ (t, x) =

0 if x < 0(λ+ 1)xp−1 − f(t, x) if 0 ≤ x ≤ u(t) = ξ(λ+ 1)up−1 − f(t, u) if u(t) = ξ < x

(5)

g−λ (t, x) =

(λ+ 1)up−1 − f(t, u) if x < u(t) = −ξ(λ+ 1)xp−2x− f(t, x) if u(t) = −ξ ≤ x ≤ 00 if 0 < x.

(6)

Both are Caratheodory functions. We set G±λ (t, x) =∫ x

0g±λ (t, s) ds and consider

the C1-functionals ϕ±λ : W → IR defined by

ϕ±λ (u) =1

p‖u′‖pp +

1

p‖u‖pp −

∫ b

0

G±λ (t, u(t)) dt

for all u ∈W .

Proposition 4. If hypotheses H hold and λ > 0, then problem (Eλ) has at leasttwo nontrivial constant sign solutions

u0 ∈ [0, u] ∩ intC+ and v0 ∈ [u, 0] ∩ (−intC+).

Proof. First we show the existence of a positive solution. From (5) it is clear thatϕ+λ is coercive. Also, using the compact embedding of W into C(T ), we check that

ϕ+λ i sequentially weakly lower semicontinuous. So, by the Weierstrass theorem, e

can find u0 ∈W such that

ϕ+λ (u0) = inf

u∈Wϕ+λ (u) = m+

λ . (7)

Hypothesis H(iii) implies that, given ε > 0, we can find δ = δ(ε) ∈ (0, ξ ≡ u) suchthat

F (t, x) ≤ ε

pxp for a.a. t ∈ T, all x ∈ [0, δ]. (8)

Let t ∈ (0, 1] be small such that tu0 ∈ (0, δ]. Then, in view of (5) and (8),

ϕ+λ (tu0) ≤ −λ

ptpup0b+

ε

ptpup0b =

1

ptpup0b(ε− λ).

Choosing ε ∈ (0, λ) (recall that λ > 0), we see that ϕ+λ (tu0) < 0. Hence,

ϕ+λ (u0) = m+

λ ≤ ϕ+λ (tu0) < 0 = ϕ+

λ (0),

that is u0 6= 0. From (7) we have that (ϕ+λ )′(u0) = 0, that implies

A(u0) + |u0|p−2u0 = Ng+λ(u0). (9)


On (9) we act with −u−0 ∈W and, taking in mind (5), we obtain ‖(u−0 )′‖pp+‖u−0 ‖pp =

0. Thus, u0 ≥ 0, u0 6= 0. Also, on (9) we act with (u0 − u)+ ∈ W . Then, thanksto (5) and (4) one has

〈A(u0), (u0 − u)+〉+∫ b

0

up−10 (u0 − u)+ dt =

∫ b

0

g+λ (t, u0)(u0 − u)+ dt

=

∫ b

0

[λup−1 − f(t, u)](u0 − u)+ dt+

∫ b

0

up−1(u0 − u)+ dt

≤〈A(u), (u0 − u)+〉+

∫ b

0

up−1(u0 − u)+ dt,

that leads to

〈A(u0)−A(u), (u0 − u)+〉+

∫ b

0

(up−10 − up−1)(u0 − u)+ dt ≤ 0,

and, from Proposition 1, it follows that |{u0 > u}| = 0, hence u0 ≤ u.So we have proved

u0 ∈ [0, u], u0 6= 0,

and, from (5) we achieve

A(u0) = λup−10 −Nf (u0),

namely {−(|u′0(t)|p−2u′0(t)

)′= λu0(t)p−1 − f(t, u0(t)) a. e. on T,

u0(0) = u0(b), u′0(0) = u′0(b).

Hence, u0 ∈ C1(T ). We set ρ = ‖u0‖∞ and let ξρ > 0 be postulated by hypothesisH(iv). Then

−(|u′0(t)|p−2u′0(t)

)′+ ξρu0(t)p−1 = (λ+ ξρ)|u0(t)|p−1 − f(t, u0(t)) ≥ 0 a. e. on T,

so that (|u′0(t)|p−2u′0(t)

)′ ≤ ξρ|u0(t)|p−1a. e. on T,

and we can apply the Vazquez maximum principle ([12]) to conclude that u0 ∈intC+.

Similarly, working this time with ϕ−λ , we produce another nontrivial constant

sign solution v0 ∈ [u, 0] ∩ (−intC+).

In fact we can show that for every λ > 0 problem (Eλ) has extremal nontrivialconstant sign solutions, i.e., there is a smallest nontrivial positive solution and abiggest nontrivial negative solution.

Proposition 5. If hypotheses H hold and λ > 0, then problem (Eλ) has a smallest

nontrivial positive solution u∗ ∈ intC+ and a biggest nontrivial negative solution

v∗ ∈ (−intC+)

Proof. First we produce the smallest nontrivial positive solution. Let S+λ be the set

of nontrivial positive solutions of (Eλ) located in the ordered interval [0, u]. FromProposition 4 and its proof, we have

S+λ 6= ∅ and S+

λ ⊆ intC+.


Let C ⊆ S+λ be a chain (i.e., a totally ordered subset of S+

λ ). From Dunford-Schwartz ([7, p. 336]), we know that we can find {un}n≥1 ⊆ C such that

inf C = infn≥1

un.

Clearly, because every un is a solution of (Eλ) and in view of H(i), {un}n≥1 ⊆ Wis bounded and so by passing to a suitable subsequence, we may assume that

un ⇀ u in W and un → u in C(T ). (10)

Since un ∈ S+λ , we have

A(un) = λup−1n −Nf (un), un ∈ [0, u] for all n ≥ 1. (11)

Hence,

〈A(un), un − u〉 = λ

∫ b

0

up−1n (un − u) dt−

∫ b

0

f(t, un)(un − u) dt,

and un ∈ [0, u] for all n ≥ 1, thus, taking in mind (10), we conclude that

un → u in W. (12)

So, if in (11) we pass to limit as n→∞ and use (12), then

A(u) = λup−1 −Nf (u), u ∈ [0, u],

that is u is a solution of (Eλ) in the order interval [0, u]. If we show that u 6= 0,then u ∈ S+

λ . Suppose that u = 0. Then un → 0 in C(T ). We set yn = un/‖un‖for every n ≥ 1. Because ‖yn‖ = 1 for all n ≥ 1, we may assume that

yn ⇀ y in W and yn → y in C(T ). (13)

From (11) we have

A(yn) = λyp−1n − Nf (un)

‖un‖p−1for all n ≥ 1. (14)

Hypotheses H(i), (iii) imply that

|f(t, x)| ≤ a(t)|x|p−1 for a.a. t ∈ T, all x ∈ [u, u], with a ∈ L1(T )+,

and because un ∈ [0, u], one has that{Nf (un)

‖un‖p−1

}n≥1

⊆ L1(T ) is uniformly integrable. (15)

So, if in (14) we act with yn − y ∈ W , pass to the limit as n → ∞ and use (13),(15), then

limn→∞

〈A(yn), yn − y〉 = 0,

and from Proposition 1 we have that

yn → y in W, ‖y‖ = 1, y ≥ 0. (16)

By virtue of (15) and using the Dunford-Pettis theorem and hypothesis H(iii), wehave (at least for a subsequence), see ([2]),

Nf (un)

‖un‖p−1⇀ 0 in L1(T ). (17)

So, if in (14) we pass to the limit as n→∞ and use (16), (17), then

A(y) = λyp−1,


namely {−(|y′(t)|p−2y′(t)

)′= λy(t)p−1 a. e. on T,

y(0) = y(b), y′(0) = y′(b).

Since λ > 0, y is either nodal or zero, both contradicting (16). Therefore u 6= 0 andso

u ∈ S+λ , u = inf C.

Because C is an arbitrary chain, from the Kuratowsky-Zorn Lemma we infer thatS+λ has a minimal element u∗. From [2] we know that S+

λ is downward direct (i.e, if

u, v ∈ S+λ , then we can find y ∈ S+

λ such that y ≤ u and y ≤ v). Therefore u∗ ∈ S+λ

is the smallest nontrivial positive solution of (Eλ).Similarly, if S−λ is the set of nontrivial negative solutions of (Eλ) in the order in-

terval [u, 0], then arguing as above, we produce v∗ ∈ (−intC+) the biggest nontrivialnegative solution of (Eλ).

4. Nodal solutions. In this section we show that for all λ > λ1 problem (Eλ)has also nodal solutions. The strategy is clear. We truncate the reaction at{v∗(t), u∗(t)} t ∈ T (here v∗ and u∗ are the two extremal constant sign solutionsof (Eλ), produced in Proposition 4) and using variational methods, we produce a

third solution (Eλ) in the order interval [v∗, u∗]. Then, assuming that λ > λ1 andusing Proposition 3, we can show that this solution is nontrivial, hence nodal. Inwhat follows we implement the strategy.

Let ρ = max{‖v∗‖∞, ‖u∗‖∞}. By virtue of hypotheses H(i), (iii) we can findαρ ∈ L1(T ) such that for a.a. t ∈ T we have

αρ(t) ≥ β > 0

and

(λ+ αρ(t))xp−1 ≥ f(t, x) for all x ∈ [0, ρ],

f(t, x) ≥ (λ+ αρ(t))|x|p−2x for all x ∈ [−ρ, 0].

Hence,

F (t, x) ≤ λ+ αρt

|x|pfor a.a. t ∈ T, all |x| ≤ ρ. (18)

We introduce the following truncation-perturbation of the reaction in problem(Eλ)

hλ(t, x) =

(λ+ αρ(t))|v∗(t)|p−2v∗(t)− f(t, v∗(t)) if x < v∗(t)(λ+ αρ(t))|x|p−2x− f(t, x)) if v∗(t) ≤ x ≤ u∗(t)(λ+ αρ(t))|u∗(t)|p−2u∗(t)− f(t, u∗(t)) if u∗(t) < x,

(19)

where v∗ ∈ (−intC+) and u∗ ∈ intC+ are the two extremal nontrivial constantsign solutions produced in Proposition 4. Also, let h±λ (t, x) = hλ(t, x±). Then

hλ, h±λ are all Caratheodory functions. We set H(t, x) =

∫ x0hλ(t, s) ds, H±(t, x) =∫ x

0h±λ (t, s) ds and introduce the C1-functionals ψλ, ψ

±λ : W → IR defined by

ψλ(u) =1

p‖u′‖pp +

1

p

∫ b

0

αρ(t)|u(t)|p dt−∫ b

0

Hλ(t, u(t)) dt,

ψ±λ (u) =1

p‖u′‖pp +

1

p

∫ b

0

αρ(t)|u(t)|p dt−∫ b

0

H±λ (t, u(t)) dt

for all u ∈W .


Proposition 6. If hypotheses H hold and λ > 0, then

Kψλ ⊆ [v∗, u∗] = {u ∈W : v∗(t) ≤ u(t) ≤ u∗(t) for all t ∈ T},Kψ+

λ= {0, u∗}, Kψ−

λ= {0, v∗}.

Proof. Let u ∈ Kψλ . Then

A(u) + αρ|u|p−2u = Nhλ(u). (20)

On (20) we act with (u− u∗)+ ∈W . Then, thanks to (19)

〈A(u), (u− u∗)+〉+∫ b

0

αρ|u|p−2u(u− u∗)+ dt =

∫ b

0

hλ(t, u)(u− u∗)+ dt

=

∫ b

0

[(λ+ αρ)up−1∗ − f(t, u∗)](u− u∗)+ dt

=〈A(u∗), (u− u∗)+〉+

∫ b

0

αρup−1∗ (u− u∗)+ dt,

that is

〈A(u)−A(u∗), (u− u∗)+〉+

∫ b

0

αρ(up−1 − up−1

∗ )(u− u∗)+ dt = 0,

hence, |{u > u∗}| = 0 and u ≤ u∗.Similarly, acting on (20) with (v∗ − u)+ ∈ W we obtain v∗ ≤ u. So, we have

proved thatu ∈ [v∗, u∗],

that implies thatKψλ ⊆ [v∗, u∗].

In a similar fashion, we show that Kψ+λ⊆ [0, u∗]. The minimality of u∗ and since

the critical points of ψ+λ are nonnegative solutions of problem (Eλ) (an immediate

consequence of the definition of h+λ (t, x)), we infer that

Kψ+λ

= {0, u∗}.

Analogously, we show that Kψ−λ

= {0, v∗}.

Proposition 7. If hypotheses H hold and λ > 0, then u∗ ∈ intC+ and v∗ ∈(−intC+) are local minimizers of the functional ψλ.

Proof. From (19) it is clear that ψ+λ is coercive. Also, it is sequentially weakly lower

semicontinuous. So, we can find u0 ∈W such that

ψ+λ (u0) = inf

u∈Wψ+λ (u). (21)

As in the proof of Proposition 4, using hypothesis H(iii), we show that

ψ+λ (u0) < 0 = ψ+

λ (0),

hence u0 6= 0. Also, from (21) and Proposition 6 we have

u0 ∈ Kψ+λ

= {0, u∗},

hence u0 = u∗ ∈ int(C+). Note that ψ+λ |C+

= ψλ|C+. So, u∗ is a C1(T )-minimizer

of ψλ. Invoking Proposition 2 it follows that u∗ is a local W -minimizer of ψλ.Similarly, working with ψ−λ |C+

, we show that v∗ ∈ (−int(C+)) is a local minimizer

of ψλ.


Now we are ready to produce a nodal solution. To achieve this we have to restrictthe range of the parameter λ.

Proposition 8. If hypotheses H hold and λ > λ1, then problem (Eλ) admits a

nodal solution y0 ∈ [v∗, u∗] ∩ C1(T ).

Proof. Without any loss of generality, we may assume that ψλ(v∗) ≤ ψλ(u∗) (theanalysis is similar if the opposite inequality holds). By virtue of Proposition 7 wehave that u∗ is a local minimizer of ψλ. So, we can find ρ0 ∈ (0, 1) small such that

ψλ(v∗) ≤ ψλ(u∗) < inf‖u−u∗‖=ρ0

ψλ(u) = ηλρ0 , ‖v∗ − u∗‖ > ρ. (22)

Since ψλ is coercive (see (19)), it satisfies the PS-condition. This fact and (22)permit the use of Theorem 2.1. So, we can find y0 ∈W such that

y0 ∈ Kψλ and ηλρ0 ≤ ψλ(y0). (23)

From (22), (23) and Proposition 6 we have

y0 ∈ [v∗, u∗] and y0 /∈ {v∗, u∗}, (24)

that is y0 ∈ C1(T ) is a solution of (Eλ) (see (19) and (23)).If we show that y0 6= 0, then from (24) and the extremality of v∗, u∗, we have

hat y0 is nodal.Since y0 is a critical point of ψλ of mountain pass type, we have

ψλ(y0) = infγ∈Γ

max0≤t≤1

ψλ(γ(t)), (25)

where Γ = {γ ∈ C([0, 1],W ) : γ(0) = v∗, γ(1) = u∗}. According to (25), in orderto show the nontriviality of y0, it suffices to produce γ∗ ∈ Γ such that ψλ(γ∗(t)) < 0for all t ∈ [0, 1].

To this end, recall that M = W ∩ ∂BLp1 , with ∂BLp

1 = {u ∈ Lp(T ) : ‖u‖p = 1}.Let MC = M∩C1(T ) = C1(T )∩∂BLp1 . We furnish M with the relative W -topology

and MC with the relative C1(T )-topology. Then MC is dense in M and it followsthat C([−1, 1],MC) is dense in C([−1, 1],M). We introduce the following sets ofpaths

Γ = {γ ∈ C([−1, 1],M) : γ(−1) = −u0, γ(1) = u0},

ΓC = {γ ∈ C([−1, 1],MC) : γ(−1) = −u0, γ(1) = u0}.Then ΓC is C([−1, 1],M)-dense in Γ.

Let ς = min{minT u∗, minT (−v∗)} > 0 (recall that u∗,−v∗ ∈ intC+). Hypothe-sis H(iii) implies that given ε > 0, we can find δ ∈ (0, ς) such that

|F (t, x)| ≤ ε

p|x|p for a.a. t ∈ T, all |x| ≤ δ. (26)

From (26) and (19) and since δ < ς, we have

Hλ(t, x) ≥ λ+ αρ(t)− εp

|x| for a.a. t ∈ T, all |x| ≤ δ. (27)

By virtue of Proposition 3 and since ΓC is dense in Γ, given ε ∈ (0, 1/2(λ − λ1))

(recall that λ > λ1), we can find γ0 ∈ ΓC such that

‖(γ0(s))′‖p ≤ λ1 + ε for all s ∈ [−1, 1]. (28)


Since γ0 ∈ ΓC , the set γ0([−1, 1]) ⊆ C1(T ) is compact. Recall that u∗ ∈ intC+ and

v∗ ∈ (−intC+) (see Proposition 5). So, we can find ξ ∈ (0, 1) small such that{|ξγ0(s)(t)| ≤ δ for all s ∈ [−1, 1], all t ∈ Tξγ0(s) ∈ [v∗, u∗] for all s ∈ [−1, 1].

}(29)

Let u ∈ γ0([−1, 1]). Then, by (27)–(29) and recalling that 2ε < λ− λ1,

ψλ(ξu) =ξp

p‖u′‖pp +

ξp

p

∫ b

0

αρ(t)|u|p dt−∫ b

0

Hλ(t, ξu) dt

≤ ξp

p[λ1 + ε− λ+ ε] < 0,

hence

ψλ|γ0 < 0, where γ0 = ξγ0. (30)

Note that γ0 is a continuous path in W connecting −ξu0 and ξu0. Next we producea continuous path W connecting ξu0 and u∗. To this end, recall that Kψ+

λ=

{0, u∗} and from the proof of Proposition 7 we have ψ+λ (u∗) < 0. The function

ψ+λ being coercive (see 19) satisfies the PS-condition. Then by virtue of the second

deformation theorem (see for example Gasinski-Papageorgiou [9, p. 628]), we can

find a deformation hλ : [0, 1]× ((ψ+λ )0 \ {0})→ (ψ+

λ )0 such that

hλ(0, ·) = idW , hλ(1, (ψ+λ )0) = u∗, (31)

ψ+λ (hλ(t, u)) ≤ ψ+

λ (u) for all t ∈ [0, 1], all u ∈ (ψ+λ )0 \ {0}. (32)

We set γ+(t) = hλ(t, ξu0)+, t ∈ [0, 1]. Evidently γ+ is a continuous path in W suchthat, because of (31),

γ+(0) = hλ(0, ξu0) = ξu0, γ+(1) = hλ(1, ξu0) = u∗

and, by (32) and (30),

ψ+λ |γ+ < 0.

If W+ = {u ∈W : u(t) ≥ 0 for all t ∈ T}, then

ψλ|W+= ψ+

λ |W+and γ+([0, 1]) ⊆W+.

So, we have

ψλ|γ+ < 0. (33)

Similarly, we produce γ− another continuous path in W connecting −ξu0 and v∗and such that

ψλ|γ− < 0. (34)

We concatenate γ−, γ0, γ+ and produce γ∗ ∈ Γ such that, thanks to (30), (33) and(34),

ψλ|γ∗ < 0,

that implies y0 6= 0 and so y0 ∈ [v∗, u∗] ∩ C1(T ) is a nodal solution of (Eλ).

Therefore summarizing the situation, we can state the following multiplicity the-orem for problem (Eλ).

Theorem 4.1. If hypotheses H hold and λ > λ1, then problem (Eλ) has at leastthree nontrivial solutions

u0 ∈ intC+, v0 ∈ (−intC+) and y0 ∈ [v∗, u∗] ∩ C1(T ) nodal.


REFERENCES

[1] S. Aizicovici, N. S. Papageorgiou and V. Staicu, Multiple nontrivial solutions for nonlinearperiodic problems with the p-Laplacian, J. Differential Equation, 243 (2007), 504–535.

[2] S. Aizicovici, N. S. Papageorgiou and V. Staicu, Existence of multiple solutions with precise

sign information for superlinear Neuamann problems, Ann. Mat. Pura Appl., 186 (2009),679–719.

[3] S. Aizicovici, N. S. Papageorgiou and V. Staicu, Nonlinear reasonant periodic problems with

concave terms, J. Math Anal. Appl., 375 (2011), 342–364.[4] S. Aizicovici, N. S. Papageorgiou and V. Staicu, Positive solutions for nonlinear periodic

problems with cnaceve terms, J. Math Anal. Appl., 381 (2011), 866–883.

[5] M. Del Pino, R. Manasevich and A. Murua, Exixtence and multiplcity of solutions withprescribed period for second order quasilinear ODE, Nonlinear Anal., 18 (1992), 79–92.

[6] P. Drabek and R. Manasevich, On the closed solution to some nonhomogeneous eigenvalue

problems with p-Laplacian, Differential Integral Equations, 12 (1999), 773–788.[7] N. Dunford and J. Schwartz, Linear Operators I, Wiley-Interscience, New York, 1958.

[8] L. Gasinski, Positive solutions for reasonant boundary value problems with the scalar p-Laplacian and a nonsmooth potential, Discrete Cont. Dynam. Systems, 17 (2007), 143–158.

[9] L. Gasinski and N. S. Papageorgiou, Nonlinear Analysis, Chapman & Hall/CRC, Boca Raton,

2006.[10] L. Gasinski and N. S. Papageorgiou, Three nontrivial soluions for periodic problems with the

p-Laplacian and a p-superlinear nonlinearity, Commun. Pure Appl. Anal., 8 (2009), 1421–

1437.[11] E. Papageorgiou and N. S. Papageorgiou, Two nontrivial solutions for quasilinear periodic

problems, Proceedings Amer. Math. Soc., 132 (2004), 429–434.

[12] J. Vazquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math.Optim., 12 (1984), 191–202.

[13] X. Yang, Multiple periodic solutions of a class of p-Laplacian, J. Math. Anal. Appl., 314

(2006), 17–29.

Received March 2013; revised July 2013.

E-mail address: [email protected]



http://www.ams.org/mathscinet-getitem?mr=MR2371798&return=pdf













mailto:[email protected]



a nonlinear eigenvalue problem for the periodic scalar $p$-laplacian

Documents