and Nikolaos S. Papageorgiou

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Nonlinear nonhomogeneous Neumann eigenvalue problems

Pasquale Candito


, Roberto Livrea


and Nikolaos S. Papageorgiou


1Department DICEAM, University of Reggio Calabria, 89100 - Reggio Calabria, Italy

2National Technical University, Department of Mathematics, Zografou Campus, Athens 15780, Greece

Received 8 April 2015, appeared 28 July 2015 Communicated by Gabriele Bonanno

Abstract. We consider a nonlinear parametric Neumann problem driven by a nonho- mogeneous differential operator with a reaction which is (p1)-superlinear near± and exhibits concave terms near zero. We show that for all small values of the parame- ter, the problem has at least five solutions, four of constant sign and the fifth nodal. We also show the existence of extremal constant sign solutions.

Keywords: superlinear reaction, concave terms, maximum principle, extremal constant sign solutions, nodal solution, critical groups.

2010 Mathematics Subject Classification: 35J20, 35J60, 35J92, 58E05.

1 Introduction

Let Ω∈RN be a bounded domain with aC2-boundary∂Ω. The aim of this work is to study the existence and multiplicity of solutions with a precise sign information, for the following nonlinear nonhomogeneous parametric (eigenvalue) Neumann problem:

diva(Du(z)) = f(z,u(z),λ) inΩ, ∂u

∂n =0 on Ω. (Pλ) Here n(·) stands for the outward unit normal on Ω. Also, a: RNRN is a continu- ous and strictly monotone map which satisfies certain other regularity conditions listed in hypothesesH(a)below. These hypotheses are general enough to incorporate as a special case several differential operators of interest, such as the p-Laplacian (1 < p < ∞), the (p,q)- Laplacian (that is, the sum of a p-Laplacian and a q-Laplacian with 1 < q < p < ∞) and the generalized p-mean curvature differential operator. The variable λ > 0 is a parameter (eigenvalue) which in general enters in the equation in a nonlinear fashion. The nonlinear- ity of the right-hand side (the reaction of the problem) f(z,x,λ) is a Carathéodory function in (z,x) ∈ ×R (that is for all x ∈ R, λ > 0, z 7→ f(z,x,λ) is measurable and for a.a.

z ∈ Ω, all λ > 0, x 7→ f(z,x,λ) is continuous). We assume that x 7→ f(z,x,λ) exhibits (p−1)-superlinear growth near ±∞, while near zero we assume the presence of a concave

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term (that is, of a(p−1)-sublinear term). So, in the reaction f(z,x,λ), we can have the com- peting effects of two different kinds of nonlinearities (“concave–convex” nonlinearities). Such problems were first investigated by Ambrosetti–Brezis–Cerami [2] who deal with semilinear (that is, p = 2) equations. Their work was extended to equations driven by the Dirichlet p-Laplacian, by García Azorero–Manfredi–Peral Alonso [9] and by Guo–Zhang [13]. In all three works the reaction has the special form

f(z,x,λ) =λ|x|q2x+|x|p2x with 1<q< p< p= ( N p

Np if p< N, + if p≥ N.

More general reactions were considered by Hu–Papageorgiou [14] and by Marano–Papa- georgiou [18]. Both papers deal with Dirichlet problems driven by the p-Laplacian. For the Neumann problem, we mention the work of Papageorgiou–Smyrlis [24], where the differential operator is

u7→ −pu+β(z)u for allu∈W1,p() (1< p<), with∆p being thep-Laplace differential operator defined by

pu=div(|∇u|p2∇u) for allu∈W1,p()

andβ∈ L(), β0, β6= 0. So, in this case the differential operator is coercive. This is not the case in problem (Pλ). Moreover, the reaction in [24] has the form

f(z,x,λ) =λ|x|q2x+g(z,x)

with 1< q< pandg(z,x)is a Carathéodory function which is (p−1)-superlinear in x∈ R. Papageorgiou–Smyrlis [24] look for positive solutions and they prove a bifurcation-type theo- rem describing the set of positive solutions as the parameterλ>0 varies.

Our approach is variational based on the critical point theory. We also use suitable trun- cation and perturbation techniques and Morse theory (critical groups).

2 Mathematical background – hypotheses

In this section, we present the main mathematical tools which we will use in the sequel and state the hypotheses on the data of problem (Pλ). We also present some straightforward but useful consequences of the hypotheses.

Let X be a Banach space and X its topological dual. By h·,·i we denote the duality brackets for the pair(X,X). Let ϕ∈ C1(X). We say thatϕsatisfies the Cerami condition (the C-condition for short), if the following is true:

Every sequence{xn}n1⊆ X s.t. {ϕ(xn)}n1Ris bounded and (1+kxnk)ϕ0(xn)→0 in X as n→∞, admits a strongly convergent subsequence.

This is a compactness type condition on the functional ϕ, more general than the Palais–

Smale condition. It compensates for the fact that the ambient space X need not be locally compact (being in general infinite dimensional). The C-condition suffices to prove a deforma- tion theorem and then from it derive the minimax theory for the critical values ofϕ. Prominent in that theory is the so-called “mountain pass theorem” (see [3]).


Theorem 2.1. If ϕ∈C1(X)satisfies the C-condition, x0,x1 ∈X withkx1−x0k> ρ>0 max{ϕ(x0),ϕ(x1)}<inf{ϕ(x):kx−x0k=ρ}=mρ

and c = infγΓmax0t1ϕ(γ(t)), where Γ = {γ ∈ C([0, 1],X) : γ(0) = x0, γ(1) = x1}, then c≥mρ and c is a critical value of ϕ.

The analysis of problem (Pλ), in addition to the Sobolev spaceW1,p(), will also involve the Banach spaceC1(¯). This is an ordered Banach space with positive cone

C+ ={u∈C1(¯):u(z)≥0 for allz∈ ¯}. This cone has a nonempty interior given by

intC+={u∈C1(¯):u(z)>0 for allz∈ ¯}. Now, let us introduce the hypotheses on the mapa(·).

Letξ ∈C1(0,∞)with ξ(t)>0 for all t>0 and assume that 0<cˆ≤


ξ(t) ≤c0 and c1tp1ξ(t)≤c2(1+tp1), (2.1) for all t>0, withc1 >0.

The hypotheses on the mapa(·)are the following:

H(a): a(y) =a0(|y|)y for all y∈RN with a0(t)>0for all t >0and

(i) a0 ∈C1(0,∞), t 7→ta0(t)is strictly increasing on(0,∞), ta0(t)→0+as t→0+and



a0(t) = c>−1;

(ii) k∇a(y)k ≤c3ξ(|y|)

|y| for all y∈RN\ {0}and some c3>0;

(iii) (∇a(y)h,h)RNξ(||y|)

y| |h|2for all y∈RN\ {0}, all h∈RN; (iv) if G0(t) =Rt

0 sa0(s)ds for all t ≥ 0, then pG0(t)−t2a0(t)≥ −γfor all t ≥0and some γ>0;

(v) there existsτ∈(1,p)such that t 7→G0(t1/τ)is convex,limt0+ G0tτ(t) =0and t2a0(t)−τG0(t)≥ct˜ p

for all t>0an somec˜>0.

Remark 2.2. EvidentlyG0 is strictly convex and strictly increasing. We setG(y) =G0(|y|)for ally∈RN. ThenGis convex and it is differentiable at everyy∈RN\ {0}. Also

∇G(y) =G00(|y|) y

|y| =a0(|y|)y =a(y) for ally∈RN\ {0}, 0∈ ∂G(0), implies that Gis the primitive of the mapa.

The convexity ofGand the fact thatG(0) =0, imply

G(y)≤(a(y),y)RN =a0(|y|)|y|2 (2.2) for all y∈RN.


The next lemma is a straightforward consequence of the above hypotheses and summarizes the main properties of the mapa, which we will use in the sequel.

Lemma 2.3. If hypotheses H(a)hold, then

(a) y7→ a(y)is continuous and strictly monotone, hence maximal monotone too;

(b) |a(y)| ≤c4(1+|y|p1)for all y∈RN and some c4>0;

(c) (a(y),y)RNpc11|y|p for all y∈RN.

Lemma2.3and (2.1), (2.2), lead to the following growth estimates for the primitive G.

Corollary 2.4. If hypotheses H(a)hold, then c1

p(p−1)|y|p ≤G(y)≤ c5(1+|y|p) for all y∈RN with c5 >0.

Example 2.5. The following maps satisfy hypothesesH(a):

(a) a(y) =|y|p2ywith 1< p<∞. This map corresponds to the p-Laplacian

pu=div(|∇u|p2∇u) for all u∈W1,p().

(b) a(y) =|y|p2y+µ|y|q2ywith 1 <q< pandµ>0. This map corresponds to a sum of a p-Laplacian and aq-Laplacian, that is:

pu+µqu for allu∈W1,p().

Such differential operators arise in many physical applications (see [23] and the refer- ences therein).

(c) a(y) = (1+|y|2)p22y with 1< p<∞. This map corresponds to the generalized p-mean curvature differential operator



for allu∈W1,p(). (d) a(y) =|y|p2y+ |y|p2y

1+|y|p with 1< p< ∞.

We introduce the following nonlinear map A: W1,p()→W1,p() defined by hA(u),yi=


(a(∇u),∇y)RNdz (2.3) for allu,y∈W1,p().

The next result is a particular case of a more general result proved by Gasinski–Papa- georgiou [11].

Proposition 2.6. If hypotheses H(a)hold, then the map A:W1,p()→ W1,p() defined by(2.3) is bounded (that is, it maps bounded sets to bounded sets), demicontinuous, strictly monotone (hence maximal monotone too) and of type (S)+, that is

un *u in W1,p() and lim sup


hA(un),un−ui ≤0⇒un→u in W1,p().


Consider a Carathéodory function f0: Ω×RRsatisfying

|f0(z,x)| ≤α(z)(1+|x|r1) for a.a.z∈, all x∈R, with α∈ L()+, 1< r < p. We set F0(z,x) =Rx

0 f0(z,s)ds and consider theC1-functional ϕ0: W1,p()→Rdefined by

ϕ0(u) =




F0(z,u(z))dz for allu∈W1,p(). In the sequel byk · k1,p we denote the norm of the Sobolev spaceW1,p(), that is

kuk1,p=kukpp+k∇ukpp1/p. The following result is due to Motreanu–Papageorgiou [21].

Proposition 2.7. If u0 ∈W1,p()is a local C1(¯)-minimizer ofϕ0, that is there existsρ0 >0such that

ϕ0(u0)≤ ϕ0(u0+h) for all h∈ C1(¯)withkhkC1(¯)ρ0,

then u0 ∈ C1,β(¯) for some β ∈ (0, 1)and it is also a local W1,p()-minimizer of ϕ0, that is there existsρ1>0such that

ϕ0(u0)≤ ϕ0(u0+h) for all h∈W1,p(¯)withkhk1,pρ1.

Remark 2.8. The first such result relating local minimizers, was proved by Brezis–Nirenberg [4], for the spacesC01(¯) = {u ∈C1(¯):u|∂Ω =0}and H10()and with G(y) = 12|y|2 for all y∈RN (it corresponds to the Dirichlet Laplacian).

Now letX be a Banach space andY2 ⊆ Y1 ⊆ X. For every integerk ≥0 by Hk(Y1,Y2)we denote the kth-singular homology group with integer coefficients for the pair(Y1,Y2). Recall that Hk(Y1,Y2) =0 for all k<0.

Given ϕ∈C1(X)andc∈ R, we introduce the following sets:

ϕc ={x∈ X: ϕ(x)≤c}, Kϕ ={x ∈X: ϕ0(x) =0}, Kcϕ ={x∈ Kϕ : ϕ(x) =c}. The critical groups ofϕat an isolated critical pointx0 ∈Xwithϕ(x0) =c(that is,x0∈Kcϕ) are defined by

Ck(ϕ,x0) =Hk(ϕc∩ U,ϕc∩ U \ {x0}) for all k≥0,

where U is a neighborhood ofx0∈ Xsuch thatKϕϕc∩ U = {x0}. The excision property of singular homology implies that the above definition of critical groups is independent of the choice of the neighborhood U.

Next we introduce the hypotheses on the reaction f(z,x,λ).

H(f): f: Ω×R×(0,+) → R is a function such that for every λ > 0, (z,x) 7→ f(z,x,λ) is Carathéodory, f(z, 0,λ) =0for a.a. z∈ and

(i) for every ρ > 0and λ > 0, there exists αρ(λ) ∈ L()+ such that λ 7→ kαρ(λ)k is bounded on bounded sets and

|f(z,x,λ)| ≤αρ(λ)(z) for a.a.z∈ Ω, all|x| ≤ρ;


(ii) if F(z,x,λ) =Rx

0 f(z,s,λ)ds, then for allλ>0



|x|p = + uniformly for a.a.z∈

and there exist r(λ)∈(p,p)with r(λ)→r0∈ (p,p)asλ→0+andηˆ(λ),η(λ)∈ L()such that


η(λ)(z)≤lim inf



|x|r(λ)−2x ≤lim sup



|x|r(λ)−2x ≤ η(λ)(z)

uniformly for a.a. z∈ andλ7→ kηˆ(λ)k,kη(λ)k are bounded on bounded sets in (0,+);

(iii) for everyλ>0, there existsθ(λ)∈ max

(r(λ)−p)Np, 1 ,p

, andβ0(λ)such that 0< β0(λ)≤lim inf



|x|θ(λ) uniformly for a.a.z ∈Ω;

(iv) for every λ > 0, there exist q(λ),µ(λ) ∈ (1,τ)(see hypotheses H(a) (v)) with q(λ) ≤ µ(λ)andδ0(λ)∈(0, 1),cˆ0(λ)>0such that q(λ)→q0∈ (1,p)asλ→0+and

0(λ)|x|µ(λ)≤ f(z,x,λ)x≤q(λ)F(z,x,λ) for a.a.zΩ, all |x| ≤δ0(λ), there existβ(λ), β1,β2>0, withβ(λ)→0+asλ→0+such that

f(z,x,λ)x≤ β(λ)|x|q(λ)+β1|x|rβ2|x|p for a.a.z∈Ω, all x∈R, with r(λ)≤r < p (see (ii)) and there exits a functionη0(·,λ)∈ L()+such that

kη0(·,λ)k →0 asλ→0+, lim sup



|x|q(λ)η0(z,λ) uniformly for a.a.z∈Ω.

Remark 2.9. Hypotheses H(f) (ii), (iii) imply that for a.a. z ∈ and all λ > 0, the reac- tion f(z,·,λ)is (p−1)-superlinear near ±∞. Usually such problems are studied using the Ambrosetti–Rabinowitz condition (see [3]). Our hypothesis here is more general and incor- porates in our framework superlinear functions with “slower” growth near±, which fail to satisfy the Ambrosetti–Rabinowitz condition (see the examples below). On this issue, see also [16], [19] and the references therein.

Example 2.10. The following functions satisfy hypotheses H(f). For the sake of simplicity we drop thez-dependence:

f1(x,λ) =λ|x|q2x+|x|r2x− |x|p2x with 1<q<τ< p<r < p,

f2(x,λ) =



|x|r2x− |x|p2x−σ(λ) ifx< −ρ(λ)

λ|x|q2x− |x|p2x if −ρ(λ)≤ x≤ρ(λ)

|x|r2x− |x|p2x+σ(λ) ifρ(λ)< x

with 1 < q < τ < p < r < p, ρ(λ) ∈ (0, 1], ρ(λ) → 0+ as λ → 0+, σ(λ) = (λρ(λ)rq)ρ(λ)q1;

f3(x,λ) =λ|x|q2x+|x|p2x

ln|x| − p−1 p

with 1<q< τ< p.

Note that f4,λ)does not satisfy the Ambrosetti–Rabinowitz condition.


We introduce the following truncations-perturbations of the reaction f(z,x,λ): fˆ+(z,x,λ) =

(0 ifx ≤0

f(z,x,λ) +β2xp1 if 0<x (2.4) and

(z,x,λ) =

(f(z,x,λ) +β2|x|p2x if x<0

0 if 0≤ x. (2.5)

Both are Carathéodory functions. We set Fˆ±(z,x,λ) =

Z x


±(z,s,λ)ds and introduce theC1-functionals ˆϕλ±: W1,p()→Rdefined by


ϕλ±(u) =


G(∇u(z))dz+ β2

p kukpp


±(z,u(z),λ)dz for allu∈W1,p(). Also, by ϕλ :W1,p()→Rwe denote the energy functional for problem (Pλ) defined by

ϕλ(u) =




F(z,u(z),λ)dz for allu∈W1,p(). Clearly ϕλ ∈C1(W1,p()).

We conclude this section by fixing our notation. For x ∈ R let x± = max{±x, 0}. Then, givenu∈W1,p(), we setu±(·) =u(·)±. We have

u=u+−u, |u|=u++u and u+,u,|u| ∈W1,p().

Giveh(z,x)a jointly measurable function (for example, a Carathéodory function), we define Nh(u)(·) =h(·,u(·)) for all u∈W1,p().

Finally by| · |N we denote the Lebesgue measure onRN.

3 Solutions of constant sign

In this section we show that forλ>0 small, problem (Pλ) has at least four nontrivial solutions of constat sign (two positive and two negative).

First we establish the compactness properties of the functionals ˆϕλ± andϕλ.

Proposition 3.1. If hypotheses H(a)and H(f)hold andλ > 0, then the functionals ϕˆλ± satisfy the C-condition.

Proof. We do the proof for the functional ˆϕλ+, the proof for ˆϕλbeing similar. So, we consider a sequence{un}inW1,p()such that

|ϕˆλ+(un)| ≤ M1 (3.1)

for some M1 >0, alln≥1,

(1+kunk1,p)(ϕˆλ+)0(un)→0 inW1,p() asn→∞. (3.2)


From (3.2) we have

|h(ϕˆλ+)0(un),hi| ≤ εnkhk1,p 1+kunk1,p for allh∈W1,p()withεn →0+. Hence

hA(un),hi+β2 Z

|un|p2unh dz−


+(z,un,λ)h dz


1+kunk1,p (3.3) for alln≥1. In (3.3) we chooseh=−un ∈W1,p(). Then, in view of (2.4),


a(−∇un),−∇un)RN dz+β2kunkppεn for alln≥1, so that, because of Lemma2.3,


p−1k∇unkpp+β2kunkppεn, that is

un →0 inW1,p(). (3.4)

Next, in (3.3) we chooseh=u+n ∈W1,p(). Then


a(∇u+n),∇u+n)RN dz+


f(z,u+n,λ)u+n dz≤εn for alln≥1. (3.5) Also, from (3.1), (3.4) and Corollary2.4, we have




pF(z,u+n,λ)dz≤ M2 for some M2 >0, alln≥1. (3.6) We add (3.5) and (3.6) and obtain


pG(∇u+n)− a(∇u+n),∇u+n)RNdz+


f(z,u+n,λ)−pF(z,un,λ)dz≤ M3, (3.7) for someM3 >0, alln≥1. Hence, H(a)(iv) assures that, for alln≥1


f(z,u+n,λ)−pF(z,u+n,λ)dz≤ Mˆ3. (3.8) HypothesesH(f)(i), (iii) imply that we can findb1(λ)∈(0,β0(λ))andc6(λ)>0 such that

b1(λ)|x|θ(λ)−c6(λ)≤ f(z,x,λ)x−pF(z,x,λ) for a.a.z∈ Ω, all x∈R. (3.9) Using (3.9) in (3.8), we obtain that

{u+n} ⊆Lθ(λ)()is bounded. (3.10) Note that in hypothesis H(f) (iii) without any loss of generality, we may assume that 1 ≤ θ(λ)<r(λ). First suppose thatN6= pand lett ∈(0, 1)be such that


r(λ) = 1−t θ(λ)+ t

p. (3.11)

The interpolation inequality (see, for example, Gasinski–Papageorgiou [10, p. 905]) implies ku+nkr(λ) ≤ ku+nk1t



Thus, from (3.10) and the Sobolev embedding theorem ku+nkrr((λ)

λ)≤ M4ku+nktr1,p(λ) for someM4>0, all n≥1. (3.12) Hypotheses H(f)(i), (ii) imply that we can findc7(λ)>0 such that

f(z,x,λ)≤c7(λ)(1+|x|r(λ)−1) for a.a. z∈Ω, all x∈R. (3.13) In (3.3) we chooseh= u+n ∈W1,p(). Then



RN dz−


f(z,u+n,λ)u+n dz≤εn for alln ≥1.

Hence, from Lemma2.3, (3.13) and (3.12), there existc8(λ),c9(λ)>0 such that for alln≥1 c1

p−1k∇u+nkpp≤ c8(λ)(1+ku+nkr(λ)


≤ c9(λ)(1+ku+nktr1,p(λ)). (3.14) Recall that u7→ kukθ(λ)+k∇ukp is an equivalent norm on the spaceW1,p()(see, for exam- ple, [10, p. 227]). Then, (3.10) and (3.14) imply

ku+nk1,pp ≤c10(λ)1+ku+nktr1,p(λ) for somec10(λ)>0, alln≥1. (3.15) From hypothesis H(f) (iii) and after a simple calculation involving (3.11), we show that tr(λ)< p. So, from (3.15) we infer that

{u+n} ⊆W1,p()is bounded. (3.16) If N = p, then by the Sobolev embedding theorem we know thatW1,p(),→ Ls()continu- ously (in fact compactly) for all s ∈ [1,∞). Then, fors > r(λ) ≥ θ(λ) ≥ 1 sufficiently large, reasoning as in (3.11) and recalling hypothesis H(f)(iii), one has that

tr(λ) = (r(λ)−θ(λ))s s−θ(λ) < p.

Therefore, the previous argument remains valid and so we reach again (3.16).

From (3.4) and (3.16) it follows that

{un} ⊆W1,p()is bounded.

At this point, we may assume that there existsu∈W1,p()such that

un*u inW1,p() and un→u in Lr(λ)(). (3.17) We return to (3.3), chooseh=un−u and pass to the limit asn→and use (3.17). Then


and Proposition 2.6 implies that un → u in W1,p(). This proves that the functional ˆϕλ+ satisfies the C-condition.

With minor changes in the proof, we can also have the following result.


Proposition 3.2. If hypotheses H(a)and H(f)hold and λ > 0, then the functional ϕλ satisfies the C-condition.

Next we show that for all small values of the parameterλ >0, the functionals ˆϕλ± satisfy the mountain pass geometry (see Theorem2.1).

Proposition 3.3. If hypotheses H(a) and H(f) hold, then there exist λ± > 0 such that for every λ∈(0,λ±), we can findρλ±>0for which we have

infh ˆ

ϕλ±(u):kuk1,p=ρλ± i

= mˆλ±>0.

Proof. By virtue of hypothesis H(f)(iv) we see that given anyλ>0 F(z,x,λ)≤ β(λ)

q(λ)|x|q(λ)+ β1


p |x|p for a.a. z∈, all x∈R. (3.18) For allu∈W1,p(), because of Corollary2.4 and (2.4) we have


ϕλ+(u) =


G(∇u)dz+ β2

p kukpp




p(p−1)k∇ukpp+ β2

p kukppβ2

p ku+kpp


F(z,u+,λ)dz. (3.19) If in (3.19) we use (3.18), we obtain


ϕλ+(u)≥ c1

p(p−1)k∇ukpp+ β2

p kukppβ(λ)


λ)β1 rku+krr

hc11−(c12(λ)kukq1,p(λ)−p+c13kukr1,pp)ikuk1,pp , (3.20) with c11, c13 > 0 independent of λ and c12(λ) → 0 as λ → 0+. We introduce the function γλ: (0,∞)→(0,∞)defined by

γλ(t) =c12(λ)tq(λ)−p+c13trp for allt>0.

Recall that 1<q(λ)< p <r(λ)≤r < p. Hence

γλ(t)→+ ast→0+and ast→+∞.

Therefore we can findt0= t0(λ)∈(0,∞)such that

γλ(t0) =min[γλ(t):t>0]. In particular,

γλ0(t0) = (q(λ)−p)c12(λ)tq0(λ)−p1+ (r−p)c13tr0p1=0, hence

t0= t0(λ) =

(p−q(λ))c12(λ) (r−p)c13

r∗ −1q(


and a simple calculation leads to

γλ(t0) = [c12(λ)]q


r∗ −q(λ)c14(λ),


with λ7→ c14(λ)bounded on bounded intervals. Note that using the hypotheses onq(·)and r, we have

γλ(t0)→0+ asλ→0. So, choosingλ+ >0 small, we have

γλ(t0)<c11 for allλ∈(0,λ+). Then, from (3.20) it follows that for allλ∈(0,λ+)we have


ϕλ+(u)≥mˆλ+ >0 for allu∈W1,p() withkuk1,p =t0(λ) =ρλ+. In a similar fashion we show the existence ofλ>0 such that


ϕλ(u)≥mˆλ >0 for allu∈W1,p() withkuk1,p =t0(λ) =ρλ, and the proof is complete.

The next proposition completes the mountain pass geometry for the functionals ˆϕλ±. It is an immediate consequence of the p-superlinear hypothesis H(f)(ii).

Proposition 3.4. If hypotheses H(a)and H(f)hold,λ>0and u∈intC+, thenϕˆλ±(tu)→ −as t→ ±∞.

Now we can use variational methods to produce constant sign solutions for problem (Pλ) whenλ>0 is small.

Proposition 3.5. If hypotheses H(a)and H(f)hold, then

(a) for everyλ∈(0,λ+)problem (Pλ) has at least two positive solutions u0, ˆu ∈intC+

withu being a local minimizer ofˆ ϕλ andϕλ(uˆ)<0< ϕλ(u0); (b) for everyλ∈(0,λ)problem (Pλ) has at least two negative solutions

v0, ˆv∈ −intC+

withv being a local minimizer ofˆ ϕλ andϕλ(vˆ)<0< ϕλ(v0);

(c) ifλ =min{λ+,λ}andλ ∈(0,λ), then problem (Pλ) has at least four nontrivial solutions of constant sign

u0, ˆu∈intC+, v0, ˆv∈ −intC+

withu, ˆˆ v local minimizers of ϕλandϕλ(vˆ),ϕλ(uˆ)<0< ϕλ(u0),ϕλ(v0).

Proof. (a) For λ ∈ (0,λ+), let ρλ+ be as postulated by Proposition 3.3 and consider ¯Bρλ + = {u ∈ W1,p() : kuk1,pρλ+}, which clearly is weakly compact inW1,p(). Moreover, since


ϕλ+is sequentially weakly lower semicontinuous inW1,p(), one has that there exists ˆu∈ B¯ρλ +

such that


ϕλ+(uˆ) =infh ˆ




On the other hand, for δ0 ∈ (0, 1) as in hypothesis H(f)(iv) and ξ ∈ (0,δ0(λ)) small (take

|ξ|<ρλ+/||1/pN ), we obtain ˆ

ϕλ+(ξ) =−



Therefore, because of Proposition3.3, we can deduce that ub∈ Bρλ

+ ={u∈W1,p():kuk1,p<ρλ+}, and

(ϕˆλ+)0(uˆ) =0.

So, it follows that

A(uˆ) +β2|uˆ|p2uˆ = Nfˆλ

+(uˆ), (3.21)

where ˆf+λ(z,x) = fˆ+(z,x,λ). On (3.21) we act with −uˆ ∈ W1,p() and using (2.4) and Corollary2.4, we obtain ˆu≥0, ˆu6=0. Then, again because of (2.4), (3.21) we have

−diva(∇uˆ(z)) = f(z, ˆu(z),λ) a.e. in Ω,

∂n =0 on∂Ω,

(see [11]). From [26], we know that ˆu ∈ L(). So, we can apply the regularity result of Lieberman [17] and infer that ˆu ∈ C+\ {0}. From hypotheses H(f)(i), (iv), we see that for everyλ>0 andρ>0, we can findξρ(λ)>0 such that

f(z,x,λ)x+ξρ(λ)|x|p ≥0 for a.a. z∈Ω, all|x| ≤ρ.

Letρ=kuˆk and let ξρ(λ)>0 as above. Then

−diva(∇uˆ(z)) +ξρ(λ)uˆ(z)p = f(z, ˆu(z),λ) +ξρ(λ)uˆ(z)p≥0 a.e. in Ω, that is

diva(∇uˆ(z))≤ξρ(λ)uˆ(z)p a.e. in Ω. (3.22) Letχ(t) =ta0(t)for allt>0. Then, from H(a)(iii)

0(t) =t2a00(t) +ta0(t)≥c1tp1, hence, by integration one has

Z t

00(s)ds≥ct˜ p for allt >0. (3.23) From (3.22), (3.23) and the strong maximum principle of Pucci–Serrin [25, p. 111] we have


u(z)>0 for allz∈ Ω.

So, we can apply the boundary point theorem of Pucci–Serrin [25, p. 120] and have ˆ

u∈intC+. (3.24)

From (2.4) it is clear that


+ = ϕˆλ+|C



From this equality and (3.24) it follows that ˆu is a local C1(¯)-minimizer of ϕλ. Invoking Proposition2.7, we have that ˆuis a localW1,p()-minimizer ofϕλ.

Now we look for the second positive solution. Propositions3.1,3.3and3.4permit the use of Theorem2.1on the functional ˆϕλ+. So, we can findu0∈W1,p()such that


ϕλ+(uˆ)<0= ϕˆλ+(0)<mˆλ+ϕˆλ+(u0) and (ϕˆλ+)0(u0) =0. (3.25) From (3.25) it follows that u0 6∈ {0, ˆu}, it solves problem (Pλ) and by the nonlinear regularity theory we haveu0 ∈C+\ {0}(see [17,26]). In fact, as above, using the results of Pucci–Serrin [25, pp. 111, 120], we conclude thatu0 ∈intC+.

(b) Working in a similar fashion, this time with the function ˆϕλ, for λ ∈ (0,λ)we produce two negative solutions for problem (Pλ)

v0, ˆv∈ −intC+.

Moreover, ˆvis a local minimizer of ϕλ and ϕλ(vˆ)<0< ϕλ(v0). (c) Follows from parts (a) and (b).

4 Nodal solutions

In this section, we produce a fifth nontrivial solution of (Pλ), withλ∈(0,λ), which is nodal (sign changing). The idea is first to generate the extremal nontrivial constant sign solutions, that is the smallest nontrivial positive solutionuλ and the biggest nontrivial negative solution vλ of (Pλ). Then look for a nontrivial solution in the order interval[vλ,uλ] = {u∈ W1,p(): vλ(z) ≤ u(z) ≤ uλ(z) a.e. in Ω} distinct from vλ and uλ. Necessarily, this solution will be nodal.

HypothesesH(f)(i), (ii), (iv) imply that we can findc15 >0 such that

f(z,x,λ)x ≥cˆ0(λ)|x|µ(λ)−c15|x|r(λ) for a.a. z∈ Ω, allx∈R, allλ∈(0,λ). (4.1) This unilateral growth estimate on the reaction f(z,·,λ)leads to the following parametric auxiliary Neumann problem





−diva(∇u(z)) =cˆ0(λ)|u(z)|µ(λ)−2u(z)−c15|u(z)|r(λ)−2u(z) inΩ,


∂n =0 on∂Ω,

1<µ(λ)< p<r(λ)< p.


For this auxiliary problem, we have the following existence and uniqueness result for nontrivial solutions of constant sign.

Proposition 4.1. If hypotheses H(a)hold andλ>0, then problem(Sλ)has a unique positive solution u¯λ ∈ intC+and since problem(Sλ) is oddv¯λ = −u¯λ ∈ −intC+ is the unique negative solution of (Sλ).

Proof. First we establish the existence of a positive solution. To this end, letψ+λ : W1,p()→R be the C1-functional defined by

ψ+λ(u) =


G(∇u(z))dz+ 1



µ(λ)+ c15





Recall that 1<µ(λ)< p<r(λ)(see (Sλ)). So, using Corollary2.4, we see thatψ+λ is coercive.

Also, using the Sobolev embedding theorem, we can check that ψλ+ is sequentially weakly lower semicontinuous. Hence, by the Weierstrass theorem, we can find ¯uλ ∈ W1,p() such that

ψλ+(u¯λ) =inf{ψλ+(u):u∈W1,p()}. (4.2) Sinceµ(λ)<r(λ), forξ ∈ (0, 1)small we haveψ+λ(ξ)<0 and so, because of (4.2),

ψ+λ(u¯λ)<0= ψλ+(0),

hence ¯uλ 6=0. From (4.2), we have that ¯uλ is a critical point ofψ+λ, namely

A(u¯λ) +|u¯λ|p2λ= cˆ0(λ)(u¯+λ)µ(λ)−1−c15(u¯+λ)r(λ)−1+ (u¯+λ)p1. (4.3) On (4.3) we act with −u¯λ ∈ W1,p() and obtain ¯uλ ≥ 0, ¯uλ 6= 0. Hence ¯uλ is a positive solution of (Sλ). Nonlinear regularity theory implies ¯uλ ∈C+\ {0}. We have

−diva(∇u¯λ(z)) =cˆ0(λ)u¯λ(z)µ(λ)−1−c15λ(z)r(λ)−1 a.e. in Ω, thus

diva(∇u¯λ(z))≤c15ku¯λkr(λ)−pλ(z)p1 a.e. inΩ, and from [25, p. 111, 120] we conclude that


uλ ∈intC+.

So, we have established the existence of a positive solution ¯uλ ∈intC+for problem (Sλ).

Next we show the uniqueness of this positive solution. To this end, consider the integral functionalσλ+: L1()→R∪ {+}defined by

σλ+(u) = (R

G(∇u1/τ)dz ifu≥0, u1/τ ∈W1,p()

+ otherwise.

Let u1,u2 ∈ domσλ+ = {u ∈ L1() : σλ+(u) < +} (the effective domain of σλ+) and let t∈ [0, 1]. We set

y= ((1−t)u1+tu2)1/τ, v1 =u1/τ1 , v2=u1/τ2 . From [5, Lemma 1], we have

k∇y(z)k ≤[(1−t)k∇v1(z)kτ+tk∇v2(z)kτ]1/τ a.e. in Ω, and exploiting the monotonicity ofG0 and hypothesis H(a)(v)

G(∇y(z)) =G0(k∇y(z)k)≤ G0


≤(1−t)G0(k∇v1(z)k) +tG0(k∇v2(z)k) for a.a.z∈Ω, that isσλ+ is convex.

Also, by Fatou’s lemma σλ+ is lower semicontinuous. Now, letu ∈ W1,p()be a positive solution of problem (Sλ). From the first part of the proof, we haveu∈ intC+. So, ifh ∈C1(¯) andt ∈(−1, 1)with |t|small, we have





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