**Nonlinear nonhomogeneous Neumann** **eigenvalue problems**

**Pasquale Candito**

^{1}

### , **Roberto Livrea**

^{B}

^{1}

### and **Nikolaos S. Papageorgiou**

^{2}

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 (p−1)-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∈_{R}^{N} be a bounded domain with aC^{2}-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:

−_{div}a(Du(z)) = f(z,u(z)_{,}*λ*) _{in}_{Ω,} ^{∂u}

*∂n* =_{0} _{on} *∂*Ω. (P* _{λ}*)
Here n(·) stands for the outward unit normal on

*∂*Ω. Also, a:

**R**

^{N}→

_{R}^{N}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) ∈

_{Ω}×

**(that is for all x ∈**

_{R}

_{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

BCorresponding author. Email: roberto.livrea@unirc.it

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|^{q}^{−}^{2}x+|x|^{p}^{−}^{2}x with 1<q< p< p^{∗}=
( _{N p}

N−p 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→ −_{∆}_{p}u+*β*(z)u for allu∈W^{1,p}(_{Ω}) (1< p<_{∞}),
with∆p being thep-Laplace differential operator defined by

∆pu=div(|∇u|^{p}^{−}^{2}∇u) for allu∈W^{1,p}(_{Ω})

and*β*∈ L^{∞}(_{Ω})_{,} *β*≥_{0,} *β*6= 0. So, in this case the differential operator is coercive. This is not
the case in problem (P* _{λ}*). Moreover, the reaction in [24] has the form

f(z,x,*λ*) =*λ*|x|^{q}^{−}^{2}x+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 parameter

*λ*>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 (P* _{λ}*). 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*∈ C^{1}(X). We say that*ϕ*satisfies the Cerami condition (the
C-condition for short), if the following is true:

Every sequence{xn}_{n}_{≥}_{1}⊆ X s.t. {*ϕ*(xn)}_{n}_{≥}_{1}⊆** _{R}**is bounded and
(1+kx

_{n}k)

*ϕ*

^{0}(x

_{n})→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 of*ϕ. Prominent*
in that theory is the so-called “mountain pass theorem” (see [3]).

**Theorem 2.1.** If *ϕ*∈C^{1}(X)satisfies the C-condition, x_{0},x_{1} ∈X withkx_{1}−x_{0}k> *ρ*>0
max{*ϕ*(x_{0})_{,}*ϕ*(x_{1})}<_{inf}{*ϕ*(x)_{:}kx−x_{0}k=*ρ*}=m_{ρ}

and c = inf*γ*∈_{Γ}max_{0}≤t≤1*ϕ*(*γ*(t)), where Γ = {*γ* ∈ C([0, 1],X) : *γ*(0) = x0, *γ*(1) = x_{1}}, then
c≥m*ρ* and c is a critical value of *ϕ.*

The analysis of problem (P* _{λ}*), in addition to the Sobolev spaceW

^{1,p}(

_{Ω}), will also involve the Banach spaceC

^{1}(

_{Ω}

^{¯}). This is an ordered Banach space with positive cone

C+ ={u∈C^{1}(_{Ω}^{¯}):u(z)≥0 for allz∈ _{Ω}^{¯}}.
This cone has a nonempty interior given by

intC+={u∈C^{1}(_{Ω}^{¯}):u(z)>0 for allz∈ _{Ω}^{¯}}.
Now, let us introduce the hypotheses on the mapa(·).

Let*ξ* ∈C^{1}(0,∞)with *ξ*(t)>0 for all t>0 and assume that
0<cˆ≤ ^{tξ}

0(t)

*ξ*(t) ≤c0 and c_{1}t^{p}^{−}^{1}≤*ξ*(t)≤c2(1+t^{p}^{−}^{1}), (2.1)
for all t>0, withc_{1} >0.

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

H(a): a(y) =a_{0}(|y|)y for all y∈_{R}^{N} with a_{0}(t)>0for all t >0and

(i) a_{0} ∈C^{1}(0,∞), t 7→ta_{0}(t)is strictly increasing on(0,∞), ta_{0}(t)→0^{+}as t→0^{+}and

tlim→0^{+}

ta^{0}_{0}(t)

a0(t) = c>−_{1;}

(ii) k∇a(y)k ≤c3*ξ*(|y|)

|y| for all y∈_{R}^{N}\ {0}and some c3>0;

(iii) (∇a(y)h,h)** _{R}**N ≥

^{ξ}^{(|}

_{|}

^{y}

^{|)}

y| |h|^{2}for all y∈_{R}^{N}\ {_{0}}, all h∈_{R}^{N};
(iv) if G_{0}(t) =Rt

0 sa_{0}(s)ds for all t ≥ 0, then pG_{0}(t)−t^{2}a_{0}(t)≥ −*γ*for all t ≥0and some
*γ*>0;

(v) there exists*τ*∈(1,p)such that t 7→G_{0}(t^{1/τ})is convex,lim_{t}_{→}_{0}^{+} ^{G}^{0}_{t}*τ*^{(}^{t}^{)} =0and
t^{2}a_{0}(t)−*τG*_{0}(t)≥ct˜ ^{p}

for all t>_{0}_{an some}c˜>_{0.}

**Remark 2.2.** EvidentlyG_{0} is strictly convex and strictly increasing. We setG(y) =G_{0}(|y|)_{for}
ally∈_{R}^{N}. ThenGis convex and it is differentiable at everyy∈_{R}^{N}\ {0}. Also

∇G(y) =G_{0}^{0}(|y|) ^{y}

|y| =a_{0}(|y|)y =a(y) for ally∈_{R}^{N}\ {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)** _{R}**N =a

_{0}(|y|)|y|

^{2}(2.2) for all y∈

_{R}^{N}

_{.}

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)| ≤c_{4}(1+|y|^{p}^{−}^{1})for all y∈_{R}^{N} and some c_{4}>0;

(c) (a(y),y)** _{R}**N ≥

_{p}

^{c}

_{−}

^{1}

_{1}|y|

^{p}for all y∈

_{R}^{N}.

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
c_{1}

p(p−1)|y|^{p} ≤G(y)≤ c_{5}(1+|y|^{p})
for all y∈_{R}^{N} with c_{5} >0.

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

(a) a(y) =|y|^{p}^{−}^{2}ywith 1< p<∞. This map corresponds to the p-Laplacian

∆pu=div(|∇u|^{p}^{−}^{2}∇u) for all u∈W^{1,p}(_{Ω}).

(b) a(y) =|y|^{p}^{−}^{2}y+*µ*|y|^{q}^{−}^{2}ywith 1 <q< pand*µ*>0. This map corresponds to a sum of
a p-Laplacian and aq-Laplacian, that is:

∆pu+*µ*∆qu for allu∈W^{1,p}(_{Ω}).

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

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

divh

(1+|∇u|^{2})^{p}^{−}^{2}^{2}∇ui

for allu∈W^{1,p}(_{Ω}).
(d) a(y) =|y|^{p}^{−}^{2}y+ ^{|}^{y}^{|}^{p}^{−}^{2}^{y}

1+|y|^{p} with 1< p< _{∞.}

We introduce the following nonlinear map A: W^{1,p}(_{Ω})→W^{1,p}(_{Ω})^{∗} defined by
hA(u),yi=

Z

Ω(a(∇u),∇y)** _{R}**Ndz (2.3)
for allu,y∈W

^{1,p}(

_{Ω})

_{.}

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:W^{1,p}(_{Ω})→ W^{1,p}(_{Ω})^{∗} 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

u_{n} *u in W^{1,p}(_{Ω}) and lim sup

n→_{∞}

hA(u_{n}),u_{n}−ui ≤0⇒u_{n}→u in W^{1,p}(_{Ω}).

Consider a Carathéodory function f_{0}: Ω×** _{R}**→

**satisfying**

_{R}|f_{0}(z,x)| ≤*α*(z)(1+|x|^{r}^{−}^{1}) for a.a.z∈_{Ω}, all x∈** _{R}**,
with

*α*∈ L

^{∞}(

_{Ω})

_{+}

_{, 1}< r < p

^{∗}. We set F

_{0}(z,x) =Rx

0 f_{0}(z,s)ds and consider theC^{1}-functional
*ϕ*_{0}: W^{1,p}(_{Ω})→** _{R}**defined by

*ϕ*_{0}(u) =

Z

ΩG(∇u(z))dz−

Z

ΩF_{0}(z,u(z))dz for allu∈W^{1,p}(_{Ω}).
In the sequel byk · k_{1,p} we denote the norm of the Sobolev spaceW^{1,p}(_{Ω}), that is

kuk_{1,p}=^{}kuk^{p}_{p}+k∇uk^{p}_{p}^{}^{1/}^{p}.
The following result is due to Motreanu–Papageorgiou [21].

**Proposition 2.7.** If u_{0} ∈W^{1,p}(_{Ω})is a local C^{1}(_{Ω}^{¯})-minimizer of*ϕ*_{0}, that is there exists*ρ*_{0} >0such
that

*ϕ*_{0}(u_{0})≤ *ϕ*_{0}(u_{0}+h) for all h∈ C^{1}(_{Ω}^{¯})withkhk_{C}1(_{Ω}^{¯}) ≤*ρ*_{0},

then u0 ∈ C^{1,β}(_{Ω}^{¯}) for some *β* ∈ (0, 1)and it is also a local W^{1,p}(_{Ω})-minimizer of *ϕ*0, that is there
exists*ρ*_{1}>0such that

*ϕ*_{0}(u_{0})≤ *ϕ*_{0}(u_{0}+h) for all h∈W^{1,p}(_{Ω}^{¯})withkhk_{1,p} ≤*ρ*_{1}.

**Remark 2.8.** The first such result relating local minimizers, was proved by Brezis–Nirenberg
[4], for the spacesC_{0}^{1}(_{Ω}^{¯}) = {u ∈C^{1}(_{Ω}^{¯}):u_{|}* _{∂Ω}* =0}and H

^{1}

_{0}(

_{Ω})and with G(y) =

^{1}

_{2}|y|

^{2}for all y∈

_{R}^{N}(it corresponds to the Dirichlet Laplacian).

Now letX be a Banach space andY_{2} ⊆ Y_{1} ⊆ X. For every integerk ≥0 by H_{k}(Y_{1},Y_{2})we
denote the kth-singular homology group with integer coefficients for the pair(Y_{1},Y2). Recall
that H_{k}(Y_{1},Y_{2}) =0 for all k<0.

Given *ϕ*∈C^{1}(X)_{and}c∈ **R, we introduce the following sets:**

*ϕ*^{c} ={x∈ X: *ϕ*(x)≤c}, K*ϕ* ={x ∈X: *ϕ*^{0}(x) =0}, K^{c}* _{ϕ}* ={x∈ K

*ϕ*:

*ϕ*(x) =c}. The critical groups of

*ϕ*at an isolated critical pointx

_{0}∈Xwith

*ϕ*(x

_{0}) =c(that is,x

_{0}∈K

^{c}

*) are defined by*

_{ϕ}C_{k}(*ϕ,*x_{0}) =H_{k}(*ϕ*^{c}∩ U,*ϕ*^{c}∩ U \ {x_{0}}) for all k≥0,

where U is a neighborhood ofx_{0}∈ Xsuch thatK* _{ϕ}*∩

*ϕ*

^{c}∩ U = {x

_{0}}. 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,+

_{∞}) →

**is a function such that for every**

_{R}*λ*> 0, (z,x) 7→ f(z,x,

*λ*) is Carathéodory, f(z, 0,

*λ*) =0for a.a. z∈

_{Ω}and

(i) for every *ρ* > 0and *λ* > 0, there exists *α** _{ρ}*(

*λ*) ∈ L

^{∞}(

_{Ω})

_{+}such that

*λ*7→ k

*α*

*(*

_{ρ}*λ*)k

_{∞}is bounded on bounded sets and

|f(z,x,*λ*)| ≤*α**ρ*(*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 all*λ*>0

x→±lim_{∞}

F(z,x,*λ*)

|x|^{p} = +_{∞} uniformly for a.a.z∈ _{Ω}

and there exist r(*λ*)∈(p,p^{∗})with r(*λ*)→r_{0}∈ (p,p^{∗})as*λ*→0^{+}and*η*ˆ_{∞}(*λ*),*η*_{∞}(*λ*)∈
L^{∞}(_{Ω})_{such that}

ˆ

*η*_{∞}(*λ*)(z)≤lim inf

x→±_{∞}

f(z,x,*λ*)

|x|^{r}^{(}^{λ}^{)−}^{2}x ≤lim sup

x→±_{∞}

f(z,x,*λ*)

|x|^{r}^{(}^{λ}^{)−}^{2}x ≤ *η*_{∞}(*λ*)(z)

uniformly for a.a. z∈ _{Ω}and*λ*7→ k*η*ˆ_{∞}(*λ*)k_{∞},k*η*_{∞}(*λ*)k_{∞} are bounded on bounded sets in
(_{0,}+_{∞});

(iii) for every*λ*>0, there exists*θ*(*λ*)∈ max

(r(*λ*)−p)^{N}_{p}, 1 ,p^{∗}

, and*β*_{0}(*λ*)such that
0< *β*_{0}(*λ*)≤lim inf

x→±_{∞}

f(z,x,*λ*)x−pF(z,x,*λ*)

|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*)and*δ*_{0}(*λ*)∈(0, 1),cˆ_{0}(*λ*)>0such that q(*λ*)→q_{0}∈ (1,p)as*λ*→0^{+}and

cˆ0(*λ*)|x|^{µ}^{(}^{λ}^{)}≤ f(_{z,}_{x,}*λ*)_{x}≤q(*λ*)_{F}(_{z,}_{x,}*λ*) _{for a.a.}_{z} ∈_{Ω,} all |x| ≤*δ*_{0}(*λ*)_{,}
there exist*β*(*λ*), *β*_{1},*β*_{2}>0, with*β*(*λ*)→0^{+}as*λ*→0^{+}such that

f(z,x,*λ*)x≤ *β*(*λ*)|x|^{q}^{(}^{λ}^{)}+*β*_{1}|x|^{r}^{∗}−*β*_{2}|x|^{p} for a.a.z∈_{Ω,} all x∈** _{R,}**
with r(

*λ*)≤r

^{∗}< p

^{∗}(see (ii)) and there exits a function

*η*

_{0}(·,

*λ*)∈ L

^{∞}(

_{Ω})

_{+}such that

k*η*_{0}(·,*λ*)k_{∞} →0 as*λ*→0^{+},
lim sup

x→0

F(z,x,*λ*)

|x|^{q}^{(}^{λ}^{)} ≤ *η*0(z,*λ*) uniformly for a.a.z∈_{Ω.}

**Remark 2.9.** Hypotheses H(f) (ii), (iii) imply that for a.a. z ∈ _{Ω} 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:

f_{1}(x,*λ*) =*λ*|x|^{q}^{−}^{2}x+|x|^{r}^{−}^{2}x− |x|^{p}^{−}^{2}x with 1<q<*τ*< p<r < p^{∗},

f_{2}(x,*λ*) =

|x|^{r}^{−}^{2}x− |x|^{p}^{−}^{2}x−*σ*(*λ*) ifx< −*ρ*(*λ*)

*λ*|x|^{q}^{−}^{2}x− |x|^{p}^{−}^{2}x if −*ρ*(*λ*)≤ x≤*ρ*(*λ*)

|x|^{r}^{−}^{2}x− |x|^{p}^{−}^{2}x+*σ*(*λ*) if*ρ*(*λ*)< x

with 1 < q < *τ* < p < r < p^{∗}, *ρ*(*λ*) ∈ (0, 1], *ρ*(*λ*) → 0^{+} as *λ* → 0^{+}, *σ*(*λ*) =
(*λ*−*ρ*(*λ*)^{r}^{−}^{q})*ρ*(*λ*)^{q}^{−}^{1}_{;}

f_{3}(x,*λ*) =*λ*|x|^{q}^{−}^{2}x+|x|^{p}^{−}^{2}x

ln|x| − ^{p}−1
p

with 1<q< *τ*< p.

Note that f_{4}(·_{,}*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*) +*β*_{2}x^{p}^{−}^{1} if 0<x (2.4)
and

fˆ−(z,x,*λ*) =

(f(z,x,*λ*) +*β*_{2}|x|^{p}^{−}^{2}x if x<0

0 if 0≤ x. (2.5)

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

Z _{x}

0

fˆ±(z,s,*λ*)ds
and introduce theC^{1}-functionals ˆ*ϕ*^{λ}_{±}: W^{1,p}(_{Ω})→** _{R}**defined by

ˆ

*ϕ*^{λ}_{±}(u) =

Z

ΩG(∇u(z))dz+ ^{β}^{2}

p kuk^{p}_{p}−

Z

Ω

Fˆ±(z,u(z),*λ*)dz for allu∈W^{1,p}(_{Ω}).
Also, by *ϕ** _{λ}* :W

^{1,p}(

_{Ω})→

**we denote the energy functional for problem (P**

_{R}*) defined by*

_{λ}*ϕ** _{λ}*(u) =

Z

ΩG(∇u(z))dz−

Z

ΩF(z,u(z),*λ*)dz for allu∈W^{1,p}(_{Ω}).
Clearly *ϕ** _{λ}* ∈C

^{1}(W

^{1,p}(

_{Ω})).

We conclude this section by fixing our notation. For x ∈ ** _{R}** let x

^{±}= max{±x, 0}. Then, givenu∈W

^{1,p}(

_{Ω}), we setu

^{±}(·) =u(·)

^{±}. We have

u=u^{+}−u^{−}, |u|=u^{+}+u^{−} and u^{+},u^{−},|u| ∈W^{1,p}(_{Ω}).

Giveh(z,x)a jointly measurable function (for example, a Carathéodory function), we define
N_{h}(u)(·) =h(·,u(·)) for all u∈W^{1,p}(_{Ω}).

Finally by| · |_{N} we denote the Lebesgue measure on**R**^{N}.

**3** **Solutions of constant sign**

In this section we show that for*λ*>0 small, problem (P* _{λ}*) has at least four nontrivial solutions
of constat sign (two positive and two negative).

First we establish the compactness properties of the functionals ˆ*ϕ*^{λ}_{±} and*ϕ** _{λ}*.

**Proposition 3.1.** If hypotheses H(a)and H(f)hold and*λ* > 0, then the functionals *ϕ*ˆ^{λ}_{±} satisfy the
C-condition.

Proof. We do the proof for the functional ˆ*ϕ*^{λ}_{+}, the proof for ˆ*ϕ*^{λ}_{−}being similar. So, we consider
a sequence{un}inW^{1,p}(_{Ω})such that

|*ϕ*ˆ^{λ}_{+}(un)| ≤ M_{1} (3.1)

for some M_{1} >0, alln≥1,

(_{1}+ku_{n}k_{1,p})(*ϕ*ˆ^{λ}_{+})^{0}(u_{n})→_{0} _{in}W^{1,p}(_{Ω})^{∗} _{as}n→_{∞.} _{(3.2)}

From (3.2) we have

|h(*ϕ*ˆ^{λ}_{+})^{0}(un),hi| ≤ ^{ε}^{n}khk_{1,p}
1+kunk_{1,p}
for allh∈W^{1,p}(_{Ω})with*ε*_{n} →0^{+}. Hence

hA(u_{n})_{,}hi+*β*_{2}
Z

Ω|u_{n}|^{p}^{−}^{2}u_{n}h dz−

Z

Ω

fˆ+(z,u_{n},*λ*)h dz

≤ ^{ε}^{n}khk_{1,p}

1+kunk_{1,p} ^{(3.3)}
for alln≥1. In (3.3) we chooseh=−u^{−}_{n} ∈W^{1,p}(_{Ω}). Then, in view of (2.4),

Z

Ω a(−∇u^{−}_{n}),−∇u^{−}_{n})^{}_{R}_{N} dz+*β*_{2}ku^{−}_{n}k^{p}_{p}≤*ε*_{n} for alln≥1,
so that, because of Lemma2.3,

c_{1}

p−1k∇u^{−}_{n}k^{p}_{p}+*β*_{2}ku^{−}_{n}k^{p}_{p} ≤*ε*_{n},
that is

u^{−}_{n} →0 inW^{1,p}(_{Ω}). (3.4)

Next, in (3.3) we chooseh=u^{+}_{n} ∈W^{1,p}(_{Ω}). Then

−

Z

Ω a(∇u^{+}_{n}),∇u^{+}_{n})^{}_{R}_{N} dz+

Z

Ω f(z,u^{+}_{n},*λ*)u^{+}_{n} dz≤*ε*n for alln≥1. (3.5)
Also, from (3.1), (3.4) and Corollary2.4, we have

Z

ΩpG(∇u^{+}_{n})dz−

Z

ΩpF(z,u^{+}_{n},*λ*)dz≤ M_{2} for some M_{2} >0, alln≥1. (3.6)
We add (3.5) and (3.6) and obtain

Z

Ω

pG(∇u^{+}_{n})− a(∇u^{+}_{n}),∇u^{+}_{n})^{}_{R}_{N}^{}dz+

Z

Ω

f(z,u^{+}_{n},*λ*)−pF(z,un,*λ*)^{}dz≤ M_{3}, (3.7)
for someM_{3} >0, alln≥1. Hence, H(a)(iv) assures that, for alln≥1

Z

Ω

f(z,u^{+}_{n},*λ*)−pF(z,u^{+}_{n},*λ*)^{}dz≤ M^{ˆ}_{3}. (3.8)
HypothesesH(f)(i), (iii) imply that we can findb_{1}(*λ*)∈(0,*β*_{0}(*λ*))andc_{6}(*λ*)>0 such that

b_{1}(*λ*)|x|^{θ}^{(}^{λ}^{)}−c_{6}(*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}} ⊆L^{θ}^{(}^{λ}^{)}(_{Ω})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

1

r(*λ*) = ^{1}−t
*θ*(*λ*)+ ^{t}

p^{∗}. (3.11)

The interpolation inequality (see, for example, Gasinski–Papageorgiou [10, p. 905]) implies
ku^{+}_{n}k_{r}_{(}_{λ}_{)} ≤ ku^{+}_{n}k^{1}^{−}^{t}

*θ*(*λ*)ku^{+}_{n}k^{t}_{p}∗.

Thus, from (3.10) and the Sobolev embedding theorem
ku^{+}_{n}k^{r}_{r}^{(}_{(}^{λ}^{)}

*λ*)≤ M_{4}ku^{+}_{n}k^{tr}_{1,p}^{(}^{λ}^{)} for someM_{4}>0, all n≥1. (3.12)
Hypotheses H(f)(i), (ii) imply that we can findc_{7}(*λ*)>0 such that

f(z,x,*λ*)≤c_{7}(*λ*)(1+|x|^{r}^{(}^{λ}^{)−}^{1}) for a.a. z∈_{Ω, all} x∈** _{R.}** (3.13)
In (3.3) we chooseh= u

^{+}

_{n}∈W

^{1,p}(

_{Ω}). Then

Z

Ω a(∇u^{+}_{n}),∇u^{+}_{n}

**R**^{N} dz−

Z

Ω f(z,u^{+}_{n},*λ*)u^{+}_{n} dz≤*ε*_{n} for alln ≥1.

Hence, from Lemma2.3, (3.13) and (3.12), there existc_{8}(*λ*),c_{9}(*λ*)>0 such that for alln≥1
c_{1}

p−1k∇u^{+}_{n}k^{p}_{p}≤ c_{8}(*λ*)(1+ku^{+}_{n}k^{r}^{(}^{λ}^{)}

r(*λ*))

≤ c_{9}(*λ*)(1+ku^{+}_{n}k^{tr}_{1,p}^{(}^{λ}^{)}). (3.14)
Recall that u7→ kuk_{θ}_{(}_{λ}_{)}+k∇uk_{p} is an equivalent norm on the spaceW^{1,p}(_{Ω})(see, for exam-
ple, [10, p. 227]). Then, (3.10) and (3.14) imply

ku^{+}_{n}k_{1,p}^{p} ≤c_{10}(*λ*)^{}1+ku^{+}_{n}k^{tr}_{1,p}^{(}^{λ}^{)}^{} for somec_{10}(*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}} ⊆W^{1,p}(_{Ω})is bounded. (3.16)
If N = p, then by the Sobolev embedding theorem we know thatW^{1,p}(_{Ω}),→ L^{s}(_{Ω})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

{u_{n}} ⊆W^{1,p}(_{Ω})is bounded.

At this point, we may assume that there existsu∈W^{1,p}(_{Ω})such that

u_{n}*u inW^{1,p}(_{Ω}) and u_{n}→u in L^{r}^{(}^{λ}^{)}(_{Ω}). (3.17)
We return to (3.3), chooseh=un−u and pass to the limit asn→_{∞}and use (3.17). Then

nlim→_{∞}hA(u_{n})_{,}u_{n}−ui=_{0,}

and Proposition 2.6 implies that u_{n} → u in W^{1,p}(_{Ω}). This proves that the functional ˆ*ϕ*^{λ}_{+}
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 *ϕ** _{λ}* satisfies the
C-condition.

Next we show that for all small values of the parameter*λ* >0, the functionals ˆ*ϕ*^{λ}_{±} 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 find*ρ*^{λ}_{±}>0for which we have

infh ˆ

*ϕ*^{λ}_{±}(u):kuk_{1,p}=*ρ*^{λ}_{±}
i

= mˆ^{λ}_{±}>0.

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

*λ*)

q(*λ*)|x|^{q}^{(}^{λ}^{)}+ ^{β}^{1}

r^{∗}|x|^{r}^{∗}− ^{β}^{2}

p |x|^{p} for a.a. z∈_{Ω}, all x∈** _{R}**. (3.18)
For allu∈W

^{1,p}(

_{Ω}), because of Corollary2.4 and (2.4) we have

ˆ

*ϕ*^{λ}_{+}(u) =

Z

ΩG(∇u)dz+ ^{β}^{2}

p kuk^{p}_{p}−

Z

Ω

Fˆ+(z,u,*λ*)dz

≥ ^{c}^{1}

p(p−1)k∇uk^{p}_{p}+ ^{β}^{2}

p kuk^{p}_{p}− ^{β}^{2}

p ku^{+}k^{p}_{p}−

Z

ΩF(z,u^{+},*λ*)dz. (3.19)
If in (3.19) we use (3.18), we obtain

ˆ

*ϕ*^{λ}_{+}(u)≥ ^{c}^{1}

p(p−1)k∇uk^{p}_{p}+ ^{β}^{2}

p kuk^{p}_{p}− * ^{β}*(

*λ*)

q(*λ*)ku^{+}k^{q}_{q}^{(}_{(}^{λ}^{)}

*λ*)− ^{β}^{1}
r^{∗}ku^{+}k^{r}_{r}^{∗}∗

≥^{h}c_{11}−(c_{12}(*λ*)kuk^{q}_{1,p}^{(}^{λ}^{)−}^{p}+c_{13}kuk^{r}_{1,p}^{∗}^{−}^{p})^{i}kuk_{1,p}^{p} , (3.20)
with c_{11}, c_{13} > 0 independent of *λ* and c_{12}(*λ*) → 0 as *λ* → 0^{+}. We introduce the function
*γ** _{λ}*: (0,∞)→(0,∞)defined by

*γ** _{λ}*(t) =c

_{12}(

*λ*)t

^{q}

^{(}

^{λ}^{)−}

^{p}+c

_{13}t

^{r}

^{∗}

^{−}

^{p}for allt>0.

Recall that 1<q(*λ*)< p <r(*λ*)≤r^{∗} < p^{∗}. Hence

*γ** _{λ}*(t)→+

_{∞}

_{as}t→0

^{+}and ast→+

_{∞.}

Therefore we can findt_{0}= t_{0}(*λ*)∈(0,∞)such that

*γ** _{λ}*(t

_{0}) =

_{min}[

*γ*

*(t)*

_{λ}_{:}t>

_{0}]. In particular,

*γ*_{λ}^{0}(t_{0}) = (q(*λ*)−p)c_{12}(*λ*)t^{q}_{0}^{(}^{λ}^{)−}^{p}^{−}^{1}+ (r^{∗}−p)c_{13}t^{r}_{0}^{∗}^{−}^{p}^{−}^{1}=0,
hence

t_{0}= t_{0}(*λ*) =

(p−q(*λ*))c_{12}(*λ*)
(r^{∗}−p)c_{13}

_{r}_{∗ −}^{1}_{q}_{(}

*λ*)

and a simple calculation leads to

*γ** _{λ}*(t

_{0}) = [c

_{12}(

*λ*)]

^{q}

(*λ*)−p

r∗ −_{q}(*λ*)c_{14}(*λ*)_{,}

with *λ*7→ c_{14}(*λ*)bounded on bounded intervals. Note that using the hypotheses onq(·)and
r^{∗}, we have

*γ** _{λ}*(t

_{0})→0

^{+}as

*λ*→0

^{.}So, choosing

*λ*

^{∗}

_{+}>0 small, we have

*γ** _{λ}*(t

_{0})<c

_{11}for all

*λ*∈(0,

*λ*

^{∗}

_{+}). Then, from (3.20) it follows that for all

*λ*∈(0,

*λ*

^{∗}

_{+})we have

ˆ

*ϕ*^{λ}_{+}(u)≥mˆ^{λ}_{+} >0 for allu∈W^{1,p}(_{Ω}) withkuk_{1,p} =t_{0}(*λ*) =*ρ*^{λ}_{+}.
In a similar fashion we show the existence of*λ*^{∗}_{−}>0 such that

ˆ

*ϕ*^{λ}_{−}(u)≥mˆ^{λ}_{−} >0 for allu∈W^{1,p}(_{Ω}) withkuk_{1,p} =t_{0}(*λ*) =*ρ*^{λ}_{−},
and the proof is complete.

The next proposition completes the mountain pass geometry for the functionals ˆ*ϕ*^{λ}_{±}. 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*>_{0}and u∈intC+, then*ϕ*ˆ^{λ}_{±}(tu)→ −_{∞}as
t→ ±_{∞.}

Now we can use variational methods to produce constant sign solutions for problem (P* _{λ}*)
when

*λ*>0 is small.

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

(a) for every*λ*∈(0,*λ*^{∗}_{+})problem (P* _{λ}*) has at least two positive solutions
u

_{0}, ˆu ∈intC+

withu being a local minimizer ofˆ *ϕ** _{λ}* and

*ϕ*

*(uˆ)<0<*

_{λ}*ϕ*

*(u*

_{λ}_{0}); (b) for every

*λ*∈(0,

*λ*

^{∗}

_{−})problem (P

*λ*) has at least two negative solutions

v0, ˆv∈ −intC+

withv being a local minimizer ofˆ *ϕ** _{λ}* and

*ϕ*

*(vˆ)<0<*

_{λ}*ϕ*

*(v*

_{λ}_{0});

(c) if*λ*^{∗} =min{*λ*^{∗}_{+},*λ*^{∗}_{−}}and*λ* ∈(0,*λ*^{∗}), then problem (P* _{λ}*) has at least four nontrivial solutions
of constant sign

u_{0}, ˆu∈intC+, v_{0}, ˆv∈ −intC+

withu, ˆˆ v local minimizers of *ϕ** _{λ}*and

*ϕ*

*(vˆ),*

_{λ}*ϕ*

*(uˆ)<0<*

_{λ}*ϕ*

*(u*

_{λ}_{0}),

*ϕ*

*(v*

_{λ}_{0}).

Proof. (a) For *λ* ∈ (0,*λ*^{∗}_{+}), let *ρ*^{λ}_{+} be as postulated by Proposition 3.3 and consider ¯B_{ρ}*λ*
+ =
{u ∈ W^{1,p}(_{Ω}) : kuk_{1,p} ≤ *ρ*^{λ}_{+}}, which clearly is weakly compact inW^{1,p}(_{Ω}). Moreover, since

ˆ

*ϕ*^{λ}_{+}is sequentially weakly lower semicontinuous inW^{1,p}(_{Ω}), one has that there exists ˆu∈ B^{¯}_{ρ}*λ*
+

such that

ˆ

*ϕ*^{λ}_{+}(uˆ) =infh
ˆ

*ϕ*^{λ}_{+}(u):kuk_{1,p}≤ *ρ*^{λ}_{+}

i≤mˆ^{λ}_{+}.

On the other hand, for *δ*_{0} ∈ (0, 1) as in hypothesis H(f)(iv) and *ξ* ∈ (0,*δ*_{0}(*λ*)) small (take

|*ξ*|<*ρ*^{λ}_{+}/|_{Ω}|^{1/p}_{N} ), we obtain
ˆ

*ϕ*^{λ}_{+}(*ξ*) =−

Z

ΩF(z,*ξ*,*λ*)dz<0.

Therefore, because of Proposition3.3, we can deduce that
ub∈ B_{ρ}*λ*

+ ={u∈W^{1,p}(_{Ω})_{:}kuk_{1,p}<*ρ*^{λ}_{+}}_{,}
and

(*ϕ*ˆ^{λ}_{+})^{0}(uˆ) =0.

So, it follows that

A(uˆ) +*β*_{2}|uˆ|^{p}^{−}^{2}uˆ = N_{f}_{ˆ}*λ*

+(uˆ), (3.21)

where ˆf_{+}* ^{λ}*(z,x) = f

^{ˆ}+(z,x,

*λ*). On (3.21) we act with −uˆ

^{−}∈ W

^{1,p}(

_{Ω}) 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, *∂*uˆ

*∂n* =0 on*∂Ω,*

(see [11]). From [26], we know that ˆu ∈ L^{∞}(_{Ω}). 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
every*λ*>0 and*ρ*>0, we can find*ξ** _{ρ}*(

*λ*)>0 such that

f(z,x,*λ*)x+*ξ**ρ*(*λ*)|x|^{p} ≥0 for a.a. z∈_{Ω, all}|x| ≤*ρ.*

Let*ρ*=kuˆk_{∞} and let *ξ**ρ*(*λ*)>0 as above. Then

−diva(∇uˆ(z)) +*ξ**ρ*(*λ*)uˆ(z)^{p} = f(z, ˆu(z),*λ*) +*ξ**ρ*(*λ*)uˆ(z)^{p}≥0 a.e. in Ω,
that is

diva(∇uˆ(z))≤*ξ**ρ*(*λ*)uˆ(z)^{p} a.e. in Ω. (3.22)
Let*χ*(t) =ta_{0}(t)for allt>0. Then, from H(a)(iii)

tχ^{0}(t) =t^{2}a^{0}_{0}(t) +ta_{0}(t)≥c_{1}t^{p}^{−}^{1},
hence, by integration one has

Z _{t}

0 sχ^{0}(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

ˆ

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

*ϕ*_{λ}_{|}_{C}

+ = *ϕ*ˆ^{λ}_{+}_{|}_{C}

+.

From this equality and (3.24) it follows that ˆu is a local C^{1}(_{Ω}^{¯})-minimizer of *ϕ** _{λ}*. Invoking
Proposition2.7, we have that ˆuis a localW

^{1,p}(

_{Ω})-minimizer of

*ϕ*

*.*

_{λ}Now we look for the second positive solution. Propositions3.1,3.3and3.4permit the use
of Theorem2.1on the functional ˆ*ϕ*^{λ}_{+}. So, we can findu_{0}∈W^{1,p}(_{Ω})such that

ˆ

*ϕ*^{λ}_{+}(uˆ)<0= *ϕ*ˆ^{λ}_{+}(0)<mˆ^{λ}_{+} ≤ *ϕ*ˆ^{λ}_{+}(u_{0}) and (*ϕ*ˆ^{λ}_{+})^{0}(u_{0}) =0. (3.25)
From (3.25) it follows that u_{0} 6∈ {0, ˆu}, it solves problem (P* _{λ}*) and by the nonlinear regularity
theory we haveu

_{0}∈C+\ {0}(see [17,26]). In fact, as above, using the results of Pucci–Serrin [25, pp. 111, 120], we conclude thatu

_{0}∈intC+.

(b) Working in a similar fashion, this time with the function ˆ*ϕ*^{λ}_{−}, for *λ* ∈ (0,*λ*^{∗}_{−})we produce
two negative solutions for problem (P* _{λ}*)

v_{0}, ˆv∈ −intC+.

Moreover, ˆvis a local minimizer of *ϕ** _{λ}* and

*ϕ*

*(vˆ)<*

_{λ}_{0}<

*ϕ*

*(v*

_{λ}_{0})

_{.}(c) Follows from parts (a) and (b).

**4** **Nodal solutions**

In this section, we produce a fifth nontrivial solution of (P* _{λ}*), with

*λ*∈(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 solutionu

^{∗}

*and the biggest nontrivial negative solution v*

_{λ}^{∗}

*of (P*

_{λ}*). Then look for a nontrivial solution in the order interval[v*

_{λ}^{∗}

*,u*

_{λ}^{∗}

*] = {u∈ W*

_{λ}^{1,p}(

_{Ω}): v

^{∗}

*(z) ≤ u(z) ≤ u*

_{λ}^{∗}

*(z) a.e. in Ω} distinct from v*

_{λ}^{∗}

*and u*

_{λ}^{∗}

*. Necessarily, this solution will be nodal.*

_{λ}HypothesesH(f)(i), (ii), (iv) imply that we can findc_{15} >0 such that

f(z,x,*λ*)x ≥cˆ_{0}(*λ*)|x|^{µ}^{(}^{λ}^{)}−c_{15}|x|^{r}^{(}^{λ}^{)} for a.a. z∈ _{Ω, all}x∈_{R, all}*λ*∈(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)) =cˆ_{0}(*λ*)|u(z)|^{µ}^{(}^{λ}^{)−}^{2}u(z)−c_{15}|u(z)|^{r}^{(}^{λ}^{)−}^{2}u(z) inΩ,

*∂u*

*∂n* =0 on*∂Ω,*

1<*µ*(*λ*)< p<r(*λ*)< p^{∗}.

(S* _{λ}*)

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 and*λ*>0, then problem(S* _{λ}*)has a unique positive solution
u¯

*∈ intC+and since problem(S*

_{λ}*) is oddv¯*

_{λ}*= −u¯*

_{λ}*∈ −intC+ is the unique negative solution of (S*

_{λ}*).*

_{λ}Proof. First we establish the existence of a positive solution. To this end, let*ψ*^{+}* _{λ}* : W

^{1,p}(

_{Ω})→

**be the C**

_{R}^{1}-functional defined by

*ψ*^{+}* _{λ}*(u) =

Z

ΩG(∇u(z))dz+ ^{1}

pkuk^{p}− ^{c}^{ˆ}^{0}(*λ*)

*µ*(*λ*)ku^{+}k^{µ}^{(}^{λ}^{)}

*µ*(*λ*)+ ^{c}^{15}

r(*λ*)ku^{+}k^{r}_{r}^{(}_{(}^{λ}^{)}

*λ*)− ^{1}

pku^{+}k^{p}_{p}.

Recall that 1<*µ*(*λ*)< p<r(*λ*)(see (S* _{λ}*)). So, using Corollary2.4, we see that

*ψ*

^{+}

*is coercive.*

_{λ}Also, using the Sobolev embedding theorem, we can check that *ψ*_{λ}^{+} is sequentially weakly
lower semicontinuous. Hence, by the Weierstrass theorem, we can find ¯u* _{λ}* ∈ W

^{1,p}(

_{Ω}) such that

*ψ*_{λ}^{+}(u¯*λ*) =inf{*ψ*_{λ}^{+}(u):u∈W^{1,p}(_{Ω})}. (4.2)
Since*µ*(*λ*)<r(*λ*), for*ξ* ∈ (0, 1)small we have*ψ*^{+}* _{λ}*(

*ξ*)<0 and so, because of (4.2),

*ψ*^{+}* _{λ}*(u¯

*)<0=*

_{λ}*ψ*

_{λ}^{+}(0),

hence ¯u* _{λ}* 6=0. From (4.2), we have that ¯u

*is a critical point of*

_{λ}*ψ*

^{+}

*, namely*

_{λ}A(u¯*λ*) +|u¯*λ*|^{p}^{−}^{2}u¯*λ*= cˆ0(*λ*)(u¯^{+}* _{λ}*)

^{µ}^{(}

^{λ}^{)−}

^{1}−c15(u¯

^{+}

*)*

_{λ}^{r}

^{(}

^{λ}^{)−}

^{1}+ (u¯

^{+}

*)*

_{λ}^{p}

^{−}

^{1}

_{.}

_{(4.3)}On (4.3) we act with −u¯

^{−}

*∈ W*

_{λ}^{1,p}(

_{Ω}) and obtain ¯u

*≥ 0, ¯u*

_{λ}*6= 0. Hence ¯u*

_{λ}*is a positive solution of (S*

_{λ}*λ*). Nonlinear regularity theory implies ¯u

*λ*∈C+\ {0}. We have

−diva(∇u¯* _{λ}*(z)) =cˆ

_{0}(

*λ*)u¯

*(z)*

_{λ}

^{µ}^{(}

^{λ}^{)−}

^{1}−c

_{15}u¯

*(z)*

_{λ}^{r}

^{(}

^{λ}^{)−}

^{1}a.e. in Ω, thus

diva(∇u¯* _{λ}*(z))≤c

_{15}ku¯

*k*

_{λ}^{r}

_{∞}

^{(}

^{λ}^{)−}

^{p}u¯

*(z)*

_{λ}^{p}

^{−}

^{1}a.e. inΩ, and from [25, p. 111, 120] we conclude that

¯

u* _{λ}* ∈intC+.

So, we have established the existence of a positive solution ¯u* _{λ}* ∈intC+for problem (S

*).*

_{λ}Next we show the uniqueness of this positive solution. To this end, consider the integral
functional*σ*_{λ}^{+}: L^{1}(_{Ω})→** _{R}**∪ {+

_{∞}}defined by

*σ*_{λ}^{+}(_{u}) =
(R

ΩG(∇u^{1/τ})dz ifu≥0, u^{1/τ} ∈W^{1,p}(_{Ω})

+_{∞} otherwise.

Let u_{1},u_{2} ∈ dom*σ*_{λ}^{+} = {u ∈ L^{1}(_{Ω}) : *σ*_{λ}^{+}(u) < +_{∞}} (the effective domain of *σ*_{λ}^{+}) and let
t∈ [_{0, 1}]_{. We set}

y= ((1−t)u_{1}+tu_{2})^{1/τ}, v_{1} =u^{1/τ}_{1} , v_{2}=u^{1/τ}_{2} .
From [5, Lemma 1], we have

k∇y(z)k ≤[(1−t)k∇v_{1}(z)k* ^{τ}*+tk∇v2(z)k

*]*

^{τ}^{1/τ}a.e. in Ω, and exploiting the monotonicity ofG

_{0}and hypothesis H(a)(v)

G(∇y(z)) =G0(k∇y(z)k)≤ G0

((1−t)k∇v_{1}(z)k* ^{τ}*+tk∇v2(z)k

*)*

^{τ}^{1/τ}

^{}

≤(1−t)G_{0}(k∇v_{1}(z)k) +tG_{0}(k∇v_{2}(z)k)
for a.a.z∈_{Ω, that is}*σ*_{λ}^{+} is convex.

Also, by Fatou’s lemma *σ*_{λ}^{+} is lower semicontinuous. Now, letu ∈ W^{1,p}(_{Ω})be a positive
solution of problem (S* _{λ}*). From the first part of the proof, we haveu∈

_{int}C+. So, ifh ∈C

^{1}(

_{Ω}

^{¯}) andt ∈(−1, 1)with |t|small, we have

u* ^{τ}*+th∈

_{int}C+ ∩

_{dom}

*σ*

_{λ}^{+}.