ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

EXISTENCE OF SOLUTIONS TO NONHOMOGENEOUS DIRICHLET PROBLEMS WITH DEPENDENCE ON THE

GRADIENT

YUNRU BAI

Communicated by Vicentiu D. Radulescu

Abstract. The goal of this article is to explore the existence of positive so- lutions for a nonlinear elliptic equation driven by a nonhomogeneous partial differential operator with Dirichlet boundary condition. This equation a con- vection term and thereaction term is not required to satisfy global growth conditions. Our approach is based on the Leray-Schauder alternative princi- ple, truncation and comparison approaches, and nonlinear regularity theory.

1. Introduction

Given a bounded domain Ω ⊂ R^{N} with C^{2}-boundary ∂Ω, 1 < p < +∞, a
continuous function a : R^{N} → R^{N}, and a nonlinear function f : Ω × R × R^{N} → R,
we consider the following nonlinear nonhomogeneous Dirichlet problem involving a
convection term:

− div a(Du(z)) = f (z, u(z), Du(z)) in Ω,

u(z) = 0 on ∂Ω, (1.1)

with u(z) > 0 in Ω.

In this article, the function a : R^{N} → R^{N} is assumed to be continuous and
strictly monotone, also satisfies certain regularity and growth conditions listed in
hypotheses (H1) below. It is worth to mention that these conditions are mild
and incorporate in our framework many classical operators of interest, for example
the p-Laplacian, 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 forcing term depends also on the gradient of the unknown function
(convection term). For this reason we are not able to apply variational methods
directly on equation (1.1).

For problems with convection terms we mention the following works: Figueiredo- Girardi-Matzeu [8], Girardi-Matzeu [23] (semilinear equations driven by the Dirich- let Laplacian), Faraci-Motreanu-Puglisi [6], Huy-Quan-Khanh [25], Iturriaga-Lorca- Sanchez [26], Ruiz [36] (nonlinear equations driven by the Dirichlet p-Laplacian),

2010 Mathematics Subject Classification. 35J92, 35P30.

Key words and phrases. Nonhomogeneous p-Laplacian operator; nonlinear regularity;

Dirichlet boundary condition; convection term; truncation; Leray-Schauder alternative.

c

2018 Texas State University.

Submitted December 8, 2017. Published May 2, 2018.

1

Averna-Motreanu-Tornatore [2], Faria-Miyagaki-Motreanu [7], Tanaka [37] (equa- tions driven by the Dirichlet (p, q)-Laplacian) and finally Gasi´nski-Papageorgiou [22] (Neumann problems driven by a differential operator of the form div(a(u)Du)).

Unlike the aforementioned works, in this paper, the convection term f does
not have any global growth condition. Instead we suppose that f (z, ·, y) ad-
mits a positive root (zero) and all the other conditions refer to the behaviour
of the function x 7→ f (z, x, y) near zero locally in y ∈ R^{N}. Our approach is
based on the Leray-Schauder alternative principle, truncation and comparison tech-
niques, nonlinear regularity theory and it is closely related to the paper Bai-
Gasi´nski-Papageorgiou in [3], where the Robin boundary value problem was con-
sidered. Finally for other problems with a general nonhomogeneous operator we
refer to Gasi´nski-Papageorgiou [14, 15, 19, 21], Papageorgiou-R˘adulescu [30, 31, 33,
34] and for particular cases of a nonhomogeneous operator we refer to Gasi´nski-
Papageorgiou [13, 16] (p(z)-Laplacian) and Gasi´nski-Papageorgiou [17, 18], and for
(p, q)-Laplacian, Papageorgiou-R˘adulescu [32].

2. Notation and preliminaries

In the study of problem (1.1), we will use the Sobolev space W_{0}^{1,p}(Ω) as well
as the ordered Banach space C_{0}^{1}(Ω) = {u ∈ C^{1}( ¯Ω) : u(z) = 0 on ∂Ω} which has
positive (order) cone

C_{0}^{1}(Ω)+=u ∈ C_{0}^{1}(Ω) : u(z) ≥ 0 in Ω}.

The interior of this cone contains the set

D+=u ∈ C_{0}^{1}(Ω) : u(z) > 0 in Ω}.

Then, we give the following notation, which will be used in the sequel. For x ∈ R,
we denote x^{±} = max{±x, 0}. Likewise, for u ∈ W_{0}^{1,p}(Ω) fixed, we use the notation
u^{±}(·) = u(·)^{±}. We have that

u^{±}∈ W_{0}^{1,p}(Ω), u = u^{+}− u^{−}, |u| = u^{+}+ u^{−}.
For u ∈ W_{0}^{1,p}(Ω) such that u(z) ≥ 0 for a.a. z ∈ Ω, we define

[0, u] = {h ∈ W_{0}^{1,p}(Ω) : 0 6 h(z) 6 u(z) for a.a. z ∈ Ω}.

Now we present the conditions on the map a(y). Assume that ζ ∈ C^{1}(0, ∞) is
such that

0 <bc 6 ζ^{0}(t)t

ζ(t) 6 c^{0} and c1t^{p−1}6 ζ(t) 6 c^{2}(1 + |t|^{p−1}) ∀t > 0, (2.1)
for some c_{1}, c_{2}> 0.

The hypotheses on the map y 7→ a(y) are as follows:

(H1) a : R^{N} → R^{N} is such that a(y) = a0(|y|)y for all y ∈ R^{N} with a0(t) > 0 for
all t > 0 and

(i) a0∈ C^{1}(0, ∞), t 7→ a0(t)t is strictly increasing on (0, ∞) and
lim

t→0^{+}

a_{0}(t)t = 0 and lim

t→0^{+}

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

a0(t) = c > −1;

(ii) there exists c3> 0 such that

|∇a(y)| 6 c3

ζ(|y|)

|y| for all y ∈ R^{N} \ {0};

(iii) for all y ∈ R^{N} \ {0} and ξ ∈ R^{N},
(∇a(y)ξ, ξ)_{R}N >ζ(|y|)

|y| |ξ|^{2};
(iv) denoting G_{0}(t) =Rt

0a_{0}(s)s ds, we can find q ∈ (1, p) satisfying
t 7→ G0(t^{1/q}) is convex on R^{+}= [0, +∞),

lim

t→0^{+}

qG0(t)

t^{q} = c^{∗}> 0,
0 6 pG^{0}(t) − a0(t)t^{2}for all t > 0.

Remark 2.1. Conditions (H1)(i), (ii) and (iii) are required by the nonlinear reg- ularity theory of Lieberman [28] and the nonlinear strong maximum principle of Pucci-Serrin [35].

Example 2.2. The following maps satisfy hypotheses (H1) (see Papageorgiou- R˘adulescu [29]).

(a) a(y) = |y|^{p−2}y with 1 < p < ∞. The operator div(a(Du)) reduces to the
p-Laplace differential operator

∆pu = div(|Du|^{p−2}Du) for all u ∈ W_{0}^{1,p}(Ω).

(b) a(y) = |y|^{p−2}y + |y|^{q−2}y with 1 < q < p < ∞. The map div(a(Du))
corresponds to the (p, q)-Laplace differential operator

∆pu + ∆qu for all u ∈ W_{0}^{1,p}(Ω).

Such operators arise in problems of mathematical physics (see Cherfils- Il’yasov [4]).

(c) a(y) = (1 + |y|^{2})^{p−2}^{2} y with 1 < p < ∞. The operator div(a(Du)) corre-
sponds to the generalized p-mean curvature differential operator

div((1 + |Du|^{2})^{p−2}^{2} Du) for all u ∈ W_{0}^{1,p}(Ω).

(d)

a(y) =

(2|y|^{γ−2}y, if |y| < 1,

|y|^{p−2}y + |y|^{q−2}y if 1 < |y|,
where 1 < q < p, γ = ^{p+q}_{2} .

On the other hand, we use hypotheses (H1) to indicate that G0 is strictly in- creasing and strictly convex. Also, we denote

G(y) = G0(|y|) for all y ∈ R^{N}.
We have

∇G(y) = G^{0}_{0}(|y|)y

|y| = a0(|y|)y = a(y) for all y ∈ R^{N} \ {0}.

So, G is the primitive of a, it is convex with G(0) = 0. Hence, one has

G(y) = G(y) − G(0) 6 (a(y), y)R^{N} for all y ∈ R^{N}. (2.2)
Such hypotheses were also considered in recent the works of Gasi´nski-O’Regan-
Papageorgiou [10], Papageorgiou-R˘adulescu [29, 30, 31] and Bai-Gasi´nski-Papa-
georgiou [3].

Under hypotheses (H1)(i), (ii) and (iii), we have the following lemma, which summarizes some of important properties for the map a(·).

Lemma 2.3 ([38, Lemma 3]). Assume that the map a(·) satisfies hypotheses (H1) (i), (ii), (iii). Then the following statements hold

(a) y 7→ a(y) is continuous and strictly monotone (hence maximal monotone);

(b) |a(y)| 6 c^{4}(1 + |y|^{p−1}) for all y ∈ R^{N}, for some c4> 0;

(c) (a(y), y)_{R}N >p−1^{c}^{1} |y|^{p} for all y ∈ R^{N}, where c1 is given in (2.1).

We have the following bilateral growth restrictions on the primitive G is estab- lished.

Lemma 2.4. Assume that the map a(·) satisfies hypotheses (H1) (i), (ii), (iii).

Then, there exists c5> 0 such that c1

p(p − 1)|y|^{p}6 G(y) 6 c^{5}(1 + |y|^{p}) for all y ∈ R^{N}.

Let W^{−1,p}^{0}(Ω) be the dual space of the Sobolev space W_{0}^{1,p}(Ω). We denote the
duality brackets between W^{−1,p}^{0}(Ω) and W_{0}^{1,p}(Ω) by h·, ·i. Also, we introduce a
nonlinear operator A : W_{0}^{1,p}(Ω) → W^{−1,p}^{0}(Ω) corresponding to map a(·) defined
by

hA(u), hi = Z

Ω

(a(Du), Dh)_{R}Ndz for all u, h ∈ W_{0}^{1,p}(Ω).

Next proposition summarizes some properties of the operator A (see Gasi´nski- Papageorgiou [12] for a more general version).

Proposition 2.5. Assume that (H1)(i), (ii) and (iii) are fulfilled. Then, the map
A : W_{0}^{1,p}(Ω) → W^{−1,p}^{0}(Ω) is continuous, bounded (thus is, maps bounded sets in
W_{0}^{1,p}(Ω) to bounded sets in W^{−1,p}^{0}(Ω)), monotone (hence maximal monotone too)
and of type (S)+, i.e.,

if un

→ u in Ww _{0}^{1,p}(Ω) and lim sup_{n→+∞}hA(un), un− ui 6 0, then
u_{n}→ u in W_{0}^{1,p}(Ω).

For 1 < q < +∞, we consider the nonlinear eigenvalue problem

−∆_{q}u(z) = bλ|u(z)|^{q−2}u(z) in Ω
u = 0 on ∂Ω.

The number bλ such that the above Dirichlet problem admits a nontrivial solutionub is called an eigenvalue of −∆qwith Dirichlet boundary condition, also the nontrivial solutionbu is an eigenfunction corresponding to bλ. From Faraci-Motreanu-Puglisi [6], we can see that there exists a smallest eigenvalue bλ1(q) > 0 such that

• bλ1(q) is positive, isolated and simple (that is, if u,b bv are eigenfunctions corresponding to bλ1(q), thenbu = ξv for some ξ ∈ R \ {0}).b

• the following variational characterization holds

λb1(q) = infnR

Ω|Du|^{q}dx
R

Ω|u|^{q}dx : u ∈ W_{0}^{1,q}(Ω) with u 6= 0o
.

In what follows, we denote bybu1(q) the positive eigenfunction normalized as kbu1(q)k^{q}_{q} =
R

Ω|u|^{q}dx = 1, which is associated to bλ1(q). One hasbu1(q) ∈ D+. Additionally, we
know that if u is an eigenfunction corresponding to an eigenvalue bλ 6= bλ_{1}(q), then
u ∈ C_{0}^{1}(Ω) changes sign (see Lieberman [27, 28]).

Let f : Ω × R × R^{N} → R. The function f is called to be Carath´eodory, if

• for all (x, y) ∈ R × R^{N}, z 7→ f (z, x, y) is measurable on Ω;

• for a.a. z ∈ Ω, (x, y) 7→ f (z, x, y) is continuous.

Such a function is automatically jointly measurable (see Hu-Papageorgiou [24, p.

142]).

For the convection term f in problem (1.1), we assume that

(H2) f : Ω × R × R^{N} → R is a Carath´eodory function such that f(z, 0, y) = 0 for
a.a. z ∈ Ω, all y ∈ R^{N} and

(i) there exists η > 0 such that

f (z, η, y) = 0 for a.a. z ∈ Ω, all y ∈ R^{N},
f (z, x, y) > 0 for a.a. z ∈ Ω, all 0 6 x 6 η, all y ∈ R^{N},
f (z, x, y) 6ec1+ec2|y|^{p} for a.a. z ∈ Ω, all 0 6 x 6 η, all y ∈ R^{N},

withec1> 0,ec2<_{p−1}^{c}^{1} ;

(ii) for every M > 0, there exists ηM ∈ L^{∞}(Ω) such that
ηM(z) > c^{∗}bλ1(q) for a.a. z ∈ Ω, ηM 6≡ c^{∗}bλ1(q),
lim inf

x→0^{+}

f (z, x, y)

x^{q−1} > η^{M}(z) uniformly for a.a. z ∈ Ω, all |y| 6 M
(here q ∈ (1, p) and c^{∗} are as in hypothesis (H1)(iv));

(iii) there exists ξ_{η}> 0 such that for a.a. z ∈ Ω, all y ∈ R^{N} the function
x 7→ f (z, x, y) + ξηx^{p−1}

is nondecreasing on [0, η], for a.a. z ∈ Ω, all y ∈ R^{N} and
λ^{p−1}f (z, 1

λx, y) 6 f (z, x, y), (2.3)

f (z, x, y) 6 λ^{p}f (z, x,1
λy)

for a.a. z ∈ Ω, all 0 6 x 6 η, all y ∈ R^{N} and all λ ∈ (0, 1).

Remark 2.6. Because the goal of the present paper is to explore the existence of nonnegative solutions, so for x 6 0, without loss of generality, we may assume that

f (z, x, y) = 0 for a.a. z ∈ Ω, all y ∈ R^{N}.

Note that (2.3) is satisfied if for example, for a.a. z ∈ Ω, all y ∈ R^{N}, the function
x 7→ ^{f (z,x,y)}_{x}p−1 is nonincreasing on (0, +∞).

Example 2.7. The following function satisfies hypotheses (H2). For the sake of simplicity we drop the z-dependence:

f (x, y, z) =

((x^{r−1}− x^{s−1})|y|^{p} if 0 6 x 6 1,
(x^{τ}ln x)|y|^{p} if 1 < x,
with 1 < r < s < q < p and τ > 1.

Finally we recall the well known Leray-Schauder alternative principle (see e.g., Gasi´nski-Papageorgiou [11, p. 827]), which will play important role to establish our main results.

Theorem 2.8. Let X be a Banach space and C ⊆ X be nonempty and convex. If ϑ : C → C is a compact map, then exactly one of the following two statements is true:

(a) ϑ has a fixed point;

(b) the set S(ϑ) = {u ∈ C : u = λϑ(u), λ ∈ (0, 1)} is unbounded.

3. Positive solutions

In this section, we explore a positive solution to nonlinear nonhomogeneous
Dirichlet problem (1.1). To this end, for v ∈ C_{0}^{1}(Ω) fixed, we first consider the
following intermediate Dirichlet problem

− div a(Du(z)) = f (z, u(z), Dv(z)), in Ω,

u(z) = 0, on ∂Ω. (3.1)

Now, we apply truncation and perturbation approaches to prove that (3.1) has at least one positive solution. So, we turn our attention to consider the following truncation-perturbation Dirichlet problem

− div a(Du(z)) + ξ_{η}u(z)^{p−1} = bf (z, u(z), Dv(z)), in Ω,

u(z) = 0, on ∂Ω, (3.2)

where bf : Ω × R × R^{N} → R is the corresponding truncation-perturbation of con-
vection term f with respect to the second variable, defined by

f (z, x, y) =b

(f (z, x, y) + ξη(x^{+})^{p−1} if x 6 η,

f (z, η, y) + ξηη^{p−1} if η < x. (3.3)
Remark 3.1. Recall that f is a Carath´eodoty function (see hypotheses (H2)). It
is obvious that the truncation-perturbation bf is a Carath´eodoty function as well.

It is obvious that if a function u : Ω → R with u = 0 on ∂Ω and 0 ≤ u(z) ≤ η for a.a. z ∈ Ω is a solution of problem (3.2), then u is also a solution of problem (3.1).

Using this fact, we will now prove the existence of a positive solution for problem (3.1).

Proposition 3.2. Assume that (H1) and (H2) are satisfied. Then problem (3.1) has a positive solution uv such that

uv∈ [0, η] ∩ D+.

Proof. To prove the existence of a nontrivial solution, we introduce the C^{1}-functional
ϕbv: W_{0}^{1,p}(Ω) → R defined by

ϕbv(u) = Z

Ω

G(Du) dz +ξη

pkuk^{p}_{p}−
Z

Ω

Fbv(z, u) dz
for all u ∈ W_{0}^{1,p}(Ω), where bF_{v} is given by

Fbv(z, x) = Z x

0

f (z, s, Dv(z)) ds.b

Combining Lemma 2.4 and definition of bf (see (3.3)), we conclude that the func-
tional ϕbv is coercive. On the other hand, the Sobolev embedding theorem and
the convexity of G reveal that the functionalϕbv is sequentially weakly lower semi-
continuous. Therefore, it allows us to use the Weierstrass-Tonelli theorem to find
uv∈ W_{0}^{1,p}(Ω) such that

ϕbv(uv) = inf

u∈W_{0}^{1,p}(Ω)ϕbv(u). (3.4)
We take M := sup_{z∈Ω}|Dv(z)| and then use hypothesis (H2)(ii) to obtain that for
any ε > 0 fixed, there exists δ ∈ (0, η] satisfying

f (z, x, y) > (ηM(z) − ε)x^{q−1} for a.a. z ∈ Ω, all x ∈ [0, δ], all |y| 6 M ;
this results in

f (z, x, Dv(z)) > (ηb ^{M}(z) − ε)x^{q−1}+ ξηx^{p−1} for a.a. z ∈ Ω, all x ∈ [0, δ]

(see (3.3)). Also, we can calculate
Fb_{v}(z, x) > 1

q(η_{M}(z) − ε)x^{q}+ξη

px^{p} for a.a. z ∈ Ω, all x ∈ [0, δ]. (3.5)
Note that G(y) = G0(|y|) for all y ∈ R^{N} and lim_{t→0}+qG0(t)

t^{q} = c^{∗}> 0 (see (H1)(iv)),
so

G(y) 6 c^{∗}+ ε

q |y|^{q} for all |y| 6 δ. (3.6)
Asub1(q) ∈ D+, we can take t ∈ (0, 1) small enough such that

tub1(q)(z) ∈ [0, δ], t|Dub1(q)(z)| 6 δ for all z ∈ Ω. (3.7) Obviously, we can obtain

ϕbv(tbu1(q)) 6 c^{∗}+ ε

q t^{q}bλ1(q) −t^{q}
q

Z

Ω

(ηM(z) − ε)bu1(q)^{q}dz
6t^{q}

q

Z

Ω

(c^{∗}λb_{1}(q) − η_{M}(z))bu_{1}(q)^{q}dz + ε(bλ_{1}(q) + 1) (3.8)
(recall that kub_{1}(q)k_{q} = 1). From η_{M}(z) > c^{∗}bλ_{1}(q) for a.a. z ∈ Ω, η_{M} 6≡ c^{∗}bλ_{1}(q)
(see (H2)(ii)) andub1(q) ∈ D+, it yields

r0= Z

Ω

(ηM(z) − c^{∗}bλ1(q))ub1(q)^{q}dz > 0.

So, (3.8) becomes

ϕbv(tub1(q)) 6 t^{q}

q(−r0+ ε(bλ1(q) + 1)).

Now, we pick ε ∈ (0, ^{r}^{0}

bλ1(q)+1) to obtainϕbv(tub1(q)) < 0. This means that ϕbv(uv) < 0 =ϕbv(0),

hence uv 6= 0. Therefore, we have proved the existence of a nontrivial solution to problem (3.1).

Next, we show that u_{v}is nonnegative. Equality (3.4) indicatesϕb^{0}_{v}(u_{v}) = 0, hence
hA(uv), hi + ξη

Z

Ω

|uv|^{p−2}uvh dz

= Z

Ω

f (z, ub v, Dv)h dz for all h ∈ W_{0}^{1,p}(Ω).

(3.9)

Inserting h = −u^{−}_{v} ∈ W_{0}^{1,p}(Ω) into (3.9) to obtain

−hA(uv), u^{−}_{v}i − ξη

Z

Ω

|uv|^{p−2}uvu^{−}_{v} dz = −
Z

Ω

f (z, ub v, Dv)u^{−}_{v} dz,
thus (see (3.3) and (H2)),

hA(uv), u^{−}_{v}i + ξη||u^{−}_{v}||^{p}_{p}≤ 0.

Combining with Lemma 2.3 and (3.3), we calculate
c_{1}

p − 1kDu^{−}_{v}k^{p}+ ξ_{η}ku^{−}_{v}k^{p}_{p}6 0,
which gives uv> 0 and u^{v}6= 0.

Furthermore, we shall illustrate that uv ∈ [0, η]. Putting h = (uv − η)^{+} ∈
W_{0}^{1,p}(Ω) into (3.9), we obtain

hA(uv), (uv− η)^{+}i + ξη

Z

Ω

u^{p−1}_{v} (uv− η)^{+}dz,

= Z

Ω

f (z, η, Dv) + ξηη^{p−1}(uv− η)^{+}dz =
Z

Ω

ξηη^{p−1}(uv− η)^{+}dz
(see (3.3) and condition (H2)(i)). We use the fact that A(η) = 0, to obtain

hA(u_{v}) − A(η), (u_{v}− η)^{+}i + ξ_{η}
Z

Ω

(u^{p−1}_{v} − η^{p−1})(u_{v}− η)^{+}dz 6 0.

However, the monotonicity of A implies uv6 η. Until now, we have verified that

u_{v}∈ [0, η] \ {0}. (3.10)

Finally, we demonstrate the regularity of u_{v}, more precisely we will show that
u_{v}∈ D+. It follows from (3.3), (3.9) and (3.10) that

hA(u_{v}), hi =
Z

Ω

f (z, u_{v}, Dv)h dz for all h ∈ W_{0}^{1,p}(Ω),
which gives

− div a(Duv(z)) = f (z, uv(z), Dv(z)) for a.a. z ∈ Ω,

u_{v}(z) = 0 on ∂Ω. (3.11)

From (3.11) and Papageorgiou-R˘adulescu [30], we have
uv∈ L^{∞}(Ω).

However, using the regularity results from Lieberman [28] (see also Fukagai-Narukawa [9]), we have

uv∈ C_{0}^{1}(Ω) \ {0}.

To conclude, we have uv∈ [0, η] ∩ C_{0}^{1}(Ω) \ {0}. Moreover, we can use the nonlinear
maximum principle, see Pucci-Serrin [35]), to conclude directly that uv∈ D+.
From the proof of Proposition 3.2, we know that problem (3.1) has a solution
uv ∈ [0, η] ∩ D+. Next, we will prove that problem (3.1) has a smallest positive
solution in the order interval [0, η]. In what follows, we denote

S_{v}= {u ∈ W_{0}^{1,p}(Ω) : u 6= 0, u ∈ [0, η] is a solution of (3.1)}.

Proposition 3.2 implies

∅ 6= Sv⊆ [0, η] ∩ D+.

Let p^{∗} be the critical Sobolev exponent corresponding to p, i.e.,
p^{∗}=

( _{N p}

N −p if p < N, +∞ if N 6 p.

For ε > 0 and r ∈ (p, p^{∗}) fixed, from hypotheses (H2)(i) and (ii), there exists
c_{6}= c_{6}(ε, r, M ) > 0 (recall that M := sup_{z∈Ω}|Dv(z)|) such that

f (z, x, Dv(z)) > (ηM(z) − ε)x^{q−1}− c_{6}x^{r−1} (3.12)
for a.a. z ∈ Ω, and all 0 6 x 6 η. This unilateral growth restriction on f (z, ·, Dv(z))
drives us to consider another auxiliary Dirichlet problem as follows:

− div a(Du(z)) = (η_{M}(z) − ε)u(z)^{q−1}− c_{6}u(z)^{r−1} in Ω,

u(z) = 0 on ∂Ω, (3.13)

with u(z) > 0 in Ω.

Proposition 3.3. If hypotheses (H1) holds, then for all ε > 0, auxiliary problem
(3.13) admits a unique positive solution u^{∗}∈ D+.

Proof. First we show the existence of positive solutions for problem (3.13). To do
so, consider the C^{1}-functional ψ : W_{0}^{1,p}(Ω) → R defined by

ψ(u) = Z

Ω

G(Du) dz +1

pku^{−}k^{p}_{p}−1
q

Z

Ω

(ηM(z) − ε)(u^{+})^{q}dz
+c6

rku^{+}k^{r}_{r} for all u ∈ W_{0}^{1,p}(Ω).

From the facts G(0) = 0, u = u^{+}− u^{−} and [11, Proposition 2.4.27], we have
Z

Ω

G(Du) dz = Z

Ω

G(Du^{+}) dz +
Z

Ω

G(−Du^{−}) dz.

So, from Lemma 2.4 we have
ψ(u) > c_{1}

p(p − 1)kDu^{+}k^{p}_{p}+c_{6}

rku^{+}k^{r}_{r}+ c_{1}

p(p − 1)kDu^{−}k^{p}_{p}+1
pku^{−}k^{p}_{p}

−1 q

Z

Ω

(ηM(z) − ε)(u^{+})^{q}dz,

hence (see, Poincar´e inequality, e.g. [11, Theorem 2.5.4, p.216])
ψ(u) > c^{7}kuk^{p}− c8(kuk^{q}+ 1),

for some c7, c8 > 0. Since q < p, it is clear that ψ is coercive. We use the
compactness of embedding W_{0}^{1,p}(Ω) ⊆ L^{p}(Ω) and the convexity of G again, to
conclude that ψ is sequentially weakly lower semicontinuous. By the Weierstrass-
Tonelli theorem, we get u^{∗}∈ W_{0}^{1,p}(Ω) such that

ψ(u^{∗}) = inf

u∈W_{0}^{1,p}(Ω)

ψ(u). (3.14)

Using the same method as in the proof of Proposition 3.2, we can take t ∈ (0, 1) and ε > 0 small enough to obtain ψ(tub1(q)) < 0. This implies (see (3.14))

ψ(u^{∗}) < 0 = ψ(0),
so, u^{∗}6= 0.

The equality (3.14) implies ψ^{0}(u^{∗}) = 0. For h ∈ W_{0}^{1,p}(Ω), one has
hA(u^{∗}), hi −

Z

Ω

((u^{∗})^{−})^{p−1}h dz =
Z

Ω

(ηM(z) − ε)((u^{∗})^{+})^{q−1}h dz

− c6

Z

Ω

((u^{∗})^{+})^{r−1}h dz.

(3.15)

Taking h = −(u^{∗})^{−}∈ W_{0}^{1,p}(Ω) into (3.15), we use Lemma 2.3 again to obtain
c1

p − 1kD(u^{∗})^{−}k^{p}_{p}+ k(u^{∗})^{−}k^{p}_{p}6 0.

So, we have u^{∗}> 0 and u^{∗}6= 0. Therefore, (3.15) reduces to
hA(u^{∗}), hi =

Z

Ω

(ηM(z) − ε)(u^{∗})^{q−1}h dz − c6

Z

Ω

(u^{∗})^{r−1}h dz
for all h ∈ W_{0}^{1,p}(Ω), this means

− div a(Du^{∗}(z)) = (η_{M} − ε)(u^{∗})(z)^{q−1}− c6(u^{∗})(z)^{r−1} for a.a. z ∈ Ω,

u^{∗}(z) = 0 on ∂Ω. (3.16)

As in the proof of Proposition 3.2, using the nonlinear regularity theory, we have
u^{∗}∈ C_{0}^{1}(Ω)+\ {0}.

Next we shall verify that u^{∗} is the unique positive solution to problem (3.13).

For this goal, we consider the integral functional j : L^{1}(Ω) → R = R ∪ {+∞}

defined by

j(u) = (R

ΩG(Du^{1/q}) dz if u > 0, u^{1/q}∈ W_{0}^{1,p}(Ω),

+∞ otherwise,

where the effective domain of the functional j is denoted by
dom j = {u ∈ L^{1}(Ω) : j(u) < +∞}.

We will show that the integral functional j is convex. Let u1, u2 ∈ dom j and
u = (1−t)u1+tu2with t ∈ [0, 1]. [5, Lemma 1] states that the function u → |Du^{1/q}|^{q}
is convex, so we have

|Du^{1/q}(z)| 6

(1 − t)|Du1(z)^{1/q}|^{q}+ t|Du2(z)^{1/q}|^{q}1/q

for a.a. z ∈ Ω.

The monotonicity of G0 and the convexity of t 7→ G0(t^{1/q}) (see hypothesis
(H1)(iv)) ensure that

G0(|Du^{1/q}(z)|) 6 G^{0} (1 − t)|Du1(z)^{1/q}|^{q}+ t|Du2(z)^{1/q}|^{q}1/q
6 (1 − t)G^{0}(|Du1(z)^{1/q}|) + tG0(|Du2(z)^{1/q}|)
for a.a. z ∈ Ω. Which leads to

G(Du^{1/q}(z)) 6 (1 − t)G(Du^{1}(z)^{1/q}) + tG(Du2(z)^{1/q}) for a.a. z ∈ Ω,
thus the map j is convex.

Suppose thateu^{∗} is another positive solution of (3.13). As we did for u^{∗}, we can
check thateu^{∗}∈ C_{0}^{1}(Ω)+\ {0}. For h ∈ C_{0}^{1}(Ω) fixed and |t| small enough, we obtain

u^{∗}+ th ∈ dom j and ue^{∗}+ th ∈ dom j.

Recalling that j is convex, it is evidently Gˆateaux differentiable at u^{∗}and atue^{∗} in
the direction h. Further, we apply the chain rule and the nonlinear Green’s identity
(see Gasi´nski-Papageorgiou [11, p. 210]) to obtain

j^{0}(u^{∗})(h) =1
q

Z

Ω

− div a(Du^{∗})

(u^{∗})^{q−1} h dz for all h ∈ C_{0}^{1}(Ω),
j^{0}(ue^{∗})(h) =1

q Z

Ω

− div a(Deu^{∗})

(eu^{∗})^{q−1} h dz for all h ∈ C_{0}^{1}(Ω).

Putting h = (u^{∗})^{q} − (ue^{∗})^{q} into the above inequalities and then subtracting the
resulting equalities, it follows from the monotonicity of j^{0} (since j is convex) that

0 6 1 q

Z

Ω

− div(Du^{∗})

(u^{∗})^{q−1} −− div a(Due^{∗})
(ue^{∗})^{q−1}

((u^{∗})^{q}− (eu^{∗})^{q}) dz

= c6

q Z

Ω

(ue^{∗})^{r−q}− (u^{∗})^{r−q}

(u^{∗})^{q}− (eu^{∗})^{q} dz

(see (3.13)), so, from q < p < r, we conclude that u^{∗} = ue^{∗}. This proves that
u^{∗}∈ C_{0}^{1}(Ω)+\ {0} is the unique positive solution for problem (3.13). We are now
to apply the nonlinear maximum principle, see Pucci-Serrin [35]), again to obtain

u^{∗}∈ D+.

Proposition 3.4. If hypotheses (H1) and (H2) hold, then u^{∗}6 u for all u ∈ S^{v}.
Proof. Let u ∈ S_{v}. We now introduce the following Carath´eodory function e :
Ω × R → R

e(z, x) =

((ηM(z) − ε)(x^{+})^{q−1}− c6(x^{+})^{r−1}+ ξη(x^{+})^{p−1} if x 6 u(z),

(ηM(z) − ε)u(z)^{q−1}− c6u(z)^{r−1}+ ξηu(z)^{p−1} if u(z) < x. (3.17)
Also, we denote

E(z, x) = Z x

0

e(z, s) ds
and consider the C^{1}-functional τ : W_{0}^{1,p}(Ω) → R defined by

τ (u) = Z

Ω

G(Du) dz +ξη

pkuk^{p}_{p}−
Z

Ω

E(z, u) dz for all u ∈ W_{0}^{1,p}(Ω).

By the definition of e (see (3.17)), we see that τ is coercive. Also, it is sequentially
weakly lower semicontinuous. Invoking the Weierstrass-Tonelli theorem, we can
findeu^{∗}∈ W_{0}^{1,p}(Ω) such that

τ (ue^{∗}) = inf

v∈W_{0}^{1,p}(Ω)

τ (v). (3.18)

As before, since q < p < r, we have

τ (ue^{∗}) < 0 = τ (0),

which impliesue^{∗}6= 0. From (3.18), we have τ^{0}(ue^{∗}) = 0, which means
hA(eu^{∗}), hi + ξη

Z

Ω

|eu^{∗}|^{p−2}eu^{∗}h dz =
Z

Ω

e(z,ue^{∗})h dz (3.19)
for all h ∈ W_{0}^{1,p}(Ω). Putting h = −(ue^{∗})^{−} ∈ W_{0}^{1,p}(Ω) into the above equality and
then using Lemma 2.3, we have

c1

p − 1kD(ue^{∗})^{−}k^{p}_{p}+ ξ_{η}k(ue^{∗})^{−}k^{p}_{p}≤ 0

(see (3.17)), soue^{∗}> 0 andue^{∗}6= 0.

On the other hand, inserting h = (ue^{∗}− u)^{+}∈ W_{0}^{1,p}(Ω) into (3.19), we obtain
hA(ue^{∗}), (eu^{∗}− u)^{+}i + ξ_{η}

Z

Ω

(eu^{∗})^{p−1}(eu^{∗}− u)^{+}dz

= Z

Ω

(η_{M}(z) − ε)u^{q−1}− c_{6}u^{r−1}+ ξ_{η}u^{p−1}(eu^{∗}− u)^{+}dz
6

Z

Ω

f (z, u, Dv)(eu^{∗}− u)^{+}dz + ξη

Z

Ω

u^{p−1}(eu^{∗}− u)^{+}dz

= hA(u), (eu^{∗}− u)^{+}i + ξη

Z

Ω

u^{p−1}(eu^{∗}− u)^{+}dz
(see (3.12), (3.17), and recall that u ∈ Sv). Therefore, we have

hA(eu^{∗}) − A(u), (ue^{∗}− u)^{+}i + ξη

Z

Ω

(ue^{∗})^{p−1}− u^{p−1}(ue^{∗}− u)^{+}dz 6 0.

Using the monotonicity of A, we deduceue^{∗}6 u. So, we have verified that

ue^{∗}∈ [0, u] \ {0}. (3.20)

Taking into account (3.17) and (3.20), we rewrite (3.19) as
hA(ue^{∗}), hi =

Z

Ω

(ηM(z) − ε)(eu^{∗})^{q−1}− c6(ue^{∗})^{r−1}h dz

for all h ∈ W_{0}^{1,p}(Ω). This combined with Proposition 3.3 giveseu^{∗}= u^{∗}, so u^{∗} 6 u,

which completes the proof.

Applying Proposition 3.4, we shall show that problem (3.1) admits a smallest positive solutionbuv ∈ [0, η] ∩ D+.

Proposition 3.5. If (H1) and (H2) are fulfilled, then problem (3.1) admits a small- est positive solution buv∈ [0, η] ∩ D+.

Proof. Invoking [24, Lemma 3.10 p. 178], we can find a decreasing sequence
{un}_{n>1}⊆ Sv such that

inf Sv= inf

n>1un. (3.21)

For all n > 1, we have hA(un), hi =

Z

Ω

f (z, un, Dv)h dz for all h ∈ W_{0}^{1,p}(Ω), (3.22)
however, from Proposition 3.4, one has

u^{∗}6 u^{n}6 η. (3.23)

Then by hypothesis (H2)(i) and Lemma 2.3, we have that the sequence {un}_{n>1}⊆
W_{0}^{1,p}(Ω) is bounded. Passing to a subsequence, we may assume that

u_{n} →^{w} bu_{v} in W_{0}^{1,p}(Ω) and u_{n} →bu_{v} in L^{p}(Ω). (3.24)
Choosing h = u_{n}−bu_{v} ∈ W_{0}^{1,p}(Ω) for (3.22), we pass to the limit as n → ∞ and
then apply (3.24) to get

n→+∞lim hA(un), un−ubvi = 0,

but the (S)+-property of A (see Proposition 2.5), results in

u_{n}→ub_{v} in W_{0}^{1,p}(Ω). (3.25)
Passing to the limit as n → +∞ in (3.22) and using (3.25) to reveal

hA(buv), hi = Z

Ω

f (z,ubv, Dv)h dz for all h ∈ W_{0}^{1,p}(Ω).

On the other hand, taking the limit as n → +∞ in (3.23), we conclude that
u^{∗}6buv 6 η.

From the above inequality, it follows that

buv∈ Sv and buv = inf Sv,

which completes the proof.

Now, we consider the set

C = {u ∈ C_{0}^{1}(Ω) : 0 6 u(z) 6 η for all z ∈ Ω}

and introduce the mapping ϑ : C → C given by ϑ(v) =ubv.

It is obvious that a fixed point of map ϑ is also a positive solution to problem (1.1).

Therefore, next, we focus our attention to produce a fixed point for ϑ. Here our approach will apply the Leray-Schauder alternative principle (see Theorem 2.8). To do so, we will need the following lemma.

Lemma 3.6. If (H1) and (H2) are satisfied, then for any sequence {vn}_{n>1} ⊆ C
with vn → v in C_{0}^{1}(Ω), and u ∈ Sv, there exists a sequence {un} ⊆ C_{0}^{1}(Ω) with
u_{n}∈ S_{v}_{n} for n > 1, such that un→ u in C_{0}^{1}(Ω).

Proof. Let {v_{n}}_{n>1} ⊆ C be such that v_{n} → v in C_{0}^{1}(Ω), and u ∈ S_{v}. First, we
consider the nonlinear Dirichlet problem

− div a(Dw(z)) + ξη|w(z)|^{p−2}w(z) = bf (z, u(z), Dvn(z)) in Ω,

w(z) = 0 on ∂Ω. (3.26)

Since u ∈ S_{v}⊆ [0, η] \ {0}, from (3.3) and hypothesis (H2)(i), we see that
f (·, u(·), Dvb n(·)) 6≡ 0 for all n > 1,

f (z, u(z), Dvb n(z)) > 0 for a.a. z ∈ Ω and all n > 1.

It is obvious that problem (3.26) has a unique positive solution u^{0}_{n}∈ D+. It follows
from (3.3), the fact that u ∈ Sv⊆ [0, η] \ {0}, and hypotheses (H2)(i), (iii) that

hA(u^{0}_{n}), (u^{0}_{n}− η)^{+}i + ξη

Z

Ω

(u^{0}_{n})^{p−1}(u^{0}_{n}− η)^{+}dz

= Z

Ω

(f (z, u, Dvn) + ξηu^{p−1})(u^{0}_{n}− η)^{+}dz
6

Z

Ω

(f (z, η, Dvn) + ξηη^{p−1})(u^{0}_{n}− η)^{+}dz

= Z

Ω

ξηη^{p−1}(u^{0}_{n}− η)^{+}dz,

hence, from A(η) = 0, we have
hA(u^{0}_{n}) − A(η), (u^{0}_{n}− η)^{+}i + ξη

Z

Ω

((u^{0}_{n})^{p−1}− η^{p−1})(u^{0}_{n}− η)^{+}dz 6 0.

However, the monotonicity of A implies u^{0}_{n}6 η. So, we conclude that
u^{0}_{n}∈ [0, η] \ {0} ∀n > 1.

Moreover the nonlinear regularity theory of Lieberman [28], and the nonlinear max- imum principle of Pucci-Serrin [35]) imply that

u^{0}_{n}∈ [0, η] ∩ D+ ∀n > 1. (3.27)
We have

− div a(Du^{0}_{n}(z)) + ξ_{η}((u^{0}_{n}(z))^{p−1}− u(z)^{p−1}) = f (z, u(z), Dv_{n}(z)) for a.a. z ∈ Ω,
u^{0}_{n}(z) = 0 on ∂Ω.

(3.28)
From (3.27)–(3.28), Lemma 2.3 and hypothesis (H2)(i), we conclude that the se-
quence {u^{0}_{n}}_{n>1}is bounded in W_{0}^{1,p}(Ω). So, on account of the nonlinear regularity
theory of Lieberman [28], we can find β ∈ (0, 1) and c9> 0 such that

u^{0}_{n} ∈ C^{1,β}(Ω) and ku^{0}_{n}k_{C}1,β(Ω)6 c^{9} ∀n > 1.

The compactness of the embedding C^{1,β}(Ω) ⊆ C^{1}(Ω) implies that there exists a
subsequence {u^{0}_{n}_{k}}_{k>1}of the sequence {u^{0}_{n}}_{n>1}such that

u^{0}_{n}

k→ue^{0} in C^{1}(Ω) as k → +∞.

Using this fact and (3.28), we have

− div a(Due^{0}(z)) + ξη((ue^{0}(z))^{p−1}− u(z)^{p−1}) = f (z, u(z), Dv(z)) for a.a. z ∈ Ω,
eu^{0}(z) = 0 on ∂Ω.

(3.29)
Recall that u ∈ S_{v}, so (3.1) holds. Taking into account (3.1) and (3.29), we have

hA(ue^{0}) − A(u), hi + ξη

Z

Ω

(ue^{0}(z)^{p−1}− u(z)^{p−1})h dz = 0

for all h ∈ W_{0}^{1,p}(Ω). Additionally, we insert h = (eu^{0}− u)^{+} and h = −(u −ue^{0})^{+}
into the above equality to obtain

eu^{0}= u ∈ S_{v}.
So, for the original sequence {u^{0}_{n}}_{n>1}, one has

u^{0}_{n}→ u in C_{0}^{1}(Ω) as n → +∞.

Next, we consider the nonlinear Dirichlet problem

− div a(Dw(z)) + ξη|w(z)|^{p−2}w(z) = bf (z, u^{0}_{n}(z), Dvn(z)) in Ω,
w(z) = 0 on ∂Ω.

As before, we verify that the above problem admits a unique solution such that
u^{1}_{n}∈ [0, η] ∩ D+ ∀n > 1.

We apply nonlinear regularity theory of Lieberman [28] again to obtain
u^{1}_{n}→ u in C_{0}^{1}(Ω) as n → +∞.

Repeating this procedure, we construct a sequence {u^{k}_{n}}_{k,n>1}such that

− div a(Du^{k}_{n}(z)) + ξ_{η}u^{k}_{n}(z)^{p−1} = bf (z, u^{k−1}_{n} (z), Dv_{n}(z)) in Ω,

u^{k}_{n}(z) = 0 on ∂Ω (3.30)

for all n, k > 1 with

u^{k}_{n}∈ [0, η] ∩ D+ ∀n, k > 1, (3.31)
u^{k}_{n}→ u in C_{0}^{1}(Ω) as n → +∞ ∀k > 1. (3.32)
For n > 1 fixed, as above, we know that the sequence {u^{k}n}_{k>1}⊆ C_{0}^{1}(Ω) is relatively
compact. Therefore, there has a subsequence {u^{k}_{n}^{m}}_{m>1} of the sequence {u^{k}_{n}}_{k>1}
satisfying

u^{k}_{n}^{m} →ue_{n} in C_{0}^{1}(Ω) as m → +∞.

This and (3.32) imply

− div a(Duen(z)) + ξηuen(z)^{p−1}= bf (z,uen(z), Dvn(z)) for a.a. z ∈ Ω,

eun(z) = 0 on ∂Ω. (3.33)

The uniqueness of the solution of (3.33) deduces that for the original sequence we have

u^{k}_{n}→uen in C_{0}^{1}(Ω) as k → +∞.

However, from (3.31), we obtain

ue_{n}∈ [0, η] ∩ D_{+} ∀n > 1,

but from (3.32) and the double limit lemma (see Aubin-Ekeland [1] or Gasi´nski-
Papageorgiou [20, p. 61]), we haveeu_{n}∈ [0, η] ∩ D+ ∀n > n0. Consequently,

eun∈ Sv ∀n > n0 and eun→ u in C_{0}^{1}(Ω),

which completes the proof of the Lemma.

Remark 3.7. Actually, if we introduce the set-valued mapping S : C^{1}(Ω) → 2^{C}^{1}^{(Ω)}
by

S(v) = Sv,

then by the above lemma, we conclude that the mapping S is lower semicontinuous.

Applying this lemma, we will prove that the map ϑ : C → C defined by ϑ(v) =ub_{v}
is compact.

Proposition 3.8. If hypotheses (H1) and (H2) are fulfilled, then the map ϑ : C → C is compact.

Proof. To end this, we shall show that ϑ is continuous and maps bounded sets in C to relatively compact subsets of C.

First, for the part of continuity of ϑ, let v ∈ C and {vn}_{n>1} ⊆ C be such that
vn → v in C_{0}^{1}(Ω), and denoteubn= ϑ(vn) for n > 1. So, we get

− div a(Dbu_{n}(z)) = f (z,ub_{n}(z), Dv_{n}(z)) for a.a. z ∈ Ω,

bun(z) = 0 on ∂Ω, (3.34)

with bu_{n} ∈ [0, η] for all n > 1. It is easy to check that {bu_{n}}_{n>1} ⊆ W_{0}^{1,p}(Ω) is
bounded. So, it follows from Lieberman [28] that there exist β ∈ (0, 1) and c_{10}> 0
satisfying

bu_{n} ∈ C^{1,β}(Ω) and kbu_{n}k_{C}1,β(Ω)6 c10 ∀n > 1.

Without loss of generality, we may assume that

ub_{n}→bu in C_{0}^{1}(Ω) as n → +∞. (3.35)
Passing to the limit in (3.34), it yields

− div a(Dbu(z)) = f (z,u(z), Dv(z))b for a.a. z ∈ Ω,

bu(z) = 0 on ∂Ω. (3.36)

By taking M > sup_{n>1}kvnk_{C}1(Ω), we apply Proposition 3.4 to obtain u^{∗}6ub_{n} ∀n >

1, hence, convergence (3.35) implies

u^{∗}6bu ∈ C_{0}^{1}(Ω)+. (3.37)
We now assert thatu = ϑ(v). Invoking Lemma 3.6, we can take a sequence {ub _{n}} ⊆
C_{0}^{1}(Ω) with un ∈ Sv_{n}, n > 1 and

un→ ϑ(v) in C_{0}^{1}(Ω) as n → +∞. (3.38)
By the definition of ϑ, we have

bun= ϑ(vn) 6 u^{n} ∀n > 1.

This combined with (3.35) and (3.38) gives u 6 ϑ(v). Recalling that (3.37), web obtain

bu = ϑ(v), therefore, ϑ is continuous.

Next we will verify that ϑ maps bounded sets in C to relatively compact subsets
of C. Assume that B ⊆ C is bounded in C_{0}^{1}(Ω). As before, we know that the set
ϑ(B) ⊆ W_{0}^{1,p}(Ω) is bounded. On the other hand, we apply the nonlinear regularity
theory of Lieberman [28] and the compactness of the embedding C_{0}^{1,s}(Ω) ⊆ C_{0}^{1}(Ω)
(with 0 < s < 1) to reveal that the set ϑ(B) ⊆ C_{0}^{1}(Ω) is relatively compact, thus ϑ

is compact.

Now we give the main result of this article.

Theorem 3.9. If (H1) and (H2) are satisfied, then problem (1.1) admits a positive solution bu, more precisely,

bu ∈ [0, η] ∩ D+. Proof. Let U (ϑ) be the set defined by

U (ϑ) = {u ∈ C : u = λϑ(u), 0 < λ < 1}.

For any u ∈ U (ϑ), we have ^{1}_{λ}u = ϑ(u), so
hA(1

λu), hi = Z

Ω

f (z,u

λ, Du)h dz for all h ∈ W_{0}^{1,p}(Ω). (3.39)
Inserting h = ^{u}_{λ} ∈ W_{0}^{1,p}(Ω) into (3.39) and taking into account Lemma 2.3, we
calculate

c1

p − 1kD(u
λ)k^{p}_{p}6

Z

Ω

f (z,u λ, Du)u

λdz 6 Z

Ω

f (z, u, Du) u
λ^{p}dz
6

Z

Ω

f (z, u, D(u

λ))u dz 6 Z

Ω ec1+ec2|D(u
λ)|^{p} dz

where the last three inequalities are obtained by using (2.3), (H2)(iii), and (H2)(i),
respectively. Considering the inequalityec2<_{p−1}^{c}^{1} (see hypothesis (H2)(i)), one has

kD(u

λ)k_{p}6 c11 for all λ ∈ (0, 1),
for some c11> 0. Hence, we have

{u

λ}_{u∈U (ϑ)}⊆ W_{0}^{1,p}(Ω) is bounded. (3.40)
From (3.39) we have

− div a(D(u

λ)(z)) = f (z,u

λ(z), Du(z)) for a.a. z ∈ Ω, u = 0 on ∂Ω.

(3.41) However, condition (H2)(iii) ensures that

f (z,u

λ, Du) 6 λ^{p}f (z,u
λ, D(u

λ)) for a.a. z ∈ Ω. (3.42) Then from (3.40)–(3.42) and the nonlinear regularity theory of Lieberman [28], we have

ku
λk_{C}1

0(Ω)6 c^{12} for all u ∈ U (ϑ),
for some c_{12}> 0, thus U (ϑ) ⊆ C_{0}^{1}(Ω) is bounded.

Recall that ϑ is compact, see Proposition 3.8, we are now in a position to apply the Leray-Schauder alternative theorem (see Theorem 2.8), to look for a function u ∈ C such thatb

u = ϑ(b u).b

Consequently, we know thatu ∈ [0, η] ∩ Db _{+} is a positive solution of (1.1).
References

[1] J.-P. Aubin, I. Ekeland; Applied Nonlinear Analysis, Wiley, New York, 1984.

[2] D. Averna, D. Motreanu, E. Tornatore; Existence and asymptotic properties for quasilinear elliptic equations with gradient dependence, Appl. Math. Lett., 61 (2016), 102–107.

[3] Y. R. Bai, L. Gasi´nski, N. S. Papageorgiou; Nonlinear nonhomogeneous Robin problems with dependence on the gradient, Bound Value Probl., 2018:17 (2018), pages 24.

[4] L. Cherfils, Y. Il’yasov; On the stationary solutions of generalized reaction diffusion equations with p&q-Laplacian, Commun. Pure Appl. Anal., 4 (2005), 9–22.

[5] J.I. D´ıaz, J. E. Sa´a; Existence et unicit´e de solutions positives pour certaines ´equations elliptiques quasilin´eaires, C. R. Acad. Sci. Paris S´er. I Math., 305 (1987), 521–524.

[6] F. Faraci, D. Motreanu, D. Puglisi; Positive solutions of quasi-linear elliptic equations with dependence on the gradient, Calc. Var. Partial Differential Equations, 54 (2015), 525–538.

[7] L. F. O. Faria, O. H. Miyagaki, D. Motreanu; Comparison and positive solutions for problems with the (p, q)-Laplacian and a convection term, Proc. Edinb. Math. Soc. (2), 57 (2014), 687–

698.

[8] D. de Figueiredo, M. Girardi, M. Matzeu; Semilinear elliptic equations with dependence on the gradient via mountain-pass techniques, Differential Integral Equations, 17 (2004), 119–

126.

[9] N. Fukagai, K. Narukawa; On the existence of multiple positive solutions of quasilinear elliptic eigenvalue problems, Ann. Mat. Pura Appl., (4), 186 (2007), 539–564.

[10] L. Gasi´nski, D. O’Regan, N. Papageorgiou; Positive solutions for nonlinear nonhomogeneous Robin problems, Z. Anal. Anwend., 34 (2015), 435–458.

[11] L. Gasi´nski, N. S. Papageorgiou; Nonlinear Analysis. Chapman & Hall/CRC, Boca Raton, FL, 2006.

[12] L. Gasi´nski, N. S. Papageorgiou; Existence and multiplicity of solutions for Neumann p- Laplacian-type equations, Adv. Nonlinear Stud., 8 (2008), 843–870.

[13] L. Gasi´nski, N.S. Papageorgiou, Anisotropic nonlinear Neumann problems, Calc. Var. Partial Differential Equations, 42 (2011), 323–354.

[14] L. Gasi´nski, N.S. Papageorgiou; Multiple solutions for nonlinear coercive problems with a nonhomogeneous differential operator and a nonsmooth potential, Set-Valued Var. Anal., 20 (2012), 417–443.

[15] L. Gasi´nski, N. S. Papageorgiou; Nonhomogeneous nonlinear Dirichlet problems with a p- superlinear reaction, Abstr. Appl. Anal., 2012, ID 918271, 28.

[16] L. Gasi´nski, N.S. Papageorgiou; A pair of positive solutions for the Dirichlet p(z)-Laplacian with concave and convex nonlinearities, J. Global Optim., 56 (2013), 1347–1360.

[17] L. Gasi´nski, N.S. Papageorgiou; Dirichlet (p, q)-equations at resonance, Discrete Contin. Dyn.

Syst., 34 (2014), 2037–2060.

[18] L. Gasi´nski, N. S. Papageorgiou; A pair of positive solutions for (p, q)-equations with com- bined nonlinearities, Commun. Pure Appl. Anal., 13 (2014), 203–215.

[19] L. Gasi´nski, N. S. Papageorgiou, On generalized logistic equations with a non-homogeneous differential operator, Dyn. Syst., 29 (2014), 190–207.

[20] L. Gasi´nski, N. S. Papageorgiou; Exercises in Analysis. Part 1, Springer, Cham, 2014.

[21] L. Gasi´nski, N. S. Papageorgiou; Positive solutions for the generalized nonlinear logistic equations, Canad. Math. Bull., 59 (2016), 73–86.

[22] L. Gasi´nski, N. S. Papageorgiou; Positive solutions for nonlinear elliptic problems with de- pendence on the gradient, J. Differential Equations, 263 (2017), 1451–1476.

[23] M. Girardi, M. Matzeu; Positive and negative solutions of a quasi-linear elliptic equation by a mountain pass method and truncature techniques, Nonlinear Anal., 59 (2004), 199–210.

[24] S. Hu, N. S. Papageorgiou, Handbook of Multivalued Analysis. Volume I: Theory, Kluwer, Dordrecht, The Netherlands, 1997.

[25] N. B. Huy, B. T. Quan, N. H. Khanh; Existence and multiplicity results for generalized logistic equations, Nonlinear Anal., 144 (2016), 77–92.

[26] L. Iturriaga, S. Lorca, J. S´anchez; Existence and multiplicity results for the p-Laplacian with a p-gradient term, NoDEA Nonlinear Differential Equations Appl., 15 (2008), 729–743.

[27] G. M. Lieberman; Boundary regularity for solutions of degenerate elliptic equations, Nonlin- ear Anal., 12 (1988), 1203–1219.

[28] G. M. Lieberman; The natural generalization of the natural conditions of Ladyzhenskaya and Ural’tseva for elliptic equations, Comm. Partial Differential Equations, 16 (1991), 311–361.

[29] N. S. Papageorgiou, V. D. R˘adulescu; Coercive and noncoercive nonlinear Neumann problems with indefinite potential, Forum Math., 28 (2016), 545–571.

[30] N. S. Papageorgiou, V. D. R˘adulescu; Nonlinear nonhomogeneous Robin problems with su- perlinear reaction term, Adv. Nonlinear Stud., 16 (2016), 737–764.

[31] N. S. Papageorgiou, V. D. R˘adulescu; Multiplicity theorems for nonlinear nonhomogeneous Robin problems, Rev. Mat. Iberoam., 33 (2017), 251–289.

[32] N. S. Papageorgiou, V. D. R˘adulescu, D. D. Repovˇs; Existence and multiplicity of solutions for resonant(p,2)-equations, Advanced Nonlinear Studies., 18 (2018), 105–129.

[33] N. S. Papageorgiou, V. D. R˘adulescu; Multiplicity of solutions for nonlinear nonhomogeneous Robin problems, Proceedings of the American Mathematical Society., 146 (2018), 601–611.

[34] N. S. Papageorgiou, V. D. R˘adulescu, D. D. Repovˇs; Robin problems with a general potential and a superlinear reaction, Journal of Differential Equations., 6 (2017), 3244–3290.

[35] P. Pucci, J. Serrin; The Maximum Principle, Birkh¨auser Verlag, Basel, 2007.

[36] D. Ruiz; A priori estimates and existence of positive solutions for strongly nonlinear prob- lems, J. Differential Equations, 199 (2004), 96–114.

[37] M. Tanaka; Existence of a positive solution for quasilinear elliptic equations with nonlinearity including the gradient, Bound. Value Probl., 2013, 2013:173.

[38] S. D. Zeng, Z. H. Liu, S. Mig´orski; Positive solutions to nonlinear nonhomogeneous inclusion problems with dependence on the gradient, Journal of Mathematical Analysis and Applica- tions, 463 (2018), 432–448.

Yunru Bai

Jagiellonian University, Faculty of Mathematics and Computer Science, ul. Lojasiewicza 6, 30-348 Krak´ow, Poland

E-mail address: yunrubai@163.com