1. Introduction
This manuscript is concerned with a study of non-homogeneous Robin problem with the
Laplacian:
is a regular bounded domain in
,
, the functions
p and
q are assumed to be continuous on
,
with
V is a weight function in the generalised space
and
is a non-negative function in
.
The operator is a generalization of the classic p-Laplace operator, but due to the fact that is non-homogeneous, it has more complicated nonlinearity than the p-Laplace operator, so, we can find some difficulties in treating these kinds of operators; for example, the Lagrange Multiplier can not be used.
In recent decades, the study of
-Laplace operators with Robin boundary conditions is very remarkable motivated by the modeling of thermo-convective flows of non-Newtonian fluids, as well as the electrorheological fluids (see [
1]), elastic mechanics (see [
2]), stationary thermo-rheological viscous flows of non-Newtonian fluids, image processing (see [
3]), and the mathematical description of the processes filtration of a barotropic gas through a porous medium (see [
4]). The reader can find recent contribution involving
-Laplace operator in [
5,
6,
7,
8].
Moreover, Robin boundary conditions has been used in many physics problems, such that electromagnetic and heat transfer problems. In addition, Robin conditions are also needed to investigate Sturm–Liouville problems which are needed in different fields in sciences and engineering. The study on elliptic equation with Robin condition starts from the celebrated paper of S.G. Deng in [
9]. The author is concerned with the problem
where
is a regular bounded domain in
,
,
p is a continuous function on
,
Using the variational methods and under appropriate conditions, the author proved the existence of
under which problem (
2) has at least two positive solutions, at least one positive solution for
and has no positive solution in the case when
After that many authors were devoted to the investigations for a variety of elliptic equations involving Robin boundary condition in a bounded domain. The readers can found a recent contributions in [
6,
10,
11,
12,
13,
14,
15]. In [
6], the author considered the Robin problem of type:
where
is a regular bounded domain in
,
,
are continuous functions on
,
and
V is an indefinite function in a generalised Lebesgue space
. The author proved several results related to the existence of a spectrum to problem (
3). Moreover, the author obtained that any
is an eigenvalue for problem (
3) using the variational method.
In this paper, we are interested to the existence of nontrivial solutions for problem (
1). Here we follow the approach as in [
6]. In fact, we use a variational approach to show that for any
, problem (
1) has a nontrivial solution. We would like to remark that the results proved here are new even for the case that our class of indefinite weights used in this paper is larger than used in [
9,
10]. Moreover, this paper generalize the work of K. Kefi [
6]. In fact, if we assume that
in
, problem (
1) becomes problem (
3) and we obtain the same result as in [
6], so the result obtained in [
6] appears as a particular case in our problem.
In what follows, the rest of the paper is articulated on two sections. First, we introduce some prerequisite properties of the generalized Lebesgue–Sobolev space
, second we state and prove the existence results concerning problem (
1).
2. Backgrounds
In the following, we start by reminding properties and definitions for
and
(see [
16]). Putting
and for
, let
, such that
and
We equip this space using the Luxemburg norm
We remark, when
,
becomes
with the standard norm
.
In the sequel, define
as the conjugate of
, such that
, we have the Hölder Inequality
Note, also, that if
,
, and
are a Lipschitz continuous functions satisfying
, then for any
and
, we have (see [
17], Proposition 2.5):
We define also the so called modular on
which is a function
defined by
and verify
Proposition 1. (See [18]) For any , one has
Proposition 2. (See [19]) Let p and q two measurable functions, such that and, for a.e. . Let , . Then, The reader can found more details for the modular in [
16,
18].
In the sequel, put
, where
We equip this space with the norm
Then,
is a Banach space which is separable and reflexive.
Putting
Then, due to Theorem 2.1 in [
9],
is a norm on
X which is equivalent to
, moreover, if we define
by
we have
Proposition 3.
Proposition 4. (Proposition 2.2 in [20]) Let , then the function is a strictly monotone, where is the derivative of ξ and is the dual space of X. Moreover, it is a continuous bounded homeomorphism and is of type , so if and , then . Now, let
We mention that if
and
for all
, so
is compact.
For a measurable function
, define the weighted space
we equip it with
here
is the measure on the boundary. Then,
is a Banach space. Note that in the particular case when
on
,
.
Proposition 5. Let . For , we have
- (1)
- (2)
- (3)
;
- (4)
.
For
, we denote by
and
. Define
where
with
.
Our conditions are the following:
Hypothesis 1. , and in , with .
Hypothesis 2. for all and is a non-negative function in
We recall the next important theorem.
Theorem 1. (See Theorem 2.1 [9]) Suppose that the cone property holds on the boundary of and , such that . Assume, also, that and for all . If andone has, is compact. Consequently, is compact where for all . Definition 1. We said that is a weak solution of problem (1) iffor any . 3. Results
In what follows, denote by a positive constants which change from line to another.
Put
,
the conjugate functions of
,
, respectively, and let
then we have:
Remark 1. Assume that conditions and hold, thenresp. for all ,resp. for all Consequently, the embeddingresp. are compact and continuous. Our result is due to Theorem 25.D in [
21].
Theorem 2. Suppose that conditions and hold. Then, for any problem (1) has a weak solution. In the following, let the functions
,
defined by:
Using Proposition 2 and Remark 1, we assert that
and
are well-defined. In fact, for all
, one has
and
The energy of the problem (
1) is
, where
First, we show some auxiliary results:
Proposition 6. Assume that the assertions and are fulfilled, then is weakly lower semi-continuous and inside the space , moreover is a critical point of if, and only if, w is a weak solution for problem (1). For the proof of Proposition 6, we advice readers to refer to K. Kefi [
6].
Now, we show the existence of a valley for the functional near the origin
Lemma 1. There exists , such that , , and , for small enough.
Proof. Due to assumption , we have In the following, let and Let such that Since , then there exists an open set verifying So
Now, let
, such that
,
in a subset
,
in
. Then,
consequently
for
with
Since
in
, then
, which completes the proof of Lemma 1. □
Lemma 2. Assume the conditions and are fulfilled, then the functional is coercive.
Proof. Let
with
, so, from the Hölder inequality (
4) and the Sobolev embedding, one has
Since
, then
, so
is coercive. □
Proof of Theorem 2 completed. has a global minimizer. In addition, due to Lemma 1 this minimizer is non-trivial which complete the proof of our result. □