For the rest of the paper we denote by
the first eigenfunction of the Steklov eigenvalue problem (1.4) corresponding to its first eigenvalue
. Furthermore, the function
stands for the unique solution of the auxiliary Neumann boundary value problem defined in (1.9). Our first lemma reads as follows.
Lemma 3.1.
Let conditions (H1)-(H2) be satisfied and let
. Then there exist constants
such that
and
are a positive supersolution and a negative subsolution, respectively, of problem (1.1).
Proof.
Setting
with a positive constant
to be specified and considering the auxiliary problem (1.9), we obtain
In order to satisfy Definition 2.4 for
, we have to show that the following inequality holds true meaning:
where
with
. Condition (H1)(f2) implies the existence of
such that
and due to (H1)(f3), we have
Hence, we get
Because of hypothesis (H2)(g2), there exists
such that
and thanks to condition (H2)(g3), we find a constant
such that
Finally, we have
Using the inequality in (3.5) to the first integral in (3.2) yields
which proves its nonnegativity if
. Applying (3.8) to the second integral in (3.2) ensures that
We take
to verify that both integrals in (3.2) are nonnegative. Hence, the function
is in fact a positive supersolution of problem (1.1). In a similar way one proves that
is a negative subsolution, where we apply the following estimates:
This completes the proof.
The next two lemmas show that constant multipliers of
may be sub- and supersolution of (1.1). More precisely, we have the following result.
Lemma 3.2.
Assume that (H1)-(H2) are satisfied. If
, then for
sufficiently small and any
the function
is a positive subsolution of problem (1.1).
Proof.
The Steklov eigenvalue problem (1.4) implies for all 
Definition 2.3 is satisfied for
provided that the inequality
is valid for all
. With regard to hypothesis (H1)(f4), we obtain, for
,
where
denotes the usual supremum norm. Thanks to condition (H2)(g1), there exists a number
such that
In case
we get
Selecting
guarantees that
is a positive subsolution.
The following lemma on the existence of a negative supersolution can be proved in a similar way.
Lemma 3.3.
Assume that (H1)-(H2) are satisfied. If
, then for
sufficiently small and any
the function
is a negative supersolution of problem (1.1).
Concerning Lemmas 3.1–3.3, we obtain a positive pair
and a negative pair
of sub- and supersolutions of problem (1.1) provided that
is sufficiently small.
In the next step we are going to prove the regularity of solutions of problem (1.1) belonging to the order intervals
and
, respectively. We also point out that
is both a subsolution and a supersolution because of the hypotheses (H1)(f1) and (H2)(g1).
Lemma 3.4.
Assume (H1)-(H2) and let
. If
(resp.,
) is a solution of problem (1.1) satisfying
in
, then it holds that
(resp.,
).
Proof.
We just show the first case; the other case acts in the same way. Let
be a solution of (1.1) satisfying
. We directly obtain the
-boundedness, and, hence, the regularity results of Lieberman in [25, Theorem
] imply that
with
. Due to assumptions (H1)(f1), (H1)(f3), (H2)(g1), and (H2)(g3), we obtain the existence of constants
satisfying
Applying (3.17) to (1.1) provides
where
is a positive constant. We set
for all
and use Vázquez's strong maximum principle (cf., [27]) which is possible because
. Hence, it holds that
in
. Finally, we suppose the existence of
satisfying
. Applying again the maximum principle yields
. However, because of
in combination with the Neumann condition in (1.1), we get
. This is a contradiction and, hence,
in
, which proves that
.
The main result in this section about the existence of extremal constant-sign solutions is given in the following theorem.
Theorem 3.5.
Assume (H1)-(H2). For every
and
, there exists a smallest positive solution
of (1.1) in the order interval
with the constant
as in Lemma 3.1. For every
and
there exists a greatest solution
in the order interval
with the constant
as in Lemma 3.1.
Proof.
Let
. Lemmas 3.1 and 3.2 guarantee that
is a subsolution of problem (1.1) and
is a supersolution of problem (1.1). Moreover, we choose
sufficiently small such that
. Applying the method of sub- and supersolution (see [28]) corresponding to the order interval
provides the existence of a smallest positive solution
of problem (1.1) fulfilling
. In view of Lemma 3.4, we have
. Hence, for every positive integer
sufficiently large, there exists a smallest solution
of problem (1.1) in the order interval
. We obtain
with some function
satisfying
.
Claim 1.
is a solution of problem (1.1).
As
and
, we obtain the boundedness of
in
and
, respectively. Definition 2.2 holds, in particular, for
and
which results in
with some positive constants
independent of
. Consequently,
is bounded in
and due to the reflexivity of
we obtain the existence of a weakly convergent subsequence of
. Because of the compact embedding
, the monotony of
, and the compactness of the trace operator
, we get for the entire sequence 
Since
solves problem (1.1), one obtains, for all
,
Setting
in (3.22) results in
Using (3.21) and the hypotheses (H1)(f3) as well as (H2)(g3) yields
which provides, by the
-property of
on
along with (3.21),
The uniform boundedness of the sequence
in conjunction with the strong convergence in (3.25) and conditions (H1)(f3) as well as (H2)(g3) admit us to pass to the limit in (3.22). This shows that
is a solution of problem (1.1).
Claim 2.
One has
.
In order to apply Lemma 3.4, we have to prove that
. Let us assume that this assertion is not valid meaning that
. From (3.19) it follows that
We set
It is clear that the sequence
is bounded in
which ensures the existence of a weakly convergent subsequence of
, denoted again by
, such that
with some function
belonging to
. In addition, we may suppose that there are functions
such that
With the aid of (3.22), we obtain for
the following variational equation:
We select
in the last equality to get
Making use of (3.17) in combination with (3.29) results in
and, respectively,
We see at once that the right-hand sides of (3.32) and (3.33) belong to
and
, respectively, which allows us to apply Lebesgue's dominated convergence theorem. This fact and the convergence properties in (3.28) show that
From (3.28), (3.31), and (3.34) we infer that
and the
-property of
corresponding to
implies that
Remark that
, which means that
. Applying (3.26) and (3.36) along with conditions (H1)(f1), (H2)(g1) to (3.30) provides
The equation above is the weak formulation of the Steklov eigenvalue problem in (1.4) where
is the eigenfunction with respect to the eigenvalue
. As
is nonnegative in
, we get a contradiction to the results of Martínez and Rossi in [22, Lemma
] because
must change sign on
. Hence,
. Applying Lemma 3.4 yields
.
Claim 3.
is the smallest positive solution of (1.1) in
.
Let
be a positive solution of (1.1) satisfying
. Lemma 3.4 immediately implies that
. Then there exists an integer
sufficiently large such that
. However, we already know that
is the smallest solution of (1.1) in
which yields
. Passing to the limit proves that
. Hence,
must be the smallest positive solution of (1.1). The existence of the greatest negative solution of (1.1) within
can be proved similarly. This completes the proof of the theorem.