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

Because of hypothesis (H2)(g2), there exists

such that

and thanks to condition (H2)(g3), we find a constant

such that

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

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

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.