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.