Sign-Changing and Extremal Constant-Sign Solutions of Nonlinear Elliptic Neumann Boundary Value Problems

Let be a bounded domain with Lipschitz boundary . We consider the following nonlinear elliptic boundary value problem. Find and constants such that

(1.1)

where is the negative -Laplacian, denotes the outer normal derivative of , and as well as are the positive and negative parts of , respectively. The nonlinearities and are some Carathéodory functions which are bounded on bounded sets. For reasons of simplification, we drop the notation for the trace operator which is used on the functions defined on the boundary .

The motivation of our study is a recent paper of the author in [1] in which problem (1.1) was treated in case . We extend this approach and prove the existence of multiple solutions for the more general problem (1.1). To be precise, the existence of a smallest positive solution, a greatest negative solution, as well as a sign-changing solution of problem (1.1) is proved by using variational and topological tools, for example, critical point theory, Mountain-Pass Theorem, and the Second Deformation Lemma. Additionally, the Fu ik spectrum for the -Laplacian takes an important part in our treatments.

Neumann boundary value problems in the form of (1.1) arise in different areas of pure and applied mathematics, for example, in the theory of quasiregular and quasiconformal mappings in Riemannian manifolds with boundary (see [2, 3]), in the study of optimal constants for the Sobolev trace embedding (see [4–7]), or at non-Newtonian fluids, flow through porus media, nonlinear elasticity, reaction diffusion problems, glaciology, and so on (see [8–11]).

The existence of multiple solutions for Neumann problems like those in the form of (1.1) has been studied by a number of authors, such as, for example, the authors of [12–15], and homogeneous Neumann boundary value problems were considered in [16, 17] and [15], respectively. Analogous results for the Dirichlet problem have been recently obtained in [18–21]. Further references can also be found in the bibliography of [1].

In our consideration, the nonlinearities and only need to be Carathéodory functions which are bounded on bounded sets whereby their growth does not need to be necessarily polynomial. The novelty of our paper is the fact that we do not need differentiability, polynomial growth, or some integral conditions on the mappings and .

First, we have to make an analysis of the associated spectrum of (1.1). The Fu ik spectrum for the -Laplacian with a nonlinear boundary condition is defined as the set of such that

(1.2)

has a nontrivial solution. In view of the identity

(1.3)

we see at once that for problem (1.2) reduces to the Steklov eigenvalue problem

(1.4)

We say that is an eigenvalue if (1.4) has nontrivial solutions. The first eigenvalue is isolated and simple and has a first eigenfunction which is strictly positive in (see [22]). Furthermore, one can show that belongs to (cf., [23, Lemma and Theorem ] or [24, Theorem ]), and along with the results of Lieberman in [25, Theorem ] it holds that . This fact combined with in yields , where denotes the interior of the positive cone in the Banach space , given by

(1.5)

Let us recall some properties of the Fu ik spectrum. If is an eigenvalue of (1.4), then the point belongs to . Since the first eigenfunction of (1.4) is positive, clearly contains the two lines and . A first nontrivial curve in through was constructed and variationally characterized by a mountain-pass procedure by Martínez and Rossi [26]. This yields the existence of a continuous path in joining and provided that is above the curve . The functional on is given by

(1.6)

Due to the fact that belongs to , there exists a variational characterization of the second eigenvalue of (1.4) meaning that can be represented as

(1.7)

where

(1.8)

The proof of this result is given in [26].

An important part in our considerations takes the following Neumann boundary value problem defined by

(1.9)

where is a constant. As pointed out in [1], there exists a unique solution of problem (1.9) which is required for the construction of sub- and supersolutions of problem (1.1).

Now, we impose the following conditions on the nonlinearities and in problem (1.1). The maps and are Carathéodory functions, which means that they are measurable in the first argument and continuous in the second one. Furthermore, we suppose the following assumptions.

(H) (f1)

(2.1)

(f2)

(2.2)

(f3) is bounded on bounded sets.

(f4) There exists such that

(2.3)

(H) (g1)

(2.4)

(g2)

(2.5)

(g3) is bounded on bounded sets.

(g4) satisfies the condition

(2.6)

for all pairs in , where is a positive constant and .

(H) Let be above the first nontrivial curve of the Fu ik spectrum constructed in [26] (see Figure 1).

Note that (H2)(g4) implies that the function fulfills a condition as in (H2)(g4), too. Moreover, we see at once that is a trivial solution of problem (1.1) because of the conditions (H1)(f1) and (H2)(g1), which guarantees that . It should be noted that hypothesis (H3) includes that (see [26] or Figure 1).

Example 2.1.

Let the functions and be given by

(2.7)

Then all conditions in (H1)(f1)–(f4) and (H2)(g1)–(g4) are fulfilled.

Definition 2.2.

A function is called a weak solution of (1.1) if the following holds:

(2.8)

Definition 2.3.

A function is called a subsolution of (1.1) if the following holds:

(2.9)

Definition 2.4.

A function is called a supersolution of (1.1) if the following holds:

(2.10)

We recall that denotes all nonnegative functions of . Furthermore, for functions satisfying , we have the relation , where stands for the well-known trace operator.

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

(3.1)

In order to satisfy Definition 2.4 for , we have to show that the following inequality holds true meaning:

(3.2)

where with . Condition (H1)(f2) implies the existence of such that

(3.3)

and due to (H1)(f3), we have

(3.4)

Hence, we get

(3.5)

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

(3.6)

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

(3.7)

Finally, we have

(3.8)

Using the inequality in (3.5) to the first integral in (3.2) yields

(3.9)

which proves its nonnegativity if . Applying (3.8) to the second integral in (3.2) ensures that

(3.10)

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:

(3.11)

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

(3.12)

Definition 2.3 is satisfied for provided that the inequality

(3.13)

is valid for all . With regard to hypothesis (H1)(f4), we obtain, for ,

(3.14)

where denotes the usual supremum norm. Thanks to condition (H2)(g1), there exists a number such that

(3.15)

In case we get

(3.16)

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

(3.17)

Applying (3.17) to (1.1) provides

(3.18)

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

(3.19)

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

(3.20)

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

(3.21)

Since solves problem (1.1), one obtains, for all ,

(3.22)

Setting in (3.22) results in

(3.23)

Using (3.21) and the hypotheses (H1)(f3) as well as (H2)(g3) yields

(3.24)

which provides, by the -property of on along with (3.21),

(3.25)

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

(3.26)

We set

(3.27)

It is clear that the sequence is bounded in which ensures the existence of a weakly convergent subsequence of , denoted again by , such that

(3.28)

with some function belonging to . In addition, we may suppose that there are functions such that

(3.29)

With the aid of (3.22), we obtain for the following variational equation:

(3.30)

We select in the last equality to get

(3.31)

Making use of (3.17) in combination with (3.29) results in

(3.32)

and, respectively,

(3.33)

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

(3.34)

From (3.28), (3.31), and (3.34) we infer that

(3.35)

and the -property of corresponding to implies that

(3.36)

Remark that , which means that . Applying (3.26) and (3.36) along with conditions (H1)(f1), (H2)(g1) to (3.30) provides

(3.37)

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.

Theorem 3.5 ensures the existence of extremal positive and negative solutions of (1.1) for all and denoted by and , respectively. Now, we introduce truncation functions and as follows:

(4.1)

For the truncation operators on apply to the corresponding traces . We just write for simplification without . Furthermore, the truncation operators are continuous, uniformly bounded, and Lipschitz continuous with respect to the second argument. By means of these truncations, we define the following associated functionals given by

(4.2)

which are well defined and belong to . Due to the truncations, one can easily show that these functionals are coercive and weakly lower semicontinuous, which implies that their global minimizers exist. Moreover, they also satisfy the Palais-Smale condition.

Lemma 4.1.

Let and be the extremal constant-sign solutions of (1.1). Then the following hold.

(i) A critical point of is a nonnegative solution of (1.1) satisfying .

(ii) A critical point of is a nonpositive solution of (1.1) satisfying .

(iii) A critical point of is a solution of (1.1) satisfying .

Proof.

Let be a critical point of meaning . We have for all

(4.3)

As is a positive solution of (1.1), it satisfies

(4.4)

Subtracting (4.4) from (4.3) and setting provide

(4.5)

Based on the definition of the truncation operators, we see that the right-hand side of the equality above is equal to zero. On the other hand, the integrals on the left-hand side are strictly positive in case which is a contradiction. Thus, we get and, hence, . The proof for acts in a similar way which shows that , and , and therefore, is a solution of (1.1) satisfying . The statements in (i) and (ii) can be shown in the same way.

An important tool in our considerations is the relation between local -minimizers and local -minimizers for -functionals. The fact is that every local -minimizer of is a local -minimizer of which was proved in similar form in [1, Proposition ]. This result reads as follows.

Proposition 4.2.

If is a local -minimizer of meaning that there exists such that

(4.6)

then is a local minimizer of in meaning that there exists such that

(4.7)

We also refer to a recent paper (see [29]) in which the proposition above was extended to the more general case of nonsmooth functionals. With the aid of Proposition 4.2, we can formulate the next lemma about the existence of local and global minimizers with respect to the functionals , and .

Lemma 4.3.

Let and . Then the extremal positive solution of (1.1) is the unique global minimizer of the functional , and the extremal negative solution of (1.1) is the unique global minimizer of the functional . In addition, both and are local minimizers of the functional .

Proof.

As is coercive and weakly sequentially lower semicontinuous, its global minimizer exists meaning that is a critical point of . Concerning Lemma 4.1, we know that is a nonnegative solution of (1.1) satisfying . Due to condition (H2)(g1), there exists a number such that

(4.8)

Choosing and applying assumption (H1)(f4), inequality (4.8) along with the Steklov eigenvalue problem in (1.4) implies that

(4.9)

From the calculations above, we see at once that , which means that . This allows us to apply Lemma 3.4 getting . Since is the smallest positive solution of (1.1) in fulfilling , it must hold that , which proves that is the unique global minimizer of . The same considerations show that is the unique global minimizer of . In order to complete the proof, we are going to show that and are local minimizers of the functional as well. The extremal positive solution belongs to , which means that there is a neighborhood of in the space satisfying . Therefore, on proves that is a local minimizer of on . Applying Proposition 4.2 yields that is also a local -minimizer of . Similarly, we see that is a local minimizer of , which completes the proof.

Lemma 4.4.

The functional has a global minimizer which is a nontrivial solution of (1.1) satisfying .

Proof.

As we know, the functional is coercive and weakly sequentially lower semicontinuous. Hence, it has a global minimizer . More precisely, is a critical point of which is a solution of (1.1) satisfying (see Lemma 4.1). The fact that (see the proof of Lemma 4.3) proves that is nontrivial meaning that .

The main result in this section about the existence of a nontrivial solution of problem (1.1) reads as follows.

Theorem 5.1.

Under hypotheses (H1)–(H3), problem (1.1) has a nontrivial sign-changing solution .

Proof.

In view of Lemma 4.4, the existence of a global minimizer of satisfying has been proved. This means that is a nontrivial solution of (1.1) belonging to . If and , then must be a sign-changing solution because is the greatest negative solution and is the smallest positive solution of (1.1), which proves the theorem in this case. We still have to show the theorem in case that either or . Let us only consider the case because the case can be proved similarly. The function is a local minimizer of . Without loss of generality, we suppose that is a strict local minimizer; otherwise, we would obtain infinitely many critical points of which are sign-changing solutions due to and the extremality of the solutions . Under these assumptions, there exists a such that

(5.1)

where . Now, we may apply the Mountain-Pass Theorem to (cf., [30]) thanks to (5.1) along with the fact that satisfies the Palais-Smale condition. This yields the existence of satisfying and

(5.2)

where

(5.3)

It is clear that (5.1) and (5.2) imply that and . Hence, is a sign-changing solution provided that . We have to show that which is fulfilled if there exists a path such that

(5.4)

Let , where , and be equipped with the topologies induced by and , respectively. Furthermore, we set

(5.5)

Because of the results of Martínez and Rossi in [26], there exists a continuous path satisfying provided that is above the curve of hypothesis (H3). Recall that the functional is given by

(5.6)

This implies the existence of such that

(5.7)

It is well known that is dense in , which implies the density of in . Thus, a continuous path exists such that

(5.8)

The boundedness of the set in ensures the existence of such that

(5.9)

Theorem 3.5 yields that . Thus, for every and any bounded neighborhood of in there exist positive numbers and satisfying

(5.10)

for all , for all , and for all . Using (5.10) along with a compactness argument implies the existence of such that

(5.11)

for all , for all , and for all . Representing in terms of , we obtain

(5.12)

In view of (5.11) we get for all and all

(5.13)

Due to hypotheses (H1)(f1) and (H2)(g1), there exist positive constants such that

(5.14)

Choosing such that and using (5.14) provide

(5.15)

Applying (5.15) to (5.13) yields

(5.16)

We have constructed a continuous path joining and . In order to construct continuous paths connecting and , respectively, and , we first denote that

(5.17)

It holds that because is a global minimizer of . By Lemma 4.1 the functional has no critical values in the interval . The coercivity of along with its property to satisfy the Palais-Smale condition allows us to apply the Second Deformation Lemma (see, e.g., [31, page 366]) to . This ensures the existence of a continuous mapping satisfying the following properties:

(i) , for all ,

(ii) , for all ,

(iii) , for all and for all .

Next, we introduce the path given by for all which is obviously continuous in joining and . Additionally, one has

(5.18)

Similarly, the Second Deformation Lemma can be applied to the functional . We get a continuous path connecting and such that

(5.19)

In the end, we combine the curves , and to obtain a continuous path joining and . Taking into account (5.16), (5.18), and (5.19), we get . This yields the existence of a nontrivial sign-changing solution of problem (1.1) satisfying which completes the proof.