Abstract
Our aim is the study of a class of nonlinear elliptic problems under Neumann conditions involving the 
-Laplacian. We prove the existence of at least three nontrivial solutions, which means that we get two extremal constant-sign solutions and one sign-changing solution by using truncation techniques and comparison principles for nonlinear elliptic differential inequalities. We also apply the properties of the Fu
ik spectrum of the 
-Laplacian and, in particular, we make use of variational and topological tools, for example, critical point theory, Mountain-Pass Theorem, and the Second Deformation Lemma.
be a bounded domain with Lipschitz boundary 
. We consider the following nonlinear elliptic boundary value problem. Find 
and constants 
such that
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 
.
. 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.
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 
.
ik spectrum for the 
-Laplacian with a nonlinear boundary condition is defined as the set 
of 
such that
problem (1.2) reduces to the Steklov eigenvalue problem
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 [
belongs to 
(cf., [
and Theorem 
] or [
]), and along with the results of Lieberman in [
] it holds that 
. This fact combined with 
in 
yields 
, where 
denotes the interior of the positive cone 
in the Banach space 
, given by
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 [
joining 
and 
provided that 
is above the curve 
. The functional 
on 
is given by
belongs to 
, there exists a variational characterization of the second eigenvalue of (1.4) meaning that 
can be represented as
is a constant. As pointed out in [
of problem (1.9) which is required for the construction of sub- and supersolutions of problem (1.1).
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.
is bounded on bounded sets.
such that
is bounded on bounded sets.
satisfies the condition
in 
, where 
is a positive constant and 
.
be above the first nontrivial curve 
of the Fu
ik spectrum constructed in [
ik spectrum
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 [
and 
be given by
is called a weak solution of (1.1) if the following holds:
is called a subsolution of (1.1) if the following holds:
is called a supersolution of (1.1) if the following holds:
denotes all nonnegative functions of 
. Furthermore, for functions 
satisfying 
, we have the relation 
, where 
stands for the well-known trace operator.
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.
. Then there exist constants 
such that 
and 
are a positive supersolution and a negative subsolution, respectively, of problem (1.1).
with a positive constant 
to be specified and considering the auxiliary problem (1.9), we obtain
, we have to show that the following inequality holds true meaning:
with 
. Condition (H1)(f2) implies the existence of 
such that
such that
such that
. Applying (3.8) to the second integral in (3.2) ensures that
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:
may be sub- and supersolution of (1.1). More precisely, we have the following result.
, then for 
sufficiently small and any 
the function 
is a positive subsolution of problem (1.1).
provided that the inequality
. With regard to hypothesis (H1)(f4), we obtain, for 
,
denotes the usual supremum norm. Thanks to condition (H2)(g1), there exists a number 
such that
we get
guarantees that 
is a positive subsolution.
, then for 
sufficiently small and any 
the function 
is a negative supersolution of problem (1.1).
and a negative pair 
of sub- and supersolutions of problem (1.1) provided that 
is sufficiently small.
and 
, respectively. We also point out that 
is both a subsolution and a supersolution because of the hypotheses (H1)(f1) and (H2)(g1).
. If 
(resp., 
) is a solution of problem (1.1) satisfying 
in 
, then it holds that 
(resp., 
).
be a solution of (1.1) satisfying 
. We directly obtain the 
-boundedness, and, hence, the regularity results of Lieberman in [
] imply that 
with 
. Due to assumptions (H1)(f1), (H1)(f3), (H2)(g1), and (H2)(g3), we obtain the existence of constants 
satisfying
is a positive constant. We set 
for all 
and use Vázquez's strong maximum principle (cf., [
. 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 
.
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.
. 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 [
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
satisfying 
.
is a solution of problem (1.1).
and 
, we obtain the boundedness of 
in 
and 
, respectively. Definition 2.2 holds, in particular, for 
and 
which results in
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 
solves problem (1.1), one obtains, for all 
,
in (3.22) results in
-property of 
on 
along with (3.21),
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).
.
. Let us assume that this assertion is not valid meaning that 
. From (3.19) it follows that
is bounded in 
which ensures the existence of a weakly convergent subsequence of 
, denoted again by 
, such that
belonging to 
. In addition, we may suppose that there are functions 
such that
the following variational equation:
in the last equality to get
and 
, respectively, which allows us to apply Lebesgue's dominated convergence theorem. This fact and the convergence properties in (3.28) show that
-property of 
corresponding to 
implies that
, which means that 
. Applying (3.26) and (3.36) along with conditions (H1)(f1), (H2)(g1) to (3.30) provides
is the eigenfunction with respect to the eigenvalue 
. As 
is nonnegative in 
, we get a contradiction to the results of Martínez and Rossi in [
] because 
must change sign on 
. Hence, 
. Applying Lemma 3.4 yields 
.
is the smallest positive solution of (1.1) in 
.
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.
and 
denoted by 
and 
, respectively. Now, we introduce truncation functions 
and 
as follows:
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
. 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.
and 
be the extremal constant-sign solutions of (1.1). Then the following hold.
of 
is a nonnegative solution of (1.1) satisfying 
.
of 
is a nonpositive solution of (1.1) satisfying 
.
of 
is a solution of (1.1) satisfying 
.
be a critical point of 
meaning 
. We have for all 
is a positive solution of (1.1), it satisfies
provide
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.
-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 [
]. This result reads as follows.
is a local 
-minimizer of 
meaning that there exists 
such that
is a local minimizer of 
in 
meaning that there exists 
such that
, and 
.
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 
.
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
and applying assumption (H1)(f4), inequality (4.8) along with the Steklov eigenvalue problem in (1.4) implies 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.
has a global minimizer 
which is a nontrivial solution of (1.1) satisfying 
.
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 
.
.
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
. Now, we may apply the Mountain-Pass Theorem to 
(cf., [
satisfies the Palais-Smale condition. This yields the existence of 
satisfying 
and
and 
. Hence, 
is a sign-changing solution provided that 
. We have to show that 
which is fulfilled if there exists a path 
such that
, where 
, and 
be equipped with the topologies induced by 
and 
, respectively. Furthermore, we set
satisfying 
provided that 
is above the curve 
of hypothesis (H3). Recall that the functional 
is given by
such that
is dense in 
, which implies the density of 
in 
. Thus, a continuous path 
exists such that
in 
ensures the existence of 
such that
. Thus, for every 
and any bounded neighborhood 
of 
in 
there exist positive numbers 
and 
satisfying
, for all 
, and for all 
. Using (5.10) along with a compactness argument implies the existence of 
such that
, for all 
, and for all 
. Representing 
in terms of 
, we obtain
and all 
such that
such that 
and using (5.14) provide
joining 
and 
. In order to construct continuous paths 
connecting 
and 
, respectively, 
and 
, we first denote 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., [
. This ensures the existence of a continuous mapping 
satisfying the following properties:
, for all 
,
, for all 
,
, for all 
and for all 
.
given by 
for all 
which is obviously continuous in 
joining 
and 
. Additionally, one has
. We get a continuous path 
connecting 
and 
such that
, 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.
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