- Research Article
- Open access
- Published:
Sign-Changing and Extremal Constant-Sign Solutions of Nonlinear Elliptic Neumann Boundary Value Problems
Boundary Value Problems volume 2010, Article number: 139126 (2010)
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.
1. Introduction
Let be a bounded domain with Lipschitz boundary
. We consider the following nonlinear elliptic boundary value problem. Find
and constants
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ1_HTML.gif)
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 Fuik spectrum for the
-Laplacian with a nonlinear boundary condition is defined as the set
of
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ2_HTML.gif)
has a nontrivial solution. In view of the identity
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ3_HTML.gif)
we see at once that for problem (1.2) reduces to the Steklov eigenvalue problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ4_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ5_HTML.gif)
Let us recall some properties of the Fuik 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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ6_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ7_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ8_HTML.gif)
The proof of this result is given in [26].
An important part in our considerations takes the following Neumann boundary value problem defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ9_HTML.gif)
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).
2. Notations and Hypotheses
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)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ10_HTML.gif)
(f2)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ11_HTML.gif)
(f3) is bounded on bounded sets.
(f4) There exists such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ12_HTML.gif)
(H) (g1)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ13_HTML.gif)
(g2)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ14_HTML.gif)
(g3) is bounded on bounded sets.
(g4) satisfies the condition
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ15_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ16_HTML.gif)
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ17_HTML.gif)
Definition 2.3.
A function is called a subsolution of (1.1) if the following holds:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ18_HTML.gif)
Definition 2.4.
A function is called a supersolution of (1.1) if the following holds:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ19_HTML.gif)
We recall that denotes all nonnegative functions of
. Furthermore, for functions
satisfying
, we have the relation
, where
stands for the well-known trace operator.
3. Extremal Constant-Sign Solutions
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ20_HTML.gif)
In order to satisfy Definition 2.4 for , we have to show that the following inequality holds true meaning:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ21_HTML.gif)
where with
. Condition (H1)(f2) implies the existence of
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ22_HTML.gif)
and due to (H1)(f3), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ23_HTML.gif)
Hence, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ24_HTML.gif)
Because of hypothesis (H2)(g2), there exists such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ25_HTML.gif)
and thanks to condition (H2)(g3), we find a constant such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ26_HTML.gif)
Finally, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ27_HTML.gif)
Using the inequality in (3.5) to the first integral in (3.2) yields
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ28_HTML.gif)
which proves its nonnegativity if . Applying (3.8) to the second integral in (3.2) ensures that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ29_HTML.gif)
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ30_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ31_HTML.gif)
Definition 2.3 is satisfied for provided that the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ32_HTML.gif)
is valid for all . With regard to hypothesis (H1)(f4), we obtain, for
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ33_HTML.gif)
where denotes the usual supremum norm. Thanks to condition (H2)(g1), there exists a number
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ34_HTML.gif)
In case we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ35_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ36_HTML.gif)
Applying (3.17) to (1.1) provides
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ37_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ38_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ39_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ40_HTML.gif)
Since solves problem (1.1), one obtains, for all
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ41_HTML.gif)
Setting in (3.22) results in
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ42_HTML.gif)
Using (3.21) and the hypotheses (H1)(f3) as well as (H2)(g3) yields
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ43_HTML.gif)
which provides, by the -property of
on
along with (3.21),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ44_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ45_HTML.gif)
We set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ46_HTML.gif)
It is clear that the sequence is bounded in
which ensures the existence of a weakly convergent subsequence of
, denoted again by
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ47_HTML.gif)
with some function belonging to
. In addition, we may suppose that there are functions
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ48_HTML.gif)
With the aid of (3.22), we obtain for the following variational equation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ49_HTML.gif)
We select in the last equality to get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ50_HTML.gif)
Making use of (3.17) in combination with (3.29) results in
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ51_HTML.gif)
and, respectively,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ52_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ53_HTML.gif)
From (3.28), (3.31), and (3.34) we infer that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ54_HTML.gif)
and the -property of
corresponding to
implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ55_HTML.gif)
Remark that , which means that
. Applying (3.26) and (3.36) along with conditions (H1)(f1), (H2)(g1) to (3.30) provides
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ56_HTML.gif)
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.
4. Variational Characterization of Extremal Solutions
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ57_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ58_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ59_HTML.gif)
As is a positive solution of (1.1), it satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ60_HTML.gif)
Subtracting (4.4) from (4.3) and setting provide
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ61_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ62_HTML.gif)
then is a local minimizer of
in
meaning that there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ63_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ64_HTML.gif)
Choosing and applying assumption (H1)(f4), inequality (4.8) along with the Steklov eigenvalue problem in (1.4) implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ65_HTML.gif)
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
.
5. Existence of Sign-Changing Solutions
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ66_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ67_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ68_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ69_HTML.gif)
Let , where
, and
be equipped with the topologies induced by
and
, respectively. Furthermore, we set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ70_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ71_HTML.gif)
This implies the existence of such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ72_HTML.gif)
It is well known that is dense in
, which implies the density of
in
. Thus, a continuous path
exists such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ73_HTML.gif)
The boundedness of the set in
ensures the existence of
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ74_HTML.gif)
Theorem 3.5 yields that . Thus, for every
and any bounded neighborhood
of
in
there exist positive numbers
and
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ75_HTML.gif)
for all , for all
, and for all
. Using (5.10) along with a compactness argument implies the existence of
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ76_HTML.gif)
for all , for all
, and for all
. Representing
in terms of
, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ77_HTML.gif)
In view of (5.11) we get for all and all
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ78_HTML.gif)
Due to hypotheses (H1)(f1) and (H2)(g1), there exist positive constants such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ79_HTML.gif)
Choosing such that
and using (5.14) provide
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ80_HTML.gif)
Applying (5.15) to (5.13) yields
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ81_HTML.gif)
We have constructed a continuous path joining
and
. In order to construct continuous paths
connecting
and
, respectively,
and
, we first denote that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ82_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ83_HTML.gif)
Similarly, the Second Deformation Lemma can be applied to the functional . We get a continuous path
connecting
and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F139126/MediaObjects/13661_2009_Article_897_Equ84_HTML.gif)
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.
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Winkert, P. Sign-Changing and Extremal Constant-Sign Solutions of Nonlinear Elliptic Neumann Boundary Value Problems. Bound Value Probl 2010, 139126 (2010). https://doi.org/10.1155/2010/139126
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DOI: https://doi.org/10.1155/2010/139126