Multiple Solutions for a Class of Semilinear Elliptic Equations with Nonlinear Boundary Conditions ()
Keywords:Multiple Solutions; Nonlinear Boundary Conditions; Local Linking Theorem
1. Introduction
In this paper, we investigate the multiple solutions for semilinear elliptic equation with nonlinear boundary conditions
(1.1)
where
is bounded domain with smooth boundary
and
is the outward normal derivative on
, and the function
satisfies C) ![](https://www.scirp.org/html/htmlimages\10-7401938x\412105ad-cf8c-4098-91da-fb9ad46237ee.png)
Problems of the above type have been discussed extensively. In 1902, Steklov (see [1]) studied the eigenvalue problem
(1.2)
Auchmuty (see [2]) considered the eigenvalue problem
(1.3)
where
satisfies the condition C) and proved that the eigenfunctions provide a complete orthonormal bases of certain closed subspace of
. Using sub and super-solutions method, Amann (see [3]), Mawhin and Schmitt (see [4]) obtained some existence results for the problem (1.1). However, since it is based on comparison techniques, the sub and super-method does not apply when the nonlinearities are compared with higher eigenvalues.
In this paper, using the Local Linking Theorem, we obtain multiple solutions for the problem (1.1), which the nonlinearites are compared with higher Neumann eigenvalue and the first Steklov eigenvalue.
2. Preliminaries and Main Results
Let
denote the Lebesgue space with the norm
, and
with the norm
. Obviously, the space
and the space
inner product are denoted by
.
is a Hilbert space under the standard inner product
(2.1)
with the associated norm
. As the function
satisfies the condition C), we define the weighted
inner product by
(2.2)
and the associated norm
. By Corollary 3.3 in [2], we obtain that the norm
is equivalent to the standard norm
. As the function
satisfies the condition C), by Equation (2.2), we can split
as a direct orthogonal sum.
Now, we state the Local Linking theorem introduced by [5].
Lemma 2.1 Let
is a reflexive Banach space,
with
satisfies the (PS) condition, if 1) there exists a constant
such that
,
2)
is bounded below and
then the functional
has at least two nontrivial critical points.
Proof. See Theorem 4 in [5].
For the problem (1.3), Auchmuty (see [2]) obtained that
![](https://www.scirp.org/html/htmlimages\10-7401938x\71f8a8a9-567f-430f-9cec-01a6106a0d64.png)
holds for all
, where
is the first Steklov eigenvalue for the problem (1.3). In [6], for the Neuman eigenvalue problem
![](https://www.scirp.org/html/htmlimages\10-7401938x\474e86a6-0423-4ad3-bdf1-0c534e5039db.png)
they obtain that the above problem has a sequence of real eigenvalues
(2.3)
with finite dimensional eigenspaces.
Assume that,
,
are Carathedory functions satisfying H1) There exist
and
such that
for all
and for a.e. ![](https://www.scirp.org/html/htmlimages\10-7401938x\8975c07d-d96f-4885-8c41-716045a03f2c.png)
H2) There exist
and
such that
for all
and for a.e. ![](https://www.scirp.org/html/htmlimages\10-7401938x\bb188617-6761-4a70-bfae-f959fc577f15.png)
H3) There exist
such that
![](https://www.scirp.org/html/htmlimages\10-7401938x\a16338f2-d145-4c74-8315-6d25fa7d4c6b.png)
and
uniformly for
with
.
H4) There exist a integer
and four constants
such that
![](https://www.scirp.org/html/htmlimages\10-7401938x\ea194ab8-668e-4e9d-8cb8-e777fe618639.png)
uniformly for
,
, where ![](https://www.scirp.org/html/htmlimages\10-7401938x\4f49833f-0b00-44e1-86ad-b4a66a07d16f.png)
![](https://www.scirp.org/html/htmlimages\10-7401938x\8ad0af9d-70c2-48fd-b671-c23580c9bb25.png)
![](https://www.scirp.org/html/htmlimages\10-7401938x\74259585-b2de-454c-9e48-caccdf119888.png)
Theorem 2.2 Suppose
satisfies C), and H1)-H4) hold, Then the problem (1.1) has at least two distinct nontrivial solutions.
3. The Proofs of Theorem 2.2
Now, we define the functional ![](https://www.scirp.org/html/htmlimages\10-7401938x\08979a0b-75a0-4a5b-9f9f-1eaf5203c927.png)
(3.1)
where ![](https://www.scirp.org/html/htmlimages\10-7401938x\284f9e34-7d24-4145-9f72-c019309c7208.png)
Since the function
satisfies H1),
satisfies H2), by the Sobolev embedding of ![](https://www.scirp.org/html/htmlimages\10-7401938x\e6cf47d7-64e7-4b8e-9529-eb4a25dce5e9.png)
into
, the continuity of the trace operator from
into
and the Holder inequalitywe obtain that the functional
is well defined. Moreover, by Lemma 2.1, and Lemma 4.2 in [7], we obtain that
, and
(3.2)
Furthermore, the functional
is weakly continuous, and
is compact. Let
in (3.2) and a simple computation, we obtain that the critical point of the functional
is the weak solution of the problem (1.1).
Lemma 3.1 (see [7]) Assume that the function
satisfies the condition C), H1) and H2) hold. If
is a (PS) sequence for the functional
, and
is bounded in
then
has a strongly convergence subsequence. i.e.
satisfies the (PS) condition.
Lemma 3.2 Assume that
satisfies the condition C), and H1)-H3) hold, the functional
is coercive on ![](https://www.scirp.org/html/htmlimages\10-7401938x\c6818495-1024-4f4a-9c9e-705590f513d7.png)
Proof By H3), we obtain that there exist some constants ![](https://www.scirp.org/html/htmlimages\10-7401938x\29eb1329-6710-4296-a62f-39f08de1a658.png)
and
,
such that
(3.3)
From H1), H2) and Equation (3.3), we obtain that there exist
such that
, (3.4)
Hence, we obtain that
(3.5)
Assume that
then using the continuity of the trace operator from
into
, we obtain either
or
where
is a positive constant.
Case 1 As
by Equation (3.5), we obtain
![](https://www.scirp.org/html/htmlimages\10-7401938x\aca66f72-1231-4d85-9169-03381e22a456.png)
Hence, we obtain that
is coercive on
since
.
Case2 As
we have
.
By H3), we obtain
, then
is coercive on
.
Hence we obtain that the functional
is bounded from below, and every (PS) sequence
is bounded in
. From Lemma 3.1, we obtain that
satisfies (PS) condition and is bounded from below.
The Proof of Theorem 2.2 We write
, where
and
.
Hence, we have
.
Since
is a finite dimensional space, by [2], we obtain that for given
there is a
such that
![](https://www.scirp.org/html/htmlimages\10-7401938x\36dcd460-605a-436e-8f0f-631807b0b4b3.png)
Hence, for each
, by H4), we have
![](https://www.scirp.org/html/htmlimages\10-7401938x\193618ba-8d48-48fe-a045-bd82e01e8c1c.png)
We have for
sufficiently small,
.
Therefore, we obtain that
.
On the other hand, let
and
then for every
with
, by H4), we obtain that
. (3.6)
Combining Equation (3.4), Equation (3.6), H1) and H2), we have
![](https://www.scirp.org/html/htmlimages\10-7401938x\93f52dc9-7c2d-435d-a6f3-91a33c009c7b.png)
From Equation (2.6), we have, for
sufficiently small
![](https://www.scirp.org/html/htmlimages\10-7401938x\1e3e054c-14db-4682-9030-5accb0ad9e6a.png)
Since
, we can choose
sufficiently small and
,
, such that the functional
.
By Lemma 3.2, we obtain that
satisfies (PS) condition and is bounded from below. If
then by Lemma 2.1,
possesses two nonzero critical point. From (3.3), we obtain that there exist two nontrivial weak solutions for the problem (1.1).
4. Conclusion
Using Local Linking Theorem, we obtain the existence of two nontrival weak solutions for the problem (1.1) which the nonlinearites
and
are compared with higher Neumann eigenvalue and the first Steklov eigenvalue.
Acknowledgements
This paper was supported by Shanghai Natural Science Foundation Project (No. 11ZR1424500) and Shanghai Leading Academic Discipline Project (No. XTKX2012).