Abstract
We consider a semilinear fourthorder elliptic equation with a righthand side nonlinearity which exhibits an asymmetric growth at +∞ and at −∞. Namely, it is linear at −∞ and superlinear at +∞. Combining variational methods with Morse theory, we show that the problem has at least two nontrivial solutions, one of which is negative.
Keywords:
fourthorder elliptic boundary value problems; multiple solutions; critical groups; Morse theory1 Introduction
Consider the following Navier boundary value problem:
where
The conditions imposed on
(H_{1})
(H_{2}) there exist
(H_{3})
(H_{4}) there exist
and
(H_{5}) there exist
the first and the last inequality are strict on sets (not necessary the same) of positive measure, and
In view of the conditions (H_{3}) and equation (3) in (H_{4}), it is clear that for all
is
where
with asymmetric nonlinearities by using the Fučík spectrum of the operator
To the authors’ knowledge, there seem to be few results on problem (1) when
This fourthorder semilinear elliptic problem can be considered as an analogue of
a class of secondorder problems which have been studied by many authors. In [12], there was a survey of results obtained in this direction. In [13], Micheletti and Pistoia showed that
In this article, under the guidance of [8], we consider multiple solutions of problem (1) with the asymmetric nonlinearity by using variational methods and Morse theory.
2 Main result and auxiliary lemmas
Let us now state the main result.
Theorem 2.1Assume conditions (H_{1})(H_{5}) hold. If
Lemma 2.2Under the assumptions of Theorem 2.1, thenIsatisfies the (PS) condition.
Proof Let
where
Without loss of generality we can assume
where
Passing to the limit we deduce from equation (11) that
for all
Now we claim that
where
for some positive constant
Therefore, if
which contradicts inequality (16). Thus
Clearly,
for all
Lemma 2.3Let
for all
Proof We claim that there exists a constant
for all
for all
for all n. It follows from the weak compactness of the unit ball of W that there exists a subsequence, say
Moreover one has
Hence we have
and
which implies that
3 Computation of the critical groups
It is well known that critical groups and Morse theory are the main tools in solving elliptic partial differential equation. Let us recall some results which will be used later. We refer the readers to the book [16] for more information on Morse theory.
Let H be a Hilbert space and
is said to be the qth critical group of I at
Let
From the deformation theorem, we see that the above definition is independent of
the particular choice of
For the convenience of our proof, we first recall two interesting results and prove two important propositions.
Proposition 3.1[18]
Under (H_{2}), if
Proposition 3.2[19]
If
Proposition 3.3If the assumptions of Theorem 2.1 hold, then
Proof Under the guidance of [8] and [18], we begin to prove this result. Let
From equations (20) and (21), we see that in order to prove the proposition, it suffices to show that
In order to prove equation (22), we proceed as follows. We define the sets
and
Consider the map
Clearly,
By equation (3) in (H_{4}), given any
Similarly, from condition (H_{3}), and by choosing
Moreover, by condition (H_{2}), we have
for some
Let
Recalling that
Using formula (4) in condition (H_{4}), we see that there exist constants
By (H_{2}) and formula (2) in condition (H_{4}), we have
for some
where C is a positive constant. Let
Then, from equation (27), we obtain
From conditions (H_{2}) and (H_{3}), we see that given
Using inequality (33), we have
for
and
Then inequality (32) implies that we can find
Moreover, the implicit function theorem implies that
By the choice of a, we have
We define the set
is a continuous deformation of
Recalling that in the first part of the proof, we established that
Combining with equation (36) leads to equation (22), which completes the proof. □
Proposition 3.4If the assumptions of Theorem 2.1 hold, then
where
Proof By condition (H_{5}), given
Since V is finite dimensional, all norms are equivalent. Thus we can find
for all
Similar to the proof of Lemma 2.3, there exists
for all
On the other hand, for given
for all
for all
From inequalities (40) and (43), we know that I has a local linking at 0. Then invoking Proposition 2.3 of Bartsch and Li [17], we obtain
4 Proof of the main result
Proof of Theorem 2.1 We consider the following problem:
where
Define a functional
where
Next, we claim that
For s small enough, it follows from inequality (45) that
and thus, by equation (44),
Since
Let
Suppose that 0 and
We know that I satisfies the (PS) condition (see Lemma 2.2). Hence choosing
(see Proposition 3.4). Because of equations (47) and (48), using Proposition 3.2, we obtain
If
If
Since
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors read and approved the final manuscript.
Acknowledgements
The authors would like to thank the referees for valuable comments and suggestions in improving this article. This study was supported by the National NSF (Grant No. 11101319) of China and Planned Projects for Postdoctoral Research Funds of Jiangsu Province (Grant No. 1301038C).
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