In this paper, we study the existence, multiplicity and non-existence of solutions to the following semilinear degenerate elliptic system of Hamiltonian type:
where , and Ω is a bounded domain in with smooth boundary . Here is the strongly degenerate elliptic operator of the form
where , and the , , satisfy some certain conditions. This operator was first introduced by Franchi and Lanconelli , and recently reconsidered and named -Laplacians by Kogoj and Lanconelli in  under an additional assumption that the operator is homogeneous of degree two with respect to a group dilation in . We denote by Q the homogeneous dimension of with respect to the group of dilations , i.e.,
see Section 2.1 for more details. The homogeneous dimension Q plays a crucial role, both in the geometry and the functional associated to the operator .
In the case of a single equation
the existence, non-existence and regularity of weak solutions to this problem were proved in . In particular, the authors established a Pohozaev-type identity, and then used it to prove the non-existence result. Later, the existence of weak solutions to this problem was proved in  for the case of non-homogeneous Dirichlet boundary conditions, and in  for the case where the nonlinearity does not satisfy the Ambrosetti–Rabinowitz condition. See also the previous works [18, 17] for related results.
For exponents lying on or above this curve, that is,
we prove the non-existence of positive classical solutions to (1.1) in -starshaped bounded domains by establishing a new Pohozaev-type identity. This new identity turns out to be a generalization of the Pohozaev-type identity in the scalar case in . For below the critical hyperbola, we prove the existence of infinitely many weak solutions. To do this, we will use the variational method and the Fountain Theorem of Bartsch and de Figueiredo. It is natural to view a weak solution to (1.1) as a critical point of the corresponding functional
A natural energy space for problem (1.1) is the Hilbert space
However, this choice of energy space will impose a strict restriction on , namely , due to the Sobolev-type embedding
(see Proposition 2.1 below). To overcome this difficulty, following the approach introduced in [7, 9], we will use the fractional Sobolev-type spaces defined by using Fourier expansions on the eigenfunctions of (see Section 2 for details). One notes that now one of the nonlinearities may have a larger growth than provided the other nonlinearity has a suitably lower growth. Another possible way to overcome this difficulty is to use the Orlicz-space approach . It is worth noticing that the results obtained in this paper are the generalizations of the corresponding results for the Laplace operator in [7, 3, 15, 13, 9, 19].
In this paper, to simplify the exposition, we only state the theorems and give the proofs for the “model problem” (1.1), although these results can be extended to a slightly more general system of the form
under some suitable assumptions of f and g.
This paper is organized as follows: In Section 2, we recall some known results and prove some important embeddings which are necessary for studying our problem. In Section 3, we prove the non-existence of positive classical solutions to (1.1) by establishing a new Pohozaev-type identity. The existence of infinitely many nontrivial weak solutions to the problem is shown in Section 4 by using the variational method on fractional Sobolev-type spaces.
2 Preliminary results
2.1 The -Laplace operator and related function spaces
As in , we consider the strongly degenerate operator of the form
where , the are continuous and , , in , where
We assume the following conditions:
, , .
holds for every and , where
There exists a constant such that
and for every .
For there exists a group of dilations
where , such that is -homogeneous of degree , i.e.,
This implies that the operator is -homogeneous of degree two, i.e.,
We denote by Q the homogeneous dimension of with respect to the group of dilations , i.e.,
We now recall some function spaces related to the -Laplace operator. Denote by , , the closure of in the norm
We define as the space of all functions u such that
with the norm
It is easy to see that and are Banach spaces. In particular, and are Hilbert spaces with the following inner products:
The following result was established in .
is continuous. Moreover, the embedding
is compact for every .
We now prove the following important result.
is continuous if .
2.2 Functional setting of the problem
We now define some functional spaces which are used to study problem (1.1).
We consider the operator
where with the homogeneous Dirichlet boundary condition. Then A is linear, positive, self-adjoint and has a compact inverse. Consequently, there exists an orthonormal basis of consisting of eigenfunctions , , of A with eigenvalues
We denote , with , the space with the inner product
We notice that, as a consequence of Lemma 2.2 and interpolation theorems, we have the following important embeddings which play an important role for our investigation.
are continuous if and , respectively. Moreover, these embeddings are compact if the corresponding inequalities are strict.
By Lemma 2.2 and since , we obtain
On the other hand, by Proposition 2.1, we have
Hence, by interpolation, the injection
is continuous. Moreover, this embedding is compact if
This completes the proof of the first statement. The second one is proved similarly. ∎
For such that , we consider a Hilbert space with the inner product
and the bilinear form
It is easy to see that B is symmetric and continuous and there exists a self-adjoint bounded linear operator with
Associated to B and L we define the quadratic form by
Using arguments as in , one can prove that the operator L only has two eigenvalues , and the eigenspaces and are
Let be an orthonormal basis of . Thus is also an orthonormal basis of . We denote
We can check that E is presented as a direct sum , , and
Therefore, for every , we have and . This implies that
Now we define the functional associated to the problem (1.1) by
One can check that Φ is well-defined on E and with
One also sees that the critical points of Φ are the weak solutions of problem (1.1) in the following sense.
We say that is a weak solution of (1.1) if
2.3 Some results of abstract critical points theory
Let X be a Hilbert space and a functional . Given a sequence of finite dimensional subspaces of X such that , , and
We say that
a sequence with , , is a -sequence if
Φ satisfies at level if every sequence -sequence has a subsequence converging to a critical point of Φ.
Theorem 2.6 ().
Assume is and satisfies the following conditions:
Φ satisfies with , and .
There exists a sequence , , such that for some ,
There exists a sequence of isomorphisms , , with for all k and n , and there exists a sequence , , such that, for and , one has
where is obtained in (Φ2).
Φ is even, i.e., .
Then Φ has an unbounded sequence of critical values.
As is noticed in , condition (Φ4) holds if the functional Φ maps bounded sets in E to bounded sets in .
3 Non-existence of positive classical solutions
In this section, we prove the non-existence result for our problem when the domain Ω is -starshaped in the following sense.
Definition 3.1 ().
A domain Ω is called -starshaped with respect to the origin if and at every point of , where ν is the outward normal vector and denotes the inner product in .
As mentioned above, we consider the vector field
and this field is the generator of the group of dilation (here, a function u is -homogeneous of degree m if and only if ).
As in , we will denote by the linear space of the functions such that and , for , exist in the weak sense of distributions in Ω and can be continuously extended to . Here .
For any , we have
where T is the vector field in (3.1), ν is the outward normal to Ω and
Integrating by parts, we have
Integrating by parts in gives
Since is -homogeneous of degree , we have . Therefore, from (2.1) we get
By similar computations, we also obtain
The following theorem is the main result of this section.
Assume , and let satisfy
If Ω is bounded and -starshaped with respect to the origin, then problem (1.1) has no nontrivial nonnegative solution .
From (1.1) we have
On the other hand, integrating by parts, we get
This is equivalent to
Now, since , the condition on implies that , at any point of for any . Therefore, on we have
Substituting this identity into (3.15), we obtain
Choosing yields , and from (3.9) we get
We infer from (3.16) that
Since Ω is -starshaped, we have on ; along with , we arrive at
and from (3.16) we get and hence .
Since on , we have or at some point of . Moreover, in Ω and on . This implies that or ; hence . ∎
4 Existence of infinitely many weak solutions
If , Ω is a smooth and bounded domain in and
then problem (1.1) has infinitely many weak solutions.
We notice that, by the fact that for all , with
the functional is even, and therefore (Φ5) is satisfied. We will check conditions (Φ1)–(Φ4) in Theorem 2.6, and the proof is divided into four steps.
Step 1: Checking condition (Φ1). By using [3, Remark 2.1], it suffices to check that a -sequence in E is bounded. This is the content of the following lemma.
Step 2: Checking condition (Φ2). We claim that if and (4.1) is satisfied, then there exists a sequence , , such that for some ,
Indeed, for any we have
We prove that
Indeed, since , i.e., is decreasing and bounded from below, there exists such that
On the other hand, for every there exists such that , and
Since E is a Hilbert space, there exist and a subsequence relabeled such that
This implies that
and since the embedding is compact, we get
Therefore, we obtain , i.e., as . Thus, (4.7) is proved.
Next, for , , we have
Choosing with , , we have
Therefore, we obtain (4.6) and this implies that condition (Φ2) holds.
Step 3: Checking condition (Φ3). We prove that there exists a sequence , , such that (Φ3) is satisfied with and .
Indeed, for each , we consider being the isomorphism defined by . It is easy to see that for all .
First, with , we denote , where , , and further , and . Then we have in , and hence, using the Hölder inequality and since Ω is bounded and , we have
Now, for each , since and are finite-dimensional subspaces, all norms corresponding on and are equivalent, so there exist positive constants , , , such that
where and .
For and one has and
we see that one of the following two inequalities occurs:
On the other hand,
Therefore, for we obtain
Since and , there exists an such that for all one has . Moreover, we also obtain
This implies that, for ,
Thus we can choose to obtain
for is given. Therefore, condition (Φ3) in Theorem 2.6 is satisfied.
Step 4: Checking condition (Φ4). Let U be any bounded subset of E, i.e.,
Since , and the embeddings and are compact, we get
This proves condition (Φ4).
The proof of Theorem 4.1 is therefore complete. ∎
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About the article
Published Online: 2017-07-28
Funding Source: National Foundation for Science and Technology Development
Award identifier / Grant number: 101.02-2015.10
This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.02-2015.10.
Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 661–678, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2016-0165.
© 2019 Walter de Gruyter GmbH, Berlin/Boston. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0