As is well known, the partial differential operator
is related with questions on mean curvature in Euclidean and Minkowski spaces, depending on whether the sign under the square root is or . For this reason, it is known as the mean curvature operator.
There is a wide literature about the general Dirichlet problem related with the mean curvature operator, which reads as follows:
When Ω is unbounded, the problem
is definitely less studied. In particular, up to our knowledge, problem (1.1) has been dealt with only when and f does not depend on x, in [12, 15, 16] for the Euclidean case and in [2, 1, 8] for the Minkowski case, respectively.
To start with this new scope of investigation, in this paper we are interested in studying the nonautonomous prescribed mean curvature equation in a classical situation, namely when we introduce the power as a potential in the equation before the nonlinearity ; consider the following problem:
where , and .
In this situation, the PDE is presented as an Hénon-type equation where the difference with respect to the classical Hénon equation consists in the fact that the Laplacian is replaced by the mean curvature operator.
The Hénon equation drew the attention of the mathematicians in recent years because of some interesting questions related with the symmetry property of its solutions. In , for example, the equation on a ball with Dirichlet boundary conditions was showed to have nonradial solutions for large values of α in the subcritical case . In  and in the papers referenced inside, one can find a number of results concerning Hénon and Hénon-type equations in bounded domains with specific attention on the conditions on α and p which guarantee the existence of nonradial solutions in spite of nice symmetry properties both of the domain and of the associated functional.
In this paper, we will focus our attention on the research of radial solutions for (P) which, in accordance with a classical definition, we will call radial ground states. We will denote by () and () problem (P) where respectively the sign “” or the sign “” appears in the differential operator.
The main result that we provide is presented in Section 2 and consists in proving the nonexistence of radial ground states to (P) when p is less than a value . Ni and Serrin  treated a variety of problems including () with , proving among other things a nonexistence result for . We emphasize the fact that the value corresponds exactly to the value when . As the authors themselves say in [15, Remark on p. 247], in their proof they use a Pohozaev-type identity (see [15, (1.6)]) besides considerations on a-priori asymptotic decaying estimates on ground state solutions.
Differently from [15, 2], in our case the presence of a potential in the equation perturbs the identity in a way that makes it useless. We overcome this difficulty by means of a suitable exploitation of some arguments which are typical in the ODE theory, comparing the graphs of ground state and sign-changing solutions in order to achieve our conclusion by a contradiction argument. However, it is worthy of note that, even if our proof is based on different tools with respect to Ni and Serrin, we are able to obtain almost the same nonexistence result as that in  for ().
In Section 3, we briefly discuss the problem of existence of ground state solutions. We point out that variational arguments as those in  work fine also for problem (P) in the Minkowski space. This fact allows us to prove the existence of a radial ground state solution when is larger than the α-critical exponent related to 2, that is,
It is easily seen that, for every , we always have the inequality . As a consequence, our study leaves as an open issue what happens for . The problem of finding ground state solutions for the Hénon prescribed mean curvature equation in the Euclidean space seems to be more complex and, at the present, it is completely open.
Let , and . Then there exists no radial solution to problem (P).
Radial solutions of (P) can be found looking for global positive solutions of the Cauchy problem defined in for some :
such that .
By standard arguments, it is easy to prove that problem (C-xi) has a unique local solution in a right neighborhood of 0. In the sequel, every time we consider any Cauchy problem whose local solution can be continued until it touches the line , we will denote by the point such that and .
Assume the following notations: We set and define the following functions:
which are of class (respectively in and in according to wether the sign is or ). We will use the notations and H every time our arguments work fine regardless of the sign. Define also the following functions in :
We are going to present some preliminary lemmas.
Let . Then there exists such that for all there exists for which the local solution of the problem
corresponding to is sign-changing and .
Consider the following boundary value problem set in the unit ball centered in 0, denoted by :
is such that, for any , the function is a local sign-changing solution of (C-xi) with
The proof follows from Lemma 2.2 by straightforward computations. ∎
If v is a radial solution of (P), then the following assertions hold:
There exist and such that for .
(i) By contradiction, let η be a positive number such that for some and for any we have . Then, since , for any we have
Of course the inequality is false for r large enough.
(ii) We know that, for every ,
Then there exist and such that, for all ,
Now we distinguish two cases:
If we are considering , then we trivially deduce .
If we are considering , then either , and then, since , we have
or just, if needed, redefining .
In any case we can assume that for suitable positive constants C and we have
Integrating in , we get the conclusion. ∎
Assume by contradiction that w is a solution as in the statement of the lemma, and the set of points in where the graphs of w and v intersect is empty.
Set and . Since v and w are decreasing in and in , respectively, we can define the functions and , the inverses of and , respectively, the first defined into , the second into .
Observe that, since and by Lemma 2.4, we obtain
Then, for every , there exist and such that we have for any .
On the other hand, since and , we have
and then, if we have taken sufficiently small, we have for .
We deduce that, necessarily, the function has a local minimum in the interval .
On the other hand, by arguments analogous to those in the proof of [13, Lemma 3.3.1], we have that the following equations hold in the interval :
If is a critical point of , computing the previous two equations in and subtracting one from the other, we should obtain the following relation (observe that ):
where we have assumed the notation . From (2.4) we deduce that, since and by our contradiction assumption we know that , the critical point must be a maximum. ∎
Proof of Theorem 2.1.
On the other hand, for any and every ,
contradicting Lemma 2.5. ∎
3 Remark on the existence in Minkowski spaces
In , the ODE radial formulation of the problem has permitted to find infinitely many infinite energy solutions by using the Erbe–Tang identity as stated in [17, Proposition 1]. Actually, this approach does not seem so suitable when because the Erbe–Tang identity becomes apparently more involved and its application not so immediate.
On the other hand, the variational approach used in  to find infinitely many finite energy solutions when seems to be fitting also for the Hénon mean curvature equation. Indeed, as far as we can see, exactly the same arguments presented in  can be repeated for () by simply adapting the variational setting, replacing the Lebesgue spaces with the weighted Lebesgue spaces
We just wish to show how to prove the compact embedding theorem in our situation (compare with [8, Lemma 2.4]). We point out that we believe that the following theorem is an already known result of the theory of weighted Sobolev and Lebesgue spaces, and actually our proof uses standard arguments.
We will assume the following notation: for any , we define the α-critical exponent related to k by .
Let and define the Sobolev spaces and obtained as the completion of smooth compactly supported radially symmetric functions with respect to the norms
Then for every we have that is embedded compactly in for any .
We will use a well-known estimate related with radial functions in Sobolev spaces, using it in the dual role of a decaying at infinity and of a growing at zero estimate.
As is well known, since
we have that there exists such that, for , we have
for all .
As a consequence, denoting by the unit ball centered at zero and by its complement, if by (3.1), we have (hereunder the constant changes from line to line)
and if by (3.2), we have
from which we deduce the continuous inclusion of in .
Now, by standard arguments based on the application of the previous uniform decaying estimates (see [6, Theorem A.I]), we deduce that the embedding is compact. ∎
Repeating the arguments in , we can prove the following result.
If , then problem () has a solution belonging to .
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About the article
Published Online: 2018-06-14
The author is partially supported by GNAMPA 2017, Project “Metodi matematici per lo studio di fenomeni fisici nonlineari”, and by Fondo R.I.L. 2015, Project “Studio di equazioni differenziali alle derivate parziali nonlineari”, Università degli Studi della Basilicata.
Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 1227–1234, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2017-0233.
© 2019 Walter de Gruyter GmbH, Berlin/Boston. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0