In this paper we deal with the existence of a ground state solution for the semilinear elliptic problem
where either and or G is a bounded domain in and . In both cases, and is a continuous function satisfying the following condition:
There exist a bounded set , with positive N-dimensional Lebesgue measure, and positive constants , and δ such that
and is the critical Sobolev exponent.
The energy functional associated with (P) is given by
and the corresponding Nehari manifold is defined by
Our goal in this paper is to obtain a critical point u of I satisfying in G and
We refer to such a critical point as a ground state solution of (P).
There are several works in the literature dealing with semilinear problems in the particular case . Let us mention some of them.
In , Pohozaev showed that the problem
does not admit a nontrivial solution if , provided that the bounded domain G is strictly star-shaped with respect to the origin in , .
In , Brezis and Nirenberg proved the following results: if , problem (P1) has a positive solution for every , where denotes the first eigenvalue of ; if , there exists such that for any problem (P1) admits a positive solution. In the particular case where G is a ball, they proved that a positive solution exists if, and only if, and also that when there exists such that (P1) does not have a solution for .
In , Coron proved that if there exist such that
and the ratio is sufficiently large, then problem (P1) with has a positive solution in .
Existence results for (P1) related to the topology of G were also obtained by Bahri, in . In , Carpio, Comte and Lewandowski obtained nonexistence results for (P1), with , in contractible non-star-shaped domains.
On the other hand, the subcritical problem
has an unbounded set of solutions in (see ).
Problem (P2) with () was studied in the papers  and . In the former, Atkinson and Peletier considered G a ball and determined the exact asymptotic behavior of the corresponding (radial) solutions , as . In , where a general bounded domain G was considered, Garcia Azorero and Peral Alonso provided an alternative for the asymptotic behavior of , as , where denotes the energy functional associated with problem (P2) and . More precisely, they showed that if
where S denotes the Sobolev constant, then converges to either a Dirac mass or a solution of the critical problem
We recall that the Sobolev constant is defined by
and given explicitly by the expression
(and also by translations, rescalings and scalar multiples of it). This function also satisfies
and is a ground state solution of problem (P) with and .
In , Kurata and Shioji studied the compactness of the embedding for a bounded domain G and a variable exponent . (For the definition and properties of see ). They showed the existence of a positive solution of (P) under the hypothesis of existence of a point , a small , and such that and
In , Alves and Souto studied the existence of nonnegative solutions for the equation
where the variable exponents and are radially symmetric functions satisfying , and
for constants .
where the constant p belongs to and has nonempty interior.
In Section 2, motivated by the results of , we use the concentration-compactness lemma by P. L. Lions and properties of the Nehari manifold to prove the existence of at least one ground state for problem (P) when and is a function satisfying condition (H1). A key point in the proof of our existence result is the achievement of the strict inequality
and we get this by exploring the “projection” on the Nehari manifold of the sequence , where and is the Nth coordinate vector.
In Section 3, we study the case where G is a bounded domain in . In this case, the argument based on the sequences of translations of the Aubin–Talenti function is not applicable. Thus, in order to achieve the inequality (1.3) we assume an additional hypothesis (H2) that is stated in terms of a subdomain U of Ω and the value
where the function is given by
and denotes the best constant of the embedding , that is,
There exists a subdomain U of Ω such that
We remark that the constant q in the statement of (H2) can be any lower bound for in the interval . Considering that the particular value is available for several domains (especially those with some kind of symmetry), we derive two lower bounds for in terms of , S and and a third, , depending only on S and . Moreover, we present sufficient conditions for to hold, when the subdomain U is either a ball or an annular-shaped domain , with . We also show that if R and are sufficiently large, then for and , respectively.
2 The semilinear elliptic problem in
In this section, we consider the semilinear elliptic problem with variable exponent
where and is a continuous function verifying hypothesis (H1).
We recall that the space is the completion of with respect to the norm
The dual space of will be denoted by .
The energy functional associated with (2.1) is given by
where . Hence, under hypothesis (H1), we can write
For a posterior use, let us estimate the second term in the above expression. For this, let and consider the set . Then
We observe from (2.2) that the functional I is well defined.
The next lemma establishes that I is of class . Since its proof is standard, it will be omitted.
Let a function satisfying (H1a). Then and
The previous lemma ensures that is a weak solution of (2.1) if, and only if, u is a critical point of I (i.e. ). We remark that a critical point u of I is nonnegative, since
where . Consequently, according to the Strong Maximum Principle, if is a critical point of I, then in .
2.1 The Nehari manifold
In this subsection we prove some properties of the Nehari manifold associated with (2.1), which is defined by
Of course, critical points of I belong to .
We say that is a ground state solution for (2.1) if and , where
In the sequel we show important properties involving the Nehari manifold, which are crucial in our approach.
Assume that (H1) holds. Then .
For an arbitrary we have
Thus, it follows from (2.2) that
where and denote positive constants that do not depend on u. Consequently,
from which we conclude that there exists such that
In view of (2.3), this implies that . ∎
Assume (H1). Then, for each with , there exists a unique such that .
We note that
Since , we have
Thus, we can see that for all sufficiently small and also that for all sufficiently large. Therefore, there exists such that
showing that .
In order to prove the uniqueness of , let us assume that satisfy . Then
Since for all , the above equality leads to the contradiction . ∎
For we have
according to (2.3). ∎
Since m is the minimum of I on , Lagrange multiplier theorem implies that there exists such that . Thus
According to the previous proposition, , and so, .
The next proposition shows that, under (H1), there exists a Palais–Smale sequence for I associated with the minimum m.
Assume (H1). There exists a sequence such that: in , and in .
According to the Ekeland variational principle (see [21, Theorem 8.5]), there exist and such that
It follows from (2.4) that
This implies that is bounded in . Hence, taking into account that
Using the fact that , we conclude from Proposition 2.6 that . Consequently, in .
We affirm that the sequence satisfies and in . Indeed, since
Now, let us fix such that . Using the fact that and , a simple computation gives
This proves the proposition. ∎
The next proposition provides a special upper bound for m.
where is the Aubin–Talenti function given by (1.2), which satisfies
A direct computation shows that and . Moreover, exploring the expression of w, we can easily check that uniformly in bounded sets and, therefore,
for any .
By Proposition 2.5, there exists such that , which means that
and then, by using (2.5) for , we can verify that the sequence is bounded,
Moreover, since ,
where we have used that the maximum value of the function is .
Combining the boundedness of the sequence with the fact that uniformly in , we can select k sufficiently large, such that in . Therefore, for this k,
since the latter integrand is strictly negative in Ω, which has positive N-dimensional Lebesgue measure. ∎
2.2 Existence of a ground state solution
Our main result in this section is the following.
We prove this theorem throughout this subsection by using the following well-known result.
Lemma 2.11 (Lions’ lemma ).
Let be a sequence in , , satisfying
Then there exist an at most countable set of indices , points and positive numbers such that
for any ,
where denotes the Dirac measure supported at .
We know from Proposition 2.8 that there exists a sequence satisfying in , and in . Since is bounded in , we can assume (by passing to a subsequence) that there exists such that in , in for and a.e. in . Moreover, and in .
We claim that . Indeed, let us suppose, by contradiction, that . We affirm that this assumption implies that the set given by Lions’ lemma is empty. Otherwise, let us fix , and as in Lions’ lemma. Let such that
and for all , where and denotes the balls centered at the origin, with radius 1 and 2, respectively.
For fixed, define
Since is bounded in , the same holds for the sequence . Thus,
According to Lions’ lemma,
it follows from (2.6) that
Now, making , we get
Combining this inequality with part (ii) of Lions’ lemma, we obtain . It follows that
Let such that and , for any . Recalling that
Since is continuous, for each , there exists such that
where . Thus,
where , for . Then, since in , for , and ϵ is arbitrary, we conclude that
Therefore, by making in (2.7), we obtain
which contradicts Proposition 2.9, showing that . Hence, it follows from Lions’ lemma that
In particular, in , so that
Since , we have
Thus, by making in the equality
we obtain , which contradicts Proposition 2.9 and proves that .
Now, combining the weak convergence
with the fact that in , we conclude that
meaning that u is a nontrivial critical point of I.
the weak convergence in and Fatou’s lemma imply that
showing that .
3 The semilinear elliptic problem in a bounded domain
In this section we consider the elliptic problem
We recall that the usual norm in is given by
We denote the dual space of by .
The energy functional associated with problem (3.1) is defined by
It belongs to and its derivative is given by
Thus, a function is a weak solution of (3.1) if, and only if, u is a critical point of I. Moreover, as in Section 2, the nontrivial critical points of I are positive in G (a consequence of the Strong Maximum Principle).
We maintain the notation of Section 2. Thus,
the Nehari manifold associated with (3.1) is defined by
We say that is a ground state solution for (3.1) if and .
We gather in the next lemma some results that can be proved as in Section 2.
Assume (H1). We claim that:
for all . (Thus, for all .)
If , then . (Thus, is a positive weak solution of ( 3.1 ).)
There exists a sequence such that in G, and in .
Unfortunately, hypothesis (H1) by itself is not sufficient to guarantee that as in Proposition 2.9. The reason is that the translation argument used in the proof of that Proposition does not apply to a bounded domain. So we assume an additional assumption (H2). In order to properly state such an assumption, we need some background information.
Let be a bounded domain and define
It is well known that if , then the infimum in (3.2) is attained by a positive function in . Actually, this follows from the compactness of the embedding .
Another well-known fact is that in the case the infimum in (3.2) coincides with the best Sobolev constant, i.e.
Moreover, in this case the infimum (3.2) is not attained if U is a proper subset of .
In the sequence we make use of the function
If , then there exists such that
is continuous (in fact it is α-Hölder continuous, for any , as proved in ) and the function
is strictly decreasing. It follows that
This latter inequality implies that
Hence, using that , we see that
Taking into account that , we can easily check that . Thus, defining
we arrive at (3.4). ∎
There exists a subdomain U of Ω such that
Let denote a positive extremal function of . Thus, in U and
Let us define the function by
For each we have
it is easy to see that and
Since and , it follows from Lemma 3.3 that
This implies that . ∎
The main result in this section is the following.
According to item (iv) of Lemma 3.2, there exists a sequence satisfying and in . Since is bounded in , there exist and a subsequence, still denoted by , such that in , in , for , and a.e. in G. Arguing as in Section 2, we can combine Lions’ lemma and Lemma 3.4 to prove that , and , showing thus that u is a ground state solution of (3.1). ∎
3.1 On hypothesis (H2)
In this subsection we present some lower bounds for the value of , defined by (3.6), which can be used as the value constant for in hypothesis (H2). Moreover, we give some examples of simple bounded domains U such that .
The value of depends on the function , which in turn depends on the function . It is well known that is the least value of λ for which the Dirichlet problem
has a nontrivial weak solution. When , this is the well-studied eigenvalue problem for and is its first eigenvalue. It follows that can be found analytically for some simple domains as balls, rectangles and other domains enjoying some kind of symmetry. For instance, if U is a ball of radius R, then
where denotes the first positive root of the first kind Bessel function of order α.
When the above problem is no longer linear and, consequently, it is more difficult to be solved analytically, even for simple domains. For this reason, determining an analytical expression for the function g on the interval is a hard task and we do not know the exact value of given by (3.6). However, the inequality (3.5) allows us to derive lower bounds for , in terms of , and S, which can be used as the value constant for in hypothesis (H2). In fact, assuming , we can easily verify that for all , where the function is defined in (3.5). Therefore, taking into account that and , there exists a unique value such that , that is
an equation that can be solved at least numerically.
A rougher but explicit lower bound for follows from the inequality
which is obtained from (3.5) by observing that . Indeed, since the function enjoys the same properties as , there exists a unique point satisfying . A simple calculation yields
Of course, .
Hence, since and , there exists a unique point satisfying . Such a point is given explicitly by
Another conclusion that follows easily from the monotonicity of the function combined with (3.3) is that if , then .
In the sequel, we present sufficient conditions for the inequality to hold when U is either a ball or an annulus. We will denote by the ball centered at y with radius . When , we will write simply .
Let . Since the Laplacian operator is invariant under translations,
Moreover, a simple scaling argument (or (3.7)) yields
So, if , then .
Let , with , for some and . Since the Laplacian operator is invariant under orthogonal transformations, we can see that
for some , where denotes the first coordinate vector. According to [15, Proposition 3.2], the function is strictly decreasing for . Therefore,
Since is the largest ball contained in , we have
Thus, we can replace the condition in (H2) by either when or when .
C. O. Alves and M. A. S. Souto, Existence of solutions for a class of problems in involving the -Laplacian, Contributions to Nonlinear Analysis, Progr. Nonlinear Differential Equations Appl. 66, Birkhäuser, Basel (2006) 17–32. Google Scholar
G. Anello, F. Faraci and A. Iannizzotto, On a problem of Huang concerning best constants in Sobolev embeddings, Ann. Mat. Pura Appl. (4) 194 (2015), no. 3, 767–779. CrossrefWeb of ScienceGoogle Scholar
A. Bahri, Critical Points at Infinity in some Variational Problems, Pitman Res. Notes Math. Ser. 182, Longman Scientific & Technical, Harlow, 1989. Google Scholar
A. Bahri and J.-M. Coron, On a nonlinear elliptic equation involving the critical Sobolev exponent: The effect of the topology of the domain, Comm. Pure Appl. Math. 41 (1988), no. 3, 253–294. CrossrefGoogle Scholar
A. Carpio Rodríguez, M. Comte and R. Lewandowski, A nonexistence result for a nonlinear equation involving critical Sobolev exponent, Ann. Inst. H. Poincaré Anal. Non Linéaire 9 (1992), no. 3, 243–261. CrossrefGoogle Scholar
J.-M. Coron, Topologie et cas limite des injections de Sobolev, C. R. Acad. Sci. Paris Sér. I Math. 299 (1984), no. 7, 209–212. Google Scholar
J. P. García Azorero and I. Peral Alonso, Existence and nonuniqueness for the p-Laplacian: Nonlinear eigenvalues, Comm. Partial Differential Equations 12 (1987), no. 12, 1389–1430. Google Scholar
J. P. García Azorero and I. Peral Alonso, On limits of solutions of elliptic problems with nearly critical exponent, Comm. Partial Differential Equations 17 (1992), no. 11–12, 2113–2126. CrossrefGoogle Scholar
K. Kurata and N. Shioji, Compact embedding from to and its application to nonlinear elliptic boundary value problem with variable critical exponent, J. Math. Anal. Appl. 339 (2008), no. 2, 1386–1394. Web of ScienceGoogle Scholar
P.-L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. I, Rev. Mat. Iberoam. 1 (1985), no. 1, 145–201. Google Scholar
J. Liu, J.-F. Liao and C.-L. Tang, Ground state solutions for semilinear elliptic equations with zero mass in , Electron. J. Differential Equations 2015 (2015), Paper No. 84. Google Scholar
S. Pohozaev, Eigenfunctions of the equation , Soviet Math. Dokl. 6 (1965), 1408–1411. Google Scholar
M. Willem, Minimax Theorems, Progr. Nonlinear Differential Equations Appl. 24, Birkhäuser, Boston, 1996. Google Scholar
About the article
Published Online: 2018-08-07
Published in Print: 2019-03-01
Funding Source: Conselho Nacional de Desenvolvimento Científico e Tecnológico
Award identifier / Grant number: 304036/2013-7
Funding Source: Conselho Nacional de Desenvolvimento Científico e Tecnológico
Award identifier / Grant number: 483970/2013-1
Award identifier / Grant number: 306590/2014-0
Funding Source: Fundação de Amparo à Pesquisa do Estado de Minas Gerais
Award identifier / Grant number: APQ-03372-16
C. O. Alves was partially supported by CNPq/Brazil (304036/2013-7) and INCT-MAT. G. Ercole was partially supported by CNPq/Brazil (483970/2013-1 and 306590/2014-0) and Fapemig/Brazil (APQ-03372-16).
Citation Information: Advances in Nonlinear Analysis, Volume 9, Issue 1, Pages 108–123, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2017-0170.
© 2020 Walter de Gruyter GmbH, Berlin/Boston. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0