1 Introduction and main results
In this note, we consider the cubic nonlinear Schrödinger equation
where is a Riemannian manifold of dimension 4, is the Laplace–Beltrami operator and , .
For , where is the standard metric on the sphere , we obtain the following result.
Assume that and is constant. Then there exists a family of positive solutions to (1.1) such that as .
It is also interesting to mention that in the case where and on (or, more generally, on a general closed manifold, where is the scalar curvature), Druet  obtained a compactness result for families of positive solutions of (1.2) with bounded energies, i.e., such that for some constant C independent of ε. Theorem 1.1, together with the result of Chen, Wei and Yan  in dimensions , shows that the energy assumption in Druet’s result is necessary at least in the case of the standard sphere.
In the case where and , the positive solutions of (1.2) have been classified by Obata  (see also ). In this case, the solutions are not bounded in but they all have the same energy. We refer to [2, 3, 10] and the references therein for results on the set of solutions of (1.2) for and . On the other hand, for on a general closed manifold, Druet  obtained pointwise a priori bounds on the set of positive solutions of (1.2). Note that if, moreover, is constant, then is the unique positive solution of (1.2), see . We refer to the books of Druet, Hebey and Robert , and Hebey  for more results on equations of type (1.1) on a closed manifold.
As in , we obtain Theorem 1.1 by proving a more general result for the case , where is the Euclidean metric on . We let be the completion of the set of smooth functions with compact support in with respect to the norm . For simplicity, we will use the notation , and . We say that the operator is coercive in if
for some constant . We have the following result.
Assume that and that is radially symmetric about the point 0. Assume, moreover, that the operator is coercive in and the function has a strict local maximum point such that . Then there exists a family of positive solutions in of (1.1) such that as .
The proof of Theorem 1.2 relies on a Lyapunov–Schmidt-type method, as in . This method for constructing solutions with infinitely many peaks was invented and successfully used in previous works by Wang, Wei and Yan [13, 14] and Wei and Yan [15, 16, 17, 18]. A specificity in our case is that the number of peaks in the construction behaves as a logarithm of the peak’s height, while it behaves as a power of the peak’s height in the higher dimensional case (see ). Due to this logarithm behavior, we need to introduce some suitable changes of variables in order to find the critical points of the reduced energy in this case (see the proof of Theorem 1.2 at the end of Section 2).
for all and . Assuming that the operator is coercive in , we can equip with the inner product
and the norm
For any and , we define
Moreover, we define
First, in Proposition 2.1 below, we solve the equation
where is the unknown function, is the orthogonal projection of onto and, as usual, for all .
We will prove the following result in Section 3.
Let be a radially symmetric function about the point 0 and such that the operator is coercive in . Then, for any , with and , there exist constants and such that for any , and , there exists a unique solution of (2.1) satisfying
Moreover, the map is continuously differentiable and if there exists a critical point of the function
then the function is a positive solution in of the equation
Then we will prove the following result in Section 4.
Let be a radially symmetric function about the point 0 and such that the operator is coercive in . Then there exist constants such that for any , with and , we have
uniformly for and , where is as in Proposition 2.1.
Proof of Theorem 1.2.
Since and is a strict local maximum point of the function , we obtain that there exists such that
For any and , we define . By applying Proposition 2.2, we obtain
uniformly for in compact subsets of . Note that the function
attains its maximal value at the point
for all and . We define
By using (2.5), we obtain that there exists such that
for all . Since , from (2.6) it follows that
uniformly for . Since and , from (2.8) it follows that
as , uniformly for . From (2.9)–(2.11) it follows that the function has a local maximum point for large k. We then obtain , and so, by applying the second part of Proposition 2.1, we obtain that the function is a positive solution of equation (2.3). Moreover, by using (2.2) together with the definition of , we easily obtain
This ends the proof of Theorem 1.2. ∎
Proof of Theorem 1.1.
By using a stereographic projection, we can see that equation (1.1) on is equivalent to the problem
3 Proof of Proposition 2.1
In this section we prove Proposition 2.1. Throughout this section, we assume that is radially symmetric about the point 0, and that the operator is coercive in .
We rewrite (2.1) as
First, we obtain the following result.
For any , with and , there exist constants and such that for any , and , is an isomorphism from to itself and
The proof of this result follows the same lines as in . ∎
We then estimate the error term . We obtain the following result.
For any , with and , there exist constants and such that
for all , , and .
For any , by integrating by parts, we obtain
By using Hölder’s inequality and Sobolev’s inequality, from (3.2) it follows that
We start with estimating the first term in (3.3). For any , we define
We then write
We observe that if , then
for all . For any , with , by using (3.4), we obtain
From (3.7) it follows that
Now, we estimate the second term in (3.4). Since , by applying Hölder’s inequality and straightforward estimates, we obtain
Proof of Proposition 2.1.
for all , and . By integrating by parts and using Hölder’s inequality, Sobolev’s inequality and straightforward estimates, we obtain
for all , and . Now, we prove that if k is large enough, then is a contraction map from to itself, i.e.,
for some constant . From Lemma 3.1 it follows that
By integrating by parts and using Hölder’s inequality, Sobolev’s inequality and (3.14), we obtain
From (3.19) it follows that
uniformly for , and . We then obtain (3.17) by putting together (3.18) and (3.20). From (3.16) and (3.17), it follows that there exists a constant such that for any , and , there exists a unique solution of (2.1). The continuous differentiability of is standard.
From (3.21) it follows that
For any and , direct calculations yield
where is a constant and if , and if . Moreover, since , we obtain
Similarly, we obtain
By using the coercivity of the operator in , we obtain that a.e. in . Then, from standard elliptic regularity theory and the strong maximum principle, it follows that is a strong positive solution in of (2.3). ∎
4 Proof of Proposition 2.2
In this section we prove Proposition 2.2. Throughout this section we assume that is radially symmetric about the point 0, and that the operator is coercive in . First, we obtain the following result.
There exist constants such that for any , with and , we have
uniformly for and .
By integrating by parts, we obtain
Direct calculations yield
uniformly for and . Moreover, straightforward estimates give
Proof of Proposition 2.2.
By integrating by parts, we obtain
This paper was written during the second author’s Ph.D. He wishes to express his gratitude to his supervisors Pengfei Guan and Jérôme Vétois for their support and guidance.
S. Brendle, Blow-up phenomena for the Yamabe equation, J. Amer. Math. Soc. 21 (2008), no. 4, 951–979. Google Scholar
L. A. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math. 42 (1989), no. 3, 271–297. CrossrefGoogle Scholar
W. Chen, J. Wei and S. Yan, Infinitely many solutions for the Schrödinger equations in with critical growth, J. Differential Equations 252 (2012), no. 3, 2425–2447. Google Scholar
O. Druet, Compactness for Yamabe metrics in low dimensions, Int. Math. Res. Not. IMRN (2004), no. 23, 1143–1191. Google Scholar
O. Druet, E. Hebey and F. Robert, Blow-Up Theory for Elliptic PDEs in Riemannian Geometry, Math. Notes 45, Princeton University Press, Princeton, 2004. Google Scholar
E. Hebey, Compactness and stability for nonlinear elliptic equations, Zur. Lect. Adv. Math., European Mathematical Society, Zürich, 2014. Google Scholar
J. Wei and S. Yan, Infinitely many positive solutions for the nonlinear Schrödinger equations in , Calc. Var. Partial Differential Equations 37 (2010), no. 3–4, 423–439. Google Scholar
J. Wei and S. Yan, Infinitely many solutions for the prescribed scalar curvature problem on , J. Funct. Anal. 258 (2010), no. 9, 3048–3081. Google Scholar
J. Wei and S. Yan, On a stronger Lazer–McKenna conjecture for Ambrosetti–Prodi type problems, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 9 (2010), no. 2, 423–457. Google Scholar
J. Wei and S. Yan, Infinitely many positive solutions for an elliptic problem with critical or supercritical growth, J. Math. Pures Appl. (9) 96 (2011), no. 4, 307–333. CrossrefWeb of ScienceGoogle Scholar
About the article
Published Online: 2017-08-18
The first author was supported by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada. The second author was supported by the China Scholarship Council.
Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 715–724, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2017-0085.
© 2019 Walter de Gruyter GmbH, Berlin/Boston. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0