Research Article | Open Access
Infinitely Many Standing Waves for the Nonlinear Chern-Simons-Schrödinger Equations
We prove the existence of infinitely many solutions of the nonlinear Chern-Simons-Schrödinger equations under a wide class of nonlinearities. This class includes the standard power-type nonlinearity with exponent . This extends the previous result which covers the exponent .
In [1, 2], Jackiw and Pi introduce a nonrelativistic model that the nonlinear Schrödinger dynamics is coupled with the Chern-Simons gauge terms as follows:Here, denotes the imaginary unit, , , for , is a complex scalar field, is a component of gauge potential and is a covariant derivative for running over , and is a parameter. The Chern-Simons gauge theory appears in the 1980s to explain electromagnetic phenomena of anyon physics such as the high temperature superconductivity or the fractional quantum Hall effect. In this paper, we are interested in standing wave solutions of (1). In , the authors introduce a standing wave ansatz of the following form:where is a phase frequency and are real valued functions on such that . Inserting (2) into (1), one may check from direct computation that (1) is reduced to the following nonlinear nonlocal elliptic equation:where . See  for its derivation. It is shown in  that (3) is an Euler-Lagrange equation of a functional,where denotes the set of radially symmetric functions in standard Sobolev space . Investigating the structure of , the authors of  obtain several existence and nonexistence results for (3), depending on the range of and . Recently, Pomponio and Ruiz  improve the results in  for the case . They find a threshold for the behavior of , depending on . They also study (3) on bounded domain in .
In this paper, we are concerned with the existence of infinitely many solutions of (3). It is proved in  that if , enjoys the symmetric mountain pass geometry and satisfies the (PS) condition so that the well-known symmetric mountain pass lemma (see ) applies to show there exist infinitely many critical points of . For , it turns out that still enjoys the symmetric mountain pass geometry although checking the (PS) condition is not easy job. One aim of this paper is to show nevertheless still admits infinitely many critical points for . Moreover, we will replace the power-type nonlinearity of (3) with more general one as follows:The structure conditions for are given by the following:(V1)Consider such that .(V2)Consider , for some .(V3)There exists some such that is monotonically increasing to as .Observe that assumptions (V1)–(V3) include the power-type nonlinearity , .
Theorem 1. Assume . Then, (5) admits infinitely many solutions.
We refer to the work of Cunha et al.  that if we insert a sufficiently small parameter into (5) as inthen much more general assumptions for , the so-called Berestycki-Lions conditions , are sufficient for guaranteeing the existence and multiplicity of solutions of (6). In our work, we assume further than the Berestycki-Lions conditions but we do not need small parameter . See also  in which Tan and Wan consider asymptotically linear nonlinearities.
To prove Theorem 1, we will apply the method employed in author’s former paper  in which the Schrödinger-Poisson equation, another nonlocal field equation similar to (5), is dealt with. Instead of generating (PS) sequences, we will show the existence and compactness of the so-called approximate solution sequences of which may be considered as more refined version of (PS) sequences. In Section 2, we give a definition of the approximate solution sequences of . Some auxiliary lemmas are also prepared in Section 2. In Section 3, we prove the compactness of approximate solution sequences. In Section 4, we construct infinitely many approximate solution sequences whose energy levels go to infinity and complete the proof of Theorem 1.
2. Mathematical Settings and Preliminaries
Let be the completion of with respect to the normThe dual space of is denoted by . Arguing similarly to , it is easy to show (5) is an Euler-Lagrange equation of the functionalIn this paper, we search for infinitely many critical points of to prove Theorem 1. To do this, we insert parameter into as follows:Here ranges over . For a sequence which converges to as , we say is an approximate solution sequence of if for all . In the following subsection, we state a variant of the famous Struwe’s monotonicity trick , which plays a crucial role in constructing approximate solution sequences.
2.1. A Variant of Struwe’s Monotonicity Trick
Let be Banach space. We say a subset is symmetric if for every . Let be a compact subset of and a closed subset of . We denote by the set of every continuous odd function such that on . Let be a closed interval in and one parameter family of even functional on . We define a minimax level byThe following theorem is a variant of so-called Struwe’s monotonicity trick . A more general version of it is given in . The property below is first proposed by Jeanjean and Toland in .
Theorem 2 (see ). Suppose that, for all ,Then, for almost every , there exists a norm-bounded (PS) sequence of at level , provided the following property for holds:(H)For given , let be a sequence strictly increasing to and a sequence in such that are all uniformly bounded above for . Then the following holds:(i) is norm-bounded in .(ii)For given , there exists such that
2.2. Some Auxiliary Lemmas
Here, we prepare some lemmas which will be necessarily used for proving the main result. Define
Lemma 3 (Lemma 3.2 in ). Let be a sequence weakly converging to some in as . Then, for each , it holds that , , and as , up to a subsequence.
Lemma 4 (Pohozaev identity). Let be a critical point of . Then one has
For each and , we define one parameter family of functions byFor fixed , we define a map by . It is easy to see that is a continuous and linear map with the inverse . Thus is a linear isomorphism.
For each and , let be a function defined by
Lemma 5. For any and , admits a unique critical point on ; that is, , such that is increasing on , attains its maximum at , and is decreasing to on .
Proof. By the change of variable, one can computeWe differentiate it with respect to to getObserve from assumption (V3) that is strictly monotonically decreasing from infinity to the positive number on and is monotonically increasing to infinity on . Therefore there is such that on , , and on . Also from assumption (V3), we deduce as . This proves the proposition.
We define a function by assigning a positive number satisfying for any nonzero . The value is defined by .
Lemma 6. The function is well-defined and continuous even map on .
Proof. To show the well-definedness of , we have to show that there exists unique satisfying for given nonzero . We note that this is equivalent to prove there is a unique solution of the equationArguing similarly to the proof of Lemma 5, we are able to see that is monotonically decreasing on for some , attains its unique local minimum at , and is monotonically increasing to infinity on . Therefore there is a unique positive zero of since . Also, the implicit function theorem says that is continuous on because . The evenness of follows from the fact that each coefficient of (20) is even. This completes the proof.
3. Compactness of Approximate Solution Sequences
In this section, we prove the compactness of an approximate solution sequence of when its energy is bounded above.
Proposition 7. Let be such that as . Let be a sequence of critical points of ; that is, . Suppose that for some , independent of . Then in for some critical point of up to a subsequence.
Proof. We divide the proof into two steps.
Step 1 (boundedness of ). We first prove that is bounded in . Arguing indirectly, suppose that is unbounded. Let , where the function is defined in Section 2. Equation (20) says is unbounded. Let so that . Then, up to a subsequence, converges weakly in and strongly in for all to some . Since is a critical point of , we see thatCombining this with the Pohozaev identity (15), we obtainThen, from the change of variable and dividing by , (22) transforms toSince and is unbounded, is bounded for but the structure assumption (V3) implies that tends to infinity as provided is not identically zero. We claim that is nonzero. Suppose is identically zero. From (19) and (22), we see thatThen, Lemma 5 implies that is the global maximum of on . Thus we see that, for each ,The last equality follows from , the convergence of to in for all , and the structure conditions (V1)-(V2). However, taking large , this makes a contradiction and shows is not identically zero. This proves the boundedness of in .
Step 2 (compactness of ). Compactness of follows from a standard procedure. Since is bounded, there exists such that converges, up to a subsequence, to weakly in and strongly in for all . Then it follows from Lemma 3 that is a critical point of . Also, it is easy to see from the boundedness of that in . Recall thatUsing Lemma 3 once again, one can observe thatwhich shows as . Therefore we have in as .
4. Construction of Approximate Solution Sequences
In this section, we construct infinitely many approximate solution sequences. Choose an orthonormal basis of . For given , let and be linear subspaces of spanned by and , respectively. We will show that enjoys a variant of symmetric mountain pass geometry (see ).
Lemma 8. There exist a sequence such that as and sequences satisfying for each and(i) for all and all ;(ii) for all and all .
Proof. We first show (i). The structure assumptions (V1)-(V2) imply thatThen,We recall the function and the linear isomorphism in Section 2. Let so that . By a change of variable, we get from (29) that, for each ,whereWe claim that, for each , as . To see this, suppose that as . Choose satisfyingSince is the unit sphere of a linear subspace of with codimension , we deduce converges to weakly in and strongly in , up to a subsequence. This however contradicts the fact thatand the claim is true. We take . Then, for any satisfying ,For each , by taking sufficiently large satisfyingwe can see that the proof of (i) is complete.
Next we show (ii). Lemma 5 says that, for each , as . Also, we see from Lemma 6 and (20) that the set is closed and bounded in finite dimensional space so it is compact. Combining these two facts with the compactness of , we can deduce easily (ii) holds.
Definewith , and given in Lemma 8. Let be the set of continuous functions satisfying on . By , we denote the set .
Lemma 9 (intersection property). For any , the intersection for every .
Proof. Choose and fix arbitrary and . Definewhere denotes the interior of in . Then is a symmetric open neighborhood of since is a continuous even map by Lemma 6 and is a continuous odd map. Equation (20) in Lemma 6 says is bounded so that is also bounded. We claim that . From the continuity of and , it holds that . Suppose that there is some such that . Then there is a neighborhood of of in such that . Choose some . From the definition of , . Since , we see that . Then, from the definition of , we have , which is a contradiction. This shows the claim is true.
Now, consider a map , where is the projection map from . Then the well-known Borsuk-Ulam theorem applies to see the continuous odd map has a vanishing point ; that is, . This means that . Therefore . The proof is complete.
Now, we are ready to prove the existence of infinitely many approximate solution sequences of . For each , we define infinitely many minimax levels as follows:It follows from Lemmas 8 and 9 that for all .
Proposition 10. For every fixed , there exists an approximate solution sequence of such that .
Proof. We invoke Theorem 2. From Lemmas 8 and 9, it holds thatLet us check enjoys property . Let be a sequence strictly increasing to some and a sequence such thatWe need to show (i) is bounded in and (ii), for given , there exists satisfyingWe first show (i). We see from (40) thatwhich shows is bounded in . Also, for given ,if is sufficiently large. This shows (ii). Therefore, there exists a subset with full measure in that, for every , there exists a bounded (PS) sequence of at level . Arguing similarly to Step 2 of Proposition 7, we also deduce converges, up to a subsequence, to some critical point of with . Since has full measure in , this completes the proof.
Completion of the Proof of Theorem 1. Now we complete the proof of Theorem 1. We first choose and fix arbitrary . Let be an approximate solution sequence of , given by Proposition 10. Take satisfyingIt follows from the compactness of thatfor sufficiently large . Then Proposition 7 applies to see converges, up to a subsequence, to some which is a critical point of . Recall that . By taking a limit , we deduce . Since is arbitrary, this shows the existence of infinitely many critical points of . This completes the proof.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
The author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2014R1A1A2054805) and also was supported by the POSCO TJ Park Science Fellowship.
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