Abstract and Applied Analysis

Volume 2013 (2013), Article ID 873948, 16 pages

http://dx.doi.org/10.1155/2013/873948

Research Article

Construction of Nodal Bubbling Solutions for the Weighted Sinh-Poisson Equation

Yibin Zhang¹ and Haitao Yang²

¹College of Science, Nanjing Agricultural University, Nanjing 210095, China

²Department of Mathematics, Zhejiang University, Hangzhou 310027, China

Received 1 June 2013; Accepted 27 August 2013

Academic Editor: Marco Donatelli

Copyright © 2013 Yibin Zhang and Haitao Yang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We consider the weighted sinh-Poisson equation in , on , where is a small parameter, , and is a unit ball in . By a constructive way, we prove that for any positive integer , there exists a nodal bubbling solution which concentrates at the origin and the other -points , , such that as , , where and is an odd integer with , or is an even integer. The same techniques lead also to a more general result on general domains.

1. Introduction

We are concerned with stationary Euler equations for an incompressible, homogeneous, and inviscid fluid on a bounded, smooth planar domain , consider

where is the velocity field, is the pressure, and is the unit outer normal vector to . Let us introduce the vorticity . By applying the curl operator to the first equation in (1), we have

On the other hand, the second equation is equivalent to rewriting the velocity field as

In return, the vorticity is expressed as in term of , the so-called stream function. Now, the ansatz  guarantees that (2) is also automatically satisfied, and then the Euler equations reduce to solving the Dirichlet elliptic problem as follows

Over the past decades, some vortex-type configurations for planar stationary turbulent Euler flows have aroused wide concern among the people (see [1–3]). Many functions have been chosen in the physical perspective to describe turbulent Euler flows with vorticity concentrated in small “blobs”. For example, on the basis of the statistical mechanics approach, Joyce and Montgomery proposed the Stuart vortex pattern with a small positive parameter to describe positive vortices (see [4–8]). Meanwhile, they also proposed the Mallier-Maslowe vortex pattern to describe coexisting positive and negative vortices (see [9, 10]). Recently, Tur and Yanovsky in [11] have used the singular ansatz  to describe vortex patterns of necklace type with -fold symmetry in rational shear flow, where and denotes the Dirac mass at . Now, we adopt another new singular ansatz in [12] with to study the corresponding vortices with concentrated vorticity. To do it, we hope to investigate the effect of the presence of the weight on the existence of nodal bubbling solutions for the weighted sinh-Poisson equation as follows:

where is a small parameter, , and is a unit ball in .

Let us first recall the two-dimensional sinh-Poisson equation as follows:

which relates to various dynamics of vorticity with respect to geophysical flows, rotating and stratified fluids, and fluid layers excited by electromagnetic forces (see [13–15] and the references therein) and the geometry of constant mean curvature surfaces studied by many works (see [16–20] and the references therein). Recently, the asymptotic behavior of solutions to (6) has been studied on a closed Riemann surface in [21, 22], and the authors applied the so-called “Pohozaev identity” and “symmetrization method,” respectively, to show that there possibly exist two different types of blow up for a family of solutions to (6). Furthermore, Grossi and Pistoia in [23] exhibited sign-changing multiple blow-up phenomena for the Dirichlet problem (6), more precisely, if and is symmetric with respect to the origin, for any integer if is small enough, there exists a family of solutions to (6), which blows up at the origin, whose positive mass is and negative mass is . This gives a complete answer to an open problem in [21]. Besides, for a similar equation, precisely the Neumann sinh-Gordon equation on a unit ball, Esposito and Wei in [24] also constructed a family of solutions with a multiple blow up at the origin. On the other hand, Bartolucci and Pistoia in [25] tried to construct blow-up solutions of (6) with Dirichlet boundary condition, and proved that for small enough, there exist at least two pairs of solutions, which change sign exactly once, concentrate in the domain, and whose nodal lines intersect the boundary. Furthermore, Bartsch et al. in [26] obtained the existence of changing sign solutions for this equation on an arbitrary bounded domain , which have three and four alternate-sign concentration points. In particular, when has an axial symmetry they proved for each there exists a nodal bubbling solution, which changes sign -times and whose alternate-sign concentration points align on the symmetric axis of . For (6) with Neumann boundary condition, Wei et al. in [27] constructed a family of solutions concentrating positively and negatively in the domain and its boundary. As for the presence of the weight , the authors in [12] showed that there exists a family of nodal bubbling solutions to (5) only involving , such that not only develops many positive and negative Dirac deltas with weight and , respectively, but also a Dirac data with weight at the origin.

We mention that an analogous blow-up analysis can be applied to multiple blow-up solutions for the Liouville equation with or without singular data as follows:

where is a smooth bounded domain in , , , , defines the Dirac mass at , and is a small parameter. For the past decades, the asymptotic analysis for blow up solutions of problem (7) has been deeply studied in the vast literature (see [28–33] and the reference therein), which exhibits the quantiaztion properties of the weak limit of as if remains uniformly bounded, and characterizes the location of concentration points as critical points of a functional in terms of Green’s function. Reciprocally, an obvious problem is how to construct solutions of (7) with these properties. In [34, 35], the authors use the asymptotic analysis to construct solutions with multiple interior concentration points for (7) with and . More generally, by a constructive way, similar results related to can also be obtained in [36–40] under some milder notions of stability of critical points. In particular, when and s are positive numbers, D’Aprile in [37] recently established a family of solutions to (7) consisting of blow up points in as long as for any , provided that the weights avoid the integers , and so the result of del Pino et al. in [39] can be extended to the case of several singular sources.

In this paper, we will continue the study of the existence of solutions to (5). We prove that there exists a family of solutions concentrating positively and negatively at the origin and outside the origin as long as . Concerning the sign-changing concentration at the origin and outside the origin, if we introduce the function defined as

with , our main result for problem (5) can be stated as follows.

Theorem 1. For any positive integer , there exists a nodal solution to problem (5) which concentrates at the origin and the other -points , , such that as ,

weakly in the sense of measures in , where is an absolute minimum point of in , is an odd integer with , or is an even integer. Moreover, for any , remains uniformly bounded on , and as ,

Theorem 1 is based on a constructive way which also works for the more generally weighted sinh-Poisson equation as follows:

for small, where is a bounded smooth domain in , with , are different singular sources in , , , and is a continuous function such that for any .

To further state our results, we need to introduce some notations. Let be Green’s function of such that for , in and on , and let be its regular part defined as . Besides, let us denote , , , , , , and , where is an integer. In what follows, we fix different points , and define

which is well defined on the following domain:

where , for , for , for , and for .

Definition 2 (see [41]). We say that is a -stable critical point of in if for any sequence of functions such that uniformly on the compact subsets of , has a critical point such that .

In particular, if is a strict local maximum or minimum point of , is a -stable critical point of .

Theorem 3. Assume that and is a -stable critical point of in with . Then, for any sufficiently small , there exists different points , , away from , so that problem (11), for , has a nodal solution such that as ,

weakly in the sense of measures in . Moreover, up to a subsequence, there exists such that

Besides, remains uniformly bounded on for any , and for any points , and , as ,

Note that for the case and (or ), Theorem 3 was partly proved in [12] (or [25]) only when and , . In contrast with the results of [12, 25], this theorem provides a more complex concentration phenomenon involving the existence of changing-sign solutions for problem (11) with both positive and negative bubbles near the singular sources , and some other discrete points. Unlike the concentration set in [12] only contains singular sources with and , and no singular source points in the domain, our concentration set also contains some singular sources with and , except for singular sources , where concentration points and singular sources coincide at the limit. As for the latter exception, which till now is a similar but very simple concentration phenomenon it appears only in [38] for the study of the Liouville equation with a singular source of integer multiplicity.

In order to obtain multiple sign-changing blow up solutions of problem (11), we use a Lyapunov-Schmidt finite-dimensional reduction scheme and convert the problem into a finite-dimensional one, for a suitable asymptotic reduced energy, related to in (12). Thus, a stable critical point of leads to the existence of multiple sign-changing blow up solutions to (11). However, in view of different signs of Green’s functions in (12), it seems very difficult to find out a stable critical point of for a general bounded domain . A simple approach can help us to overcome this difficulty by imposing the very strong symmetry condition on the domain of the problem, namely, we use the symmetry of the unit ball to reduce the problem of finding solutions of (5) to that of finding an absolute minimum point of defined in (8), and so we get the existence of nodal bubbling solutions for (5) in Theorem 1. On the other hand, motivated by the obtained results in [23], we believe that Theorem 1 should be valid for a general domain than a unit ball. More precisely, we suspect that, if and is symmetric with respect to the origin, it is possible to construct a family of sign-changing blow up solutions whose maxima and minima are located alternately at the origin and the vertices of a regular polygon, and so Theorem 1 will be a consequence of this general result.

It is important to point out that to prove the above results, we need to use classification solutions of the Liouville-type equation to construct approximate solutions of problem (5) (or (11)) as follows:

In complex notations, a complete classification of the solutions of (17) takes the following form:

where , if and if (see [33, 42–44]). Using classification solutions scaled up and projected to satisfy the Dirichlet boundary condition up to a right order, the approximate solutions can be built up as a summation of these initial approximations with some suitable signs. Thus, the nodal bubbling solutions can be constructed as a small additive perturbation of these approximations through the so-called “localized energy method,” which combined the Lyapunov-Schmidt finite dimensional reduction and variational techniques. Here, we follow [12, 25], but we will overcome some of the difficulties that the nodal concentration phenomenon brings by delicate analysis.

2. Construction of the Approximate Solution

In this section, we will provide a first approximation for the solutions of problem (11). Given a sufficiently small but fixed number , let us first fix different points , , and assume that points , where

Suppose that , , are positive numbers to be chosen later, we define

The ansatz is

where , is a linear operator such that for any , in , and on . By harmonicity, we easily get that

where .

Consider that the scaling of solution to problem (11) is as follows:

where , then satisfies

We will seek solutions of problem (25) of the form , where

In terms of , problem (11) (or (25)) becomes

where

is the “error term”:

and denotes the following “nonlinear term”:

Finally, in order to make sufficiently small, we choose the parameters , as

so that from Appendix, we have

.

3. The Linearized Problem and the Nonlinear Problem

In this section, we will first prove the bounded invertibility of the linearized operator under suitable orthogonality conditions. Let us define

The key fact to develop a satisfactory solvability theory for the operator is that formally approaches under dilations and translations, and any bounded solution of in is a linear combination of , , and for (see [34, 45, 46]), or proportional to for (see [35, 36, 41]). Let us denote

where is a smooth, nonincreasing cut-off function such that for a large but fixed number , if , and if .

Additionally, we consider the following Banach space:

with the norm

where and .

Given that and , we consider the linear problem of finding a function and scalars , , , such that

Our main interest in this problem is its bounded solvability for any , uniform in small and points , as the following result states.

Proposition 4. There exist positive numbers and such that for any , there is a unique solution , scalars , , , to problem (38) for all and points , which satisfies

Moreover, the map is , precisely for , ,

where .

We begin by stating a priori estimate for solutions of (38) satisfying orthogonality conditions with respect to , , and , , .

Lemma 5. There exist positive numbers and such that for any points , and any solution to the following equation:

with the orthogonality conditions

one has

for all .

Proof. We have the following steps.

Step  1. We first construct a suitable barrier. To realize it, we claim that for small enough, there exist and

positive and uniformly bounded so that

where and . Take

Thus, we have , and

By (32),

So, if is taken small and fixed, by (47)-(48), it follows that for ,

Let be a positive, bounded solution of in and on . Set . Then, is a positive, uniformly bounded function in such that

Consider

where is a sufficiently large constant. Obviously, is a positive and uniformly bounded function. Moreover, in view of (48)–(50), it is easy to check that satisfies the estimate (45), and the claim follows.

Step  2. Consider the following “inner norm”:

Let us claim that there exists a constant , such that

Let us take

where is chosen larger if necessary. Then, for ,

for ,

and for ,

From the maximum principle, it follows that on , which provides the estimate (53).

Step  3. We prove the priori estimate (43) by contradiction. Let us assume the existence of a sequence , and points , functions with , solutions with , such that (41)-(42) hold. From the estimate (53), for some , namely, for some . To simplify the notation, let us denote and . Set and . By (32), satisfies

for . Obviously, for , in . Since is bounded in and , the elliptic regularity theory implies that converges uniformly over compact sets near the origin to a bounded nontrivial solution of in . Then, is proportional to for (see [35, 36, 41]), or a linear combination of , , and for (see [34, 45, 46]). However, our assumed conditions (42) on and pass to the limit and yield for , or for , , which implies that . This is absurd because is nontrivial.

We will give next a priori estimate for the solutions of (41) satisfying orthogonality conditions with respect to , , .

Lemma 6. There exist positive numbers and such that for any points , and any solution to (41) with the following orthogonality conditions:

one has

for all .

Proof. Consider the radial solution , for the following equation:

where is a large number. Then, is explicitly given by

Let and be radial smooth cut-off functions on so that

Set

Besides, let us define

where

and is chosen such that for ,

and for ,

Thus,

and satisfies the orthogonality conditions (42). By (43),

Multiplying (69) by , and integrating by parts, it follows that

where . Then, for , by (70) and (71),

Let us claim that for , there exists some constant independent of , such that

Once these estimates (73)-(74) are proven, it easily follows that

This, together with (65), (70), and (73), easily gives the estimate (60) of .

Proofs of (73) and (74). Let us first define that

For , by (32),

For , , , and . Then,

Furthermore,

Integrating by parts the first term of , it follows that

Integrating by parts the first term again, it also follows that

Then,

For , . Then,

Note that , , and . Then,

Besides,

Some simple computations show that

Then, there exists some constant independent of and such that

For , . Then,

Note that , , and . Then,

Furthermore,

As a result, the estimates (73)-(74) can be easily derived from (77)–(91).

Proof of Proposition 4. We first establish the a priori estimate (39). Testing (38) against , , , and integrating by parts, it follows that

Observe that

Then,

Note that . This, together with (92)–(94), implies that

By (60),

Combining this with (95), we find that the a priori estimate (39) holds. Furthermore,

which implies that there exists a unique trivial solution to problem (38) with .

Next, we prove the solvability of problem (38). Consider the following Hilbert space:

endowed with the usual norm . Problem (38) is equivalent to that of finding such that

By Fredholm’s alternative, this is equivalent to the uniqueness of solutions to this problem, which is guaranteed by (39). As a consequence, there exists a unique solution , scalars , , , for problem (38) with , where is a continuous linear map satisfying .

Finally, we give the a priori estimate (40). Let us Differentiate (38) with respect to the parameters , , . Formally, should satisfy

where (still formally) . The orthogonality conditions now become

We consider the constants , , , defined as

By (31), it follows that uniformly holds on , which implies that for , ,

Define

We then have

Thus, can be uniquely expressed as

Furthermore, elliptic regularity theory implies that is differentiable with respect to , , . Note that , , , and for . This, together with (39) and (97)–(106), implies that

which implies that the estimate (40) holds.

Now, we solve the auxiliary nonlinear problem: for , we find a function and scalars , , , such that

The following result can be proved through arguments similar to these of [39].

Proposition 7. There exist positive numbers and such that for any and points , there is a unique solution , scalars , , , of problem (108), which satisfies

Moreover, the map is , precisely for , ,

where .

4. Variational Reduction

In what follows, we only need to find a solution of problem (27) (or (25)) with , and hence to problem (108) if points satisfy

To realize it, we consider the energy functional associated with problem (25), namely,

We define

where is defined in (26) and is the unique solution of problem (108). Critical points of correspond to solutions of (111) for small , as the following result states.

Lemma 8. For any points , the functional is of class . Moreover, for small enough, if , then points satisfy (112).

Proof. Observe that for , ,

where

Then, if , we have

Set

A simple computation shows that

This, together with the estimate (110) of , implies that

where

Set

Thus, (116) can be rewritten as: for any , ,

This is a diagonal dominant system and we thus get for all .

Next, we give a precise asymptotic expansion of defined in (113).

Lemma 9. The following precise asymptotic expansion holds

uniformly for any points , where is defined in (12).

Proof. Let us first give a priori estimate of , where . Using , a Taylor expansion and an integration by parts, it follows that

Note that , , and = . These, together with (109), imply that , and so .

Next, we only need to give an asymptotic expansion of . Observe that

By (22),

where

Note that

and for ,

Then, for any , by (128)–(130),

By (128),

Making the complex changes of variables with , we get

On the other hand, by (128)-(129), we have

Thus, by (131)–(134),

Besides, by (32),

Using the choice for ’s by (31), together with (125), (135), and (136), it follows that

This, together with (12), easily gives the asymptotic expansion (123) of .

5. Proofs of Theorems

Proof of Theorem 3. According to Lemma 8, we only need to find a critical point of the function, consider

By (123), uniformly holds on . By Definition 2, there exists a critical point of such that . Moreover, up to a subsequence, there exists points such that as , and . Thus, is a family of solutions of problem (25) (or (27)). Set for any . As a result, is a family of solutions of problem (11) with the qualitative properties predicted by the theorem, as it can be easily shown.

Proof of Theorem 1. Set , , and for . If ,

and if ,

We will seek a nodal solution for problem (5) with the concentration points 0 and , . Note that

By Theorem 3, we can reduce the problem of finding solutions of (5) to that of finding a -stable critical point of the function defined in (8). Obviously, , which implies that has an absolute minimum point in . Then, is a -stable critical point of .

Appendix

Proofs of (32) and (33). For , by (22)-(23),

which, together with the definition (31) of , implies that

Furthermore, for , a direct computation shows that

Similarly, for ,

On the other hand, if , for any , it is easy to check that

This, together with (A.3), implies (32).

Next, by our definitions,

So, if , for all ,

while if ,

As a result, combining (A.4)-(A.5) with (A.7)-(A.8), we get (33).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

We would like to thank the anonymous referees for their helpful comments which improved this paper. This research is supported by the National Natural Science Foundation of China (Grant no. 11171214) and the foundation of Nanjing Agricultural University (Grant no. LXYQ201300106).

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