Construction of Nodal Bubbling Solutions for the Weighted Sinh-Poisson Equation
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.
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  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  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  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 . Besides, for a similar equation, precisely the Neumann sinh-Gordon equation on a unit ball, Esposito and Wei in  also constructed a family of solutions with a multiple blow up at the origin. On the other hand, Bartolucci and Pistoia in  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  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  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  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  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  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 ). 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  (or ) 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  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  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 , 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
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
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
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 .