We consider the weighted sinh-Poisson equation Δu + 2ε²|x|^2αsinh u = 0 in B₁(0), u = 0 on ∂B₁(0), where ε > 0 is a small parameter, α ∈ (−1, +∞)∖{0}, and B₁(0) is a unit ball in ℝ². By a constructive way, we prove that for any positive integer m, there exists a nodal bubbling solution u_ε which concentrates at the origin and the other m-points , l = 2, …, m + 1, such that as ε → 0, , where λ ∈ (0,1) and m is an odd integer with (1 + α)(m + 2) − 1 > 0, or m 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 Ω, consider

()

where w is the velocity field, p is the pressure, and ν is the unit outer normal vector to ∂Ω. Let us introduce the vorticity ω = curl w. 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 w as

()

In return, the vorticity ω is expressed as ω = −Δψ in term of ψ, the so-called stream function. Now, the ansatz ω = ω(ψ) 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 ω(ψ) have been chosen in the physical perspective to describe turbulent Euler flows with vorticity ω concentrated in small “blobs”. For example, on the basis of the statistical mechanics approach, Joyce and Montgomery proposed the Stuart vortex pattern ω(ψ) = ε²e^ψ with a small positive parameter ε to describe positive vortices (see [4–8]). Meanwhile, they also proposed the Mallier-Maslowe vortex pattern ω(ψ) = 2ε²sinh ψ to describe coexisting positive and negative vortices (see [9, 10]). Recently, Tur and Yanovsky in [11] have used the singular ansatz ω(ψ) = ε²e^ψ − 4πNδ_q to describe vortex patterns of necklace type with N + 1-fold symmetry in rational shear flow, where N ∈ ℕ and δ_q denotes the Dirac mass at q ∈ Ω. Now, we adopt another new singular ansatz in [12] ω(ψ) = 2ε² | x|^2αsinh ψ with α ∈ (−1, +∞)∖{0} to study the corresponding vortices with concentrated vorticity. To do it, we hope to investigate the effect of the presence of the weight |x|^2α on the existence of nodal bubbling solutions for the weighted sinh-Poisson equation as follows:

()

where ε > 0 is a small parameter, α ∈ (−1, +∞)∖{0}, and B₁(0) is a unit ball in ℝ².

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 0 ∈ Ω and Ω is symmetric with respect to the origin, for any integer k if ε is small enough, there exists a family of solutions to (6), which blows up at the origin, whose positive mass is 4πk(k − 1) and negative mass is 4πk(k + 1). 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 ε > 0 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 Ω, which have three and four alternate-sign concentration points. In particular, when Ω has an axial symmetry they proved for each N ∈ ℕ there exists a nodal bubbling solution, which changes sign N-times and whose alternate-sign concentration points align on the symmetric axis of Ω. 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 |x|^2α, the authors in [12] showed that there exists a family of nodal bubbling solutions u_ε to (5) only involving 0 < α ∉ ℕ, such that 2ε² | x|^2αsinh u_ε not only develops many positive and negative Dirac deltas with weight 8π and −8π, respectively, but also a Dirac data with weight 8π(1 + α) 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 Ω is a smooth bounded domain in ℝ², M ≥ 0, α_i > −1, q_i ∈ Ω,

defines the Dirac mass at q_i, and ε > 0 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 ε²e^u as ε → 0 if ε²∫_Ω e^u 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 M ≥ 0 and 0 < α_i ∉ ℕ. More generally, by a constructive way, similar results related to −1 < α_i can also be obtained in [36–40] under some milder notions of stability of critical points. In particular, when M ≥ 1 and α_is are positive numbers, D’Aprile in [37] recently established a family of solutions to (7) consisting of

blow up points in Ω∖{q₁, …, q_M} as long as N_i < 1 + α_i for any i, provided that the weights α_i 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 u_ε concentrating positively and negatively at the origin and outside the origin as long as α ∈ (−1, +∞)∖{0}. Concerning the sign-changing concentration at the origin and outside the origin, if we introduce the function f(λ):(0,1) → ℝ defined as

()

with

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

Theorem 1. For any positive integer m, there exists a nodal solution u_ε to problem (5) which concentrates at the origin and the other m-points , l = 2, …, m + 1, such that as ε → 0,

()

weakly in the sense of measures in

, where λ₀ is an absolute minimum point of f(λ) in (0,1), m is an odd integer with (1 + α)(m + 2) − 1 > 0, or m is an even integer. Moreover, for any δ > 0, u_ε remains uniformly bounded on

, and as ε → 0,

()

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

()

for ε > 0 small, where Ω is a bounded smooth domain in ℝ², M = n + k with M ≥ 0, {q₁, …, q_M} are different singular sources in Ω, {α₁, …, α_n}⊂(−1, +∞)∖(ℕ ∪ {0}), {α_n+1, …, α_M} ⊂ ℕ, and c : Ω → ℝ is a continuous function such that c(q_i) > 0 for any i = 1, …, M.

To further state our results, we need to introduce some notations. Let G(x, y) be Green’s function of Δ_x such that for y ∈ Ω, −Δ_xG(x, y) = δ_y(x) in Ω and G(x, y) = 0 on ∂Ω, and let H(x, y) be its regular part defined as H(x, y) = G(x, y) + (1/2π)log | x − y|. Besides, let us denote

, Ω^′ = {x ∈ Ω : c(x) > 0}, Γ = {q₁, …, q_n}, J₁ = {1, …, n}, J₂ = {n + 1, …, n + k}, J₃ = {n + k + 1, …, n + k + m}, and Δ_k+m = {(q_n+1, …, q_n+k+m) : q_i = q_j for some i ≠ j}, where m ≥ 0 is an integer. In what follows, we fix n different points q_i, i ∈ J₁, and define

()

which is well defined on the following domain:

()

where q = (q_n+1, …, q_M+m), α_i = 0 for i ∈ J₃,

for

, d_i = 8π(1 + α_i) for

, and a_i = ±1 for

Definition 2 (see [41].)We say that q^* is a C⁰-stable critical point of in Λ_k,m if for any sequence of functions ψ_j such that uniformly on the compact subsets of Λ_k,m, ψ_j has a critical point ξ_j such that .

In particular, if q^* is a strict local maximum or minimum point of , q^* is a C⁰-stable critical point of .

Theorem 3. Assume that {n, k, m} ⊂ ℕ ∪ {0} and is a C⁰-stable critical point of in Λ_k,m with k + m ≥ 1. Then, for any sufficiently small ε > 0, there exists different points q_ε,l ∈ Ω^′∖Γ, l ∈ J₂ ∪ J₃, away from ∂Ω ∪ Γ, so that problem (11), for q_l = q_ε,l, l ∈ J₂, has a nodal solution u_ε such that as ε → 0,

()

weakly in the sense of measures in

. Moreover, up to a subsequence, there exists

such that

()

Besides, u_ε remains uniformly bounded on for any δ > 0, and for any points q_l, l ∈ J₁, and , as ε → 0,

()

Note that for the case n = 1 and k = 0 (or n = k = 0), Theorem 3 was partly proved in [12] (or [25]) only when c(x) = 1 and α_l ∈ (0, +∞)∖ℕ, l ∈ J₁. 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 q_l, l ∈ J₁ ∪ J₂, and some other discrete points. Unlike the concentration set in [12] only contains singular sources q_l with α_l ∈ (0, +∞)∖ℕ and l ∈ J₁, and no singular source points in the domain, our concentration set also contains some singular sources q_l with α_l ∈ (−1,0) and l ∈ J₁, except for singular sources q_l, l ∈ J₂, 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 Ω. 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 B₁(0) to reduce the problem of finding solutions of (5) to that of finding an absolute minimum point of f(λ) 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 0 ∈ Ω and Ω 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

, ξ ∈ ℂ if α ∈ ℕ ∪ {0} and ξ = 0 if α ∈ (−1, +∞)∖(ℕ ∪ {0}) (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 δ > 0, let us first fix n different points q_i, i ∈ J₁, and assume that points q = (q_n+1, …, q_n+k+m) ∈ Λ_k,m(δ), where

()

Suppose that μ_i,

, are positive numbers to be chosen later, we define

()

The ansatz is

()

where a_i = ±1,

is a linear operator such that for any u ∈ H¹(Ω), ΔPu = Δu in Ω, and Pu = 0 on ∂Ω. By harmonicity, we easily get that

()

where

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

()

where Ω_ε = (1/ε)Ω, then v satisfies

()

We will seek solutions of problem (25) of the form v = V_q + ϕ, where

()

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

()

where

()

R_q is the “error term”:

()

and N(ϕ) denotes the following “nonlinear term”:

()

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

, as

()

so that from Appendix, we have

()

z_i : = (1/v_iρ_i)(εy − q_i).

3. The Linearized Problem and the Nonlinear Problem

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

()

The key fact to develop a satisfactory solvability theory for the operator L is that L formally approaches L_i under dilations and translations, and any bounded solution of L_i(ϕ) = 0 in ℝ² is a linear combination of z_i0, z_i1, and z_i2 for i ∈ J₂ ∪ J₃ (see [34, 45, 46]), or proportional to z_i0 for i ∈ J₁ (see [35, 36, 41]). Let us denote

()

where χ(r) is a smooth, nonincreasing cut-off function such that for a large but fixed number R₀ > 0, χ(r) = 1 if r ≤ R₀, and χ(r) = 0 if r ≥ R₀ + 1.

Additionally, we consider the following Banach space:

()

with the norm

()

where α ∈ (−1, α₀) and

Given that h ∈ 𝒞_* and q ∈ Λ_k,m(δ), we consider the linear problem of finding a function ϕ and scalars c_ij, i ∈ J₂ ∪ J₃, j = 1,2, such that

()

Our main interest in this problem is its bounded solvability for any h ∈ 𝒞_*, uniform in small ε and points q ∈ Λ_k,m(δ), as the following result states.

Proposition 4. There exist positive numbers ε₀ and C such that for any h ∈ 𝒞_*, there is a unique solution ϕ ∈ L^∞(Ω_ε), scalars c_ij ∈ ℝ, i ∈ J₂ ∪ J₃, j = 1,2, to problem (38) for all ε < ε₀ and points q ∈ Λ_k,m(δ), which satisfies

()

Moreover, the map q^′ ↦ ϕ is C¹, precisely for l ∈ J₂ ∪ J₃, s = 1,2,

()

where q^′ : = (1/ε)q = ((1/ε)q_n+1, …, (1/ε)q_n+k+m).

We begin by stating a priori estimate for solutions of (38) satisfying orthogonality conditions with respect to Z_i0, i ∈ J₁, and Z_ij, i ∈ J₂ ∪ J₃, j = 0,1, 2.

Lemma 5. There exist positive numbers ε₀ and C such that for any points q ∈ Λ_k,m(δ), h ∈ 𝒞_* and any solution ϕ to the following equation:

()

with the orthogonality conditions

()

one has

()

for all ε < ε₀.

Proof. We have the following steps.

Step 1. We first construct a suitable barrier. To realize it, we claim that for ε > 0 small enough, there exist R_d > 0 and

()

positive and uniformly bounded so that

()

where −1 < α < α₀ and

. Take

()

Thus, we have 1/2 ≤ g_1i(y) ≤ 1, and

()

By (32),

()

So, if d is taken small and fixed, by (47)-(48), it follows that for |z_i | ≥ R_d,

()

Let be a positive, bounded solution of in Ω and on ∂Ω. Set . Then, g₂ is a positive, uniformly bounded function in Ω_ε such that

()

Consider

()

where C₂ > 0 is a sufficiently large constant. Obviously, ψ is a positive and uniformly bounded function. Moreover, in view of (48)–(50), it is easy to check that ψ satisfies the estimate (45), and the claim follows.

Step 2. Consider the following “inner norm”:

()

Let us claim that there exists a constant C₃ > 0, such that

()

Let us take

()

where C₄ > 0 is chosen larger if necessary. Then, for

()

for y ∈ ∂Ω_ε,

()

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 ε_j → 0, and points , functions h_j with , solutions ϕ_j with , such that (41)-(42) hold. From the estimate (53), for some κ > 0, namely, for some i. To simplify the notation, let us denote ε = ε_j and . Set and . By (32), satisfies

()

for

. Obviously, for β ∈ [1, −(1/α)],

. 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 ℝ². Then,

is proportional to z_i0 for i ∈ J₁ (see [35, 36, 41]), or a linear combination of z_i0, z_i1, and z_i2 for i ∈ J₂ ∪ J₃ (see [34, 45, 46]). However, our assumed conditions (42) on i and ϕ_j pass to the limit and yield

for i ∈ J₁, or

for i ∈ J₂ ∪ J₃, l = 0,1, 2, 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 Z_ij, i ∈ J₂ ∪ J₃, j = 1,2.

Lemma 6. There exist positive numbers ε₀ and C such that for any points q ∈ Λ_k,m(δ), h ∈ 𝒞_* and any solution ϕ to (41) with the following orthogonality conditions:

()

one has

()

for all ε < ε₀.

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

()

where R > R₀ + 1 is a large number. Then, ψ_i(r) is explicitly given by

()

Let η₁ and η₂ be radial smooth cut-off functions on ℝ² so that

()

Set

()

Besides, let us define

()

where

()

and d_i is chosen such that for i ∈ J₁,

()

and for i ∈ J₂ ∪ J₃,

()

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 C > 0 independent of ε, 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 ϕ.

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

()

For r_i : = |z_i | ≤ R, by (32),

()

For R < r_i ≤ R + 1, , 1 − ψ_i = O(1/|log ε|), and |∇ψ_i | = O(1/|log ε|). Then,

()

Furthermore,

()

Integrating by parts the first term of I₂, it follows that

()

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

()

Then,

()

For R + 1 < r_i ≤ δ(4v_iρ_i) ⁻¹, . Then,

()

Note that , |∇ψ_i| ≤ C/(|z_i|log (1/ε)), and . Then,

()

Besides,

()

Some simple computations show that

()

Then, there exists some constant C > 0 independent of ε and R such that

()

For δ(4v_iρ_i) ⁻¹ < r_i ≤ δ(v_iρ_i) ⁻¹, . Then,

()

Note that ψ_i = O(1/|log ε|), |∇ψ_i| = O(ρ_i/|log ε|), 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 η_2iZ_ij, i ∈ J₂ ∪ J₃, j = 1,2, and integrating by parts, it follows that

()

Observe that

()

Then,

()

Note that 〈h, η_2iZ_ij〉 = O(∥h∥_*). 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 h ≡ 0.

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 ϕ ∈ ℍ 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 ϕ = T(h), scalars c_ij, i ∈ J₂ ∪ J₃, j = 1,2, for problem (38) with h ∈ 𝒞_*, where T : 𝒞_* ↦ L^∞(Ω_ε) is a continuous linear map satisfying ∥T(h)∥_∞ ≤ C(log (1/ε))∥h∥_*.

Finally, we give the a priori estimate (40). Let us Differentiate (38) with respect to the parameters , l ∈ J₂ ∪ J₃, s = 1,2. Formally, should satisfy

()

where (still formally)

. The orthogonality conditions now become

()

We consider the constants b_ij, i ∈ J₂ ∪ J₃, j = 1,2, defined as

()

By (31), it follows that uniformly holds on Λ_k,m(δ), which implies that for i ∈ J₂ ∪ J₃, j = 1,2,

()

Define

()

We then have

()

Thus, can be uniquely expressed as

()

Furthermore, elliptic regularity theory implies that ϕ = T(h) is differentiable with respect to , l ∈ J₂ ∪ J₃, s = 1,2. Note that , , , and for i ≠ l. This, together with (39) and (97)–(106), implies that

()

which implies that the estimate (40) holds.

Now, we solve the auxiliary nonlinear problem: for q ∈ Λ_k,m(δ), we find a function ϕ and scalars c_ij, i ∈ J₂ ∪ J₃, j = 1,2, such that

()

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

Proposition 7. There exist positive numbers ε₀ and C such that for any ε < ε₀ and points q ∈ Λ_k,m(δ), there is a unique solution ϕ ∈ L^∞(Ω_ε), scalars c_ij ∈ ℝ, i ∈ J₂ ∪ J₃, j = 1,2, of problem (108), which satisfies

()

Moreover, the map q^′ ↦ ϕ is C¹, precisely for l ∈ J₂ ∪ J₃, s = 1,2,

()

where

4. Variational Reduction

In what follows, we only need to find a solution of problem (27) (or (25)) with k + m ≥ 1, and hence to problem (108) if points q ∈ Λ_k,m(δ) satisfy

()

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

()

We define

()

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

Lemma 8. For any points q ∈ Λ_k,m(δ), the functional F_ε(q) is of class C¹. Moreover, for ε > 0 small enough, if D_qF_ε(q) = 0, then points q ∈ Λ_k,m(δ) satisfy (112).

Proof. Observe that for l ∈ J₂ ∪ J₃, s = 1,2,

()

where

()

Then, if D_qF_ε(q) = 0, 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 l ∈ J₂ ∪ J₃, s = 1,2,

()

This is a diagonal dominant system and we thus get c_ij = 0 for all i, j.

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

Lemma 9. The following precise asymptotic expansion holds

()

uniformly for any points q ∈ Λ_k,m(δ), where

is defined in (12).

Proof. Let us first give a priori estimate of θ_ε(q), where θ_ε(q) = F_ε(q) − J_ε(V_q). Using DJ_ε(V_q + ϕ)[ϕ] = 0, a Taylor expansion and an integration by parts, it follows that

()

Note that , , and = O(∥ϕ∥_∞). These, together with (109), imply that θ_ε(q) = O(ρ² | log ε|), and so F_ε(q) = J_ε(V_q) + O(ρ² | log ε|).

Next, we only need to give an asymptotic expansion of J_ε(V_q). Observe that

()

By (22),

()

where

()

Note that

()

and for i ≠ j,

()

Then, for any i ≠ j, 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 μ_i’s by (31), together with (125), (135), and (136), it follows that

()

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

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 Λ_k,m(δ). By Definition 2, there exists a critical point q_ε of F_ε such that . Moreover, up to a subsequence, there exists points such that as ε → 0, and . Thus, is a family of solutions of problem (25) (or (27)). Set for any x ∈ Ω. As a result, u_ε 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 q₁ = (0,0), c(x) = 1, and a_l = (−1) ^l−1 for l = 1, …, m + 1. If α ∈ ℕ,

()

and if α ∈ (−1, +∞)∖(ℕ ∪ {0}),

()

We will seek a nodal solution for problem (5) with the concentration points 0 and , l = 2, …, m + 1. Note that

()

By Theorem 3, we can reduce the problem of finding solutions of (5) to that of finding a C⁰-stable critical point of the function f(λ):(0,1) → ℝ defined in (8). Obviously, lim _λ→0+f(λ) = lim _λ→1−f(λ) = +∞, which implies that f(λ) has an absolute minimum point λ₀ in (0,1). Then, λ₀ is a C⁰-stable critical point of f(λ).

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).

Appendix

Proofs of (32) and (33). For |z_i | ≤ δ(v_iρ_i) ⁻¹, by (22)-(23),

()

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

()

Furthermore, for |z_i | ≤ δ(v_iρ_i) ⁻¹, a direct computation shows that

()

Similarly, for |z_i | ≤ δ(v_iρ_i) ⁻¹,

()

On the other hand, if |z_i | ≥ δ(v_iρ_i) ⁻¹, for any , it is easy to check that

()

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

Next, by our definitions,

()

So, if |z_i | ≥ δ(v_iρ_i) ⁻¹, for all i,

()

while if |z_i | ≤ δ(v_iρ_i) ⁻¹,

()

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

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All articles

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

Abstract

1. Introduction

2. Construction of the Approximate Solution

3. The Linearized Problem and the Nonlinear Problem

4. Variational Reduction

5. Proofs of Theorems

Conflict of Interests

Acknowledgments

Appendix

References

References

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

Abstract

1. Introduction

2. Construction of the Approximate Solution

3. The Linearized Problem and the Nonlinear Problem

4. Variational Reduction

5. Proofs of Theorems

Conflict of Interests

Acknowledgments

Appendix

References

References

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