Sign-changing two-peak solutions for an elliptic free boundary problem related to confined plasmas

By a perturbative argument, we construct solutions for a plasma-type problem with two opposite-signed sharp peaks at levels $1$ and $-\gamma$, respectively, where $0<\gamma<1$. We establish some physically relevant qualitative properties for such solutions, including the connectedness of the level sets and the asymptotic location of the peaks as $\gamma\to0^+$.


Introduction
Motivated by the description of equilibrium states for plasmas in a tokamak [21], we consider the following problem: where γ is a positive constant, ε > 0 is a small positive number and Ω ⊂ R 2 is a smooth bounded domain. Problem (1.1) generalizes the classical plasma problem: −ε 2 ∆u = (u − 1) + in Ω, u = 0 on ∂Ω, as derived in [21] and extensively analyzed in [6,7,17,21], to the case of a nonlinearity indefinite in sign. From the physical point of view, problem (1.1) corresponds to the case where the tokamak contains two plasmas ionized with charges 1 and −γ, respectively. The unknown function u corresponds to the magnetic potential and ε > 0 is a constant depending on the constitution of the plasmas. Problem (1.1) admits a variational characterization. Indeed, solutions to (1.1) correspond to critical points for the functional Our first aim in this article is to construct sign-changing solutions with exactly two sharp peaks, via a perturbative Lyapunov-Schmidt argument as developed in [10,14]. Our next aim, containing the more innovative aspects, is to derive some new qualitative properties of solutions, including the connectedness of the level sets and the asymptotic location of the peaks as γ → 0 + . We recall that problems of the form −∆u = f (u) in Ω, u = 0 on ∂Ω are also of central interest in the context of steady incompressible Euler flows, see, e.g., [9,11,20] and the references therein. In this context, such sign-changing solutions are related to the Mallier-Maslowe counter rotating vortices [4,18,19]. Sign-changing solutions with exactly two peaks were also constructed for a two-dimensional elliptic problem with large exponent in [15].
Unlike the above mentioned models, problem (1.1) involves the nonlinearity f (t) = (t − 1) + − (−t − γ) + which is only Lipschitz continuous. Consequently, major technical difficulties arise due to the "free boundaries" {u = 1} and {u = −γ}, particularly in establishing some regularity properties that are essential in order to obtain solutions as critical points of continuously differentiable functionals. Such difficulties were overcome in [10] in the case of one-sided peak solutions. A key ingredient to this end is to establish that such free boundaries have zero measure, see Proposition 5.1 below. Here, we shall obtain this key property by a new simple ad hoc argument involving the Faber-Krahn inequality, suitably tailored to the case of two-peak sign-changing solutions. We note that our perturbative construction does not provide any variational characterization of solutions, which is often employed in establishing the zero measure of free boundaries. Our arguments also suggest relations to the "twisted eigenfunctions", recently analyzed in [12,13].
After constructing the peak solutions in Theorem 2.1, we analyze the asymptotic location of the peaks as γ → 0 + . Roughly speaking, our result states that for the physically relevant solutions corresponding to minima of the Kirchhoff-Routh Hamiltonian, as γ → 0 + , the positive peak approaches a harmonic center z ∈ Ω and the negative peak escapes to a point p ∈ ∂Ω which maximizes the outward normal derivative of the Green's function with pole at z. This type of property was introduced in [19], where the case of a convex domain Ω is considered. Here, we remove the convexity assumption on Ω, see Theorem 2.2.
This article is organized as follows. In Section 2 we introduce some notation and we precisely state the main results. In Section 3 we define the ansatz for solutions in terms of the basic cell functions U ε,a,z defined in (2.2). We also establish some necessary estimates for the approximate solutions. Section 4 contains the linear theory and the Lyapunov-Schmidt reduction by which we obtain a solution u ε to a "projected problem" for every fixed choice of peak points z 1 , z 2 ∈ Ω, z 1 = z 2 . Thus, we reduce problem (1.1) to a four-dimensional problem. The results in this section rely on an approach introduced [10,14]. Therefore, some of the proofs in this section are only outlined for the reader's convenience. In Section 5 we prove the crucial zero-measure property for the free boundaries {u ε = 1} and {u ε = −γ}, where u ε is the solution to the projected problem obtained in Section 4. In Section 6 we insert u ε into the variational functional for (1.1) and we check that critical points for the resulting function K ε (z 1 , z 2 ) yield the desired solutions to the full equation (1.1); we also establish the connectedness of the level sets thus establishing Theorem 2.1. In Section 7 we analyze the limit profile of the "minimal" solutions as γ → 0 + , as stated in Theorem 2.2.

Statement of the main results
In order to state our results precisely, we introduce some notation. Let s > 0 be defined by 405 is the first zero of J 0 , the first Bessel function of the first kind. Then, the first eigenvalue of −∆ in B s (0) subject to zero Dirichlet boundary conditions is λ 1 (B s (0)) = 1. Let ϕ 1 > 0 be the first eigenfunction of −∆ in B s (0) satisfying ϕ 1 (0) = 1. Namely, ϕ 1 satisfies: For simplicity, in what follows we identify ϕ with its radial profile; namely, we denote ϕ(s) = ϕ(x) |x|=s = J 0 (s).
We denote by U the "basic cell function" defined on the whole space R 2 by We note that any constant multiple of U satisfies the equation We shall obtain the desired peak solutions as perturbations of an approximate solution defined in terms of suitably rescaled translations of U . More precisely, let R > 0 be a sufficiently large constant so that R > diam Ω and consequently Ω ⊂ B R (z) for all z ∈ Ω. We define, for any ε, a > 0: Then U ε,a satisfies the Dirichlet problem Thus, we obtain the family of cell functions {U ε,a,z : ε, a > 0, z ∈ R 2 }. We note that as ε → 0 the function U ε,a,z develops a sharp peak at the point z at level a. Moreover, as ε → 0, we have U ε,a,z → 0 in L ∞ loc (Ω \ {z}) and U ε,a,z (z) → a. We denote byḠ the Green's function defined by and by g(x, z) the regular part ofḠ, defined by We note thatḠ(x, z) = ln R |x−z| − g(x, z) and, by the choice of R, we have g(x, z) ≥ 0 for all x, z ∈ Ω. We denote by h : Ω → R the Robin's function: We remark that with this notation we have We denote by H γ the Kirchhoff-Routh type Hamiltonian (see, e.g., [20]) defined by for (z 1 , z 2 ) ∈ M. We note that H γ is bounded from below on M. We denote by P : H 1 (Ω) → H 1 0 (Ω) the standard projection operator and we denote by catM the Lusternik-Schnirelmann category of M.
Our aim is to establish the following results.
Theorem 2.1. For every γ > 0 there exists ε γ > 0 such that for all 0 < ε < ε γ problem (1.1) admits at least catM sign-changing two-peak solutions of the form It may be worth observing that, although L ∞ (Ω)-bounded, the family of solutions {u ε } ε>0 obtained in Theorem 2.1 presents a lack of compactness in C α -sense.
In order to state our second result, we recall that z ∈ Ω is a harmonic center of Ω if it is a minimum point of the Robin's function h(z) = g(z, z), see [3]. If Ω is convex, then z is unique [8].
Theorem 2.2 implies that for peak solutions corresponding to minima of H γ , for small values of γ the positive peak is approximately located at a harmonic center z ∈ Ω and the negative peak is near the boundary ∂Ω.
Notation. Henceforth, for any measurable set S ⊂ R 2 we denote by mS the two-dimensional Lebesgue measure of S and by 1 S the characteristic function of S. All integrals are taken with respect to the Lebesgue measure; when the integration variable is clear from the context, we may omit it. We denote · p = · L p (Ω) . We denote by C > 0 a general constant, whose actual value may vary from line to line. Let Ω ⊂ R 2 be an open set, for a given function V defined on Ω × Ω and Z = (z 1 , z 2 ) ∈ Ω × Ω we use the notation ∂V ∂z i,j to denote the partial derivative of V with respect to the component j of the variable i.

Ansatz and properties of approximate solutions
The aim of this section is to define suitable approximate solutions W ε to problem (1.1) in terms of the basic cell functions U ε,a,z , for ε, a > 0, z ∈ R 2 , defined in (2.2). We also establish some basic properties of W ε which will be needed in the sequel.
We recall form [10] that for any a > 0 and ε > 0, the problem has as unique solution U ε,a (x) defined by where . Correspondingly, we obtain the family {U ε,a,z : z ∈ Ω, ε, a > 0} of peak solutions defined in (2.2). Since the functions U ε,a,z (x) do not satisfy the zero Dirichlet boundary condition on ∂Ω, as usual we define their projections on H 1 0 (Ω). We recall that, given u ∈ H 1 (Ω), the projection of u into H 1 0 (Ω), denoted P u, is the unique weak solution to ∆P u = ∆u in Ω, P u = 0 on ∂Ω.
In particular, in view of (3.1), P U ε,a,z (x) satisfies Moreover, for all ε > 0 sufficiently small so that B sε (z) ⊂ Ω, we have that U ε,a,z coincides with a Newtonian potential on ∂Ω, namely The following is readily checked.
We have, for every z ∈ Ω: and more precisely We seek solutions whose form is approximately the difference of two cell functions of the form (2.2). Let Z = (z 1 , z 2 ) ∈ M. We make the following Ansatz. The solution u to problem (1.1) is of the form: (3.5) for some δ > 0.
Choice of a i = a ε,i , i = 1, 2. The pair of constants (a ε,1 , a ε,2 ) is chosen as the (unique) solution to the following linear system: Namely, In order to justify the choice of a i = a ε,i , i = 1, 2 in terms of γ, ε, z 1 , z 2 and Ω, we consider the "error term" and we estimate ℓ ε near z 1 and near z 2 . Let us first observe that in order to approximate a solution of (1.1) it is reasonable to choose a 1 ≃ 1 and a 2 ≃ γ therefore we suppose 0 < a i < K for a given K > 0.
Taking x = z 1 we fit a 1 by requiring that ℓ ε (z 1 ) = 0. We consequently derive which yields the first equation in (3.7). The second equation is obtained similarly.
We note that the Ansatz and the choice of (a ε,1 , a ε,2 ) show that the interaction between P U 1 and P U 2 is essentially negligible. Moreover, we have the following expansions where U is defined in (2.1). In view of (3.7) we have the following.
Lemma 3.2. The following identities hold: Proof. We exploit the explicit expressions of P U 1 , P U 2 and the definitions of a 1 , a 2 as in (3.7). In B sε (z 1 ) we have: where we used the first equation in (3.7) in the last inequality.
Similarly, in B sε (z 2 ) we have: where we used the second equation in (3.7) in the last inequality.
The third identity readily follows observing that in The choice of a 1 , a 2 in (3.7) also implies the following estimates. Lemma 3.3. For any L > 0 fixed constant we have, for ε sufficiently small, the following expansions for W ε : Proof. The proof is readily derived from Lemma 3.2 and a Taylor expansion at z 1 and z 2 .
Using Lemma 3.3 we now provide a quantitative estimate of the sets where the approximate solution W ε takes values less than −γ, between −γ and 1 and greater than 1. These estimates will be useful in the study of the linearized problem for the finite dimensional reduction scheme.
Using (3.10) and the previous estimates we deduce that and the claim follows by choosing T > M s γ 1 and ε 0 sufficiently small. If y ∈ B sε(1−T ε) (z 2 ) we proceed in a similar way using (3.11) and the fact that To get the estimate of W ε away from the points z i we begin by noticing that, if y ∈ Ω \ (B εσ (z 1 ) ∪ B εσ (z 2 )), for someσ < 1 to be chosen later, by Lemma 3.1, we can write, for any ε sufficiently small such that εσ > sε, It follows that, by choosing for instanceσ < γ 1 /2γ 2 and ε sufficiently small, we can write Moreover, for y in the annulus B εσ (z 1 ) \ B sε(1+T εσ) (z 1 ), using (3.3) for P U ε,z 1 ,a 1 (y), (3.4) for P U ε,z 2 ,a 2 (y) and (3.7), we can write, for sufficiently small ε, We can choose T > 4M γ 2 γ 1 in order to assure that the last term in the previous inequality is negative. In a similar way we can choose T sufficiently large to have, for any y ∈ B εσ (z 2 ) \ B sε(1+T εσ) (z 2 ), Finally, to get the claim, we choose σ <σ and ε 0 small enough to satisfy the previous estimates and B sε(1+T εσ) (z i ) ⊂ B sε(1+ε σ ) (z i ).

Reduction to a four-dimensional problem
Our aim in this section is to reduce problem (1.1) to a four-dimensional problem depending on Z = (z 1 , z 2 ), via a Lyapunov-Schmidt argument; that is, we shall solve a "projection" of problem (1.1) for any given Z = (z 1 , z 2 ) ∈ M satisfying (3.6), see Proposition 4.1 below. The content of this section follows [10,14] closely; therefore, some proofs are only outlined.
Let us remark that the previous system is solvable since, in view of the expansions (3.8)-(3.9), the elements of the matrix A, A i,j,h,k for i, j, h, k ∈ {1, 2}, satisfy the following orthogonality properties: With this notation, our main result in this section is the following.
The remaining part of this section is devoted to the proof of Proposition 4.1. We note that the linearization of the problem We define L ε : Following the contradiction argument used in the proof of [10, Lemma 3.3], we have the following: Let p > 1 be fixed. There are constants c 0 > 0 and ε 0 > 0 such that for any (4.5) Consequently, Q ε L ε : E ε,Z → F ε,Z is one-to-one and onto, and for every v ∈ F ε,Z we have . Proof. The proof follows [10] closely. We outline the main ideas for the reader's convenience. We first establish (4.5). Arguing by contradiction, we assume that there exist ε n → 0, Z n = (z 1,n , z 2,n ) satifying (3.6) and u n ∈ E εn,Zn , u n ∞ = 1, such that Q εn L εn u n = 0 in Ω \ ∪ 2 j=1 B Lεn (z j,n ) and Q εn L εn u n p ≤ ε 2/p n /n. Let (b 1,n , b 2,n ) ∈ R 2×2 be such that Q εn L εn u n = L εn u n − B(b 1,n , b 2,n ).
where we have defined Then, similarly as in [10], using Lemma 3.4, (3.8) (3.9) and (4.3) , we derive b j,h,n = O ε 1+σ n | ln ε n | . Consequently, in virtue of (3.9), we have  + z 1,n ), where Ω n = {y : ε n y + z 1,n ∈ Ω}. Since u 1,n ∞ = u n ∞ = 1, we have that u 1,n is bounded in W 2,p loc (R 2 ) and there exists u 1 such that u 1,n → u 1 in C 2 loc (R 2 ). On the other hand, in view of Lemma 3.4 we have that f εn (ε n y + z 1,n ) → 1 Bs(0) . We conclude that u 1 satisfies the problem −∆u 1 − 1 Bs(0) u 1 = 0, u 1 ∈ L ∞ (R 2 ). Now Proposition 3.1 in [10] implies that for some c 1 , c 2 ∈ R, where U is the basic cell function given by (2.1). On the other hand, taking limits in the orthogonality condition Ω ∆ ∂V εn,Zn,i ∂z i,n u n = 0, which holds true since u n ∈ E εn,Zn , we derive c 1 = c 2 = 0. We conclude that for any L > 0 By a similar rescaling argument at z 2,n we also obtain that , for any L ≥ 2s. Now estimate (4.5) yields u = 0 and the asserted one-to-one property follows.
We are left to check the surjectivity property. We note that by continuity of the projection operator P , we have ∂V j ∂z j,h = P ∂U j ∂z j,h ∈ H 1 0 (Ω), j, h = 1, 2. In particular, for any u ∈ E ε,Z we have It follows that Q ε ∆u = ∆u for all u ∈ E ε,Z and it is easy to see that it is one to one and onto from E ε,Z to F ε,Z . Now, let v ∈ F ε,Z . We check that the equation Q ε L ε u = v admits a solution in E ε,Z . Equivalently, we seek a solution to The operator on the r.h.s. above is of the form ε 2 I+compact operator. Then, by the Fredholm alternative, we conclude that Q ε L ε : E ε,Z → F ε,Z is onto.
We now define a fixed point problem which is equivalent to (4.4). We recall that the error term ℓ ε = ℓ ε (x) is defined by We define the higher order error R ε as follows: We observe that if ω ∈ W 2,p (Ω) ∩ H 1 0 (Ω) satisfies L ε (ω) = ℓ ε + R ε (ω) then u = W ε + ω is a solution to problem (1.1). We note that a solution ω to readily yields a solution to (4.4). In view of the invertibility property of Q ε L ε , as stated in Lemma 4.3, we define the operator G ε : E ε,Z → F ε,Z by setting Then, the projected problem (4.6) is equivalent to the fixed point problem Arguing similarly as in the proof of [10, Proposition 3.6] one can prove that (4.7) has a unique solution. More precisely, we have the following Lemma which together with the previous observations provides the proof of Proposition 4.1.
Lemma 4.4. There exists an ε 0 > 0 such that for any ε ∈ (0, ε 0 ) and for any Z satisfying (3.6), equation (4.7) has a unique solution ω ε ∈ E ε,Z with Proof. The proof follows [10] closely. We outline it for the reader's convenience. We define We start by checking that G ε is a map from M to M . Since the action of Q ε is localized in for L ≥ 2s. Therefore, we may use estimate (4.5) to obtain Similarly as in [10], we estimate Since ω ∈ M , we have ω ∞ ≤ ε and we conclude that In particular, G ε maps M into M . By similar arguments, we can also check the contraction property: It follows that for all sufficiently small ε > 0 there exists a fixed point ω ε ∈ M for G ε . Moreover, in view of (4.9), we obtain the asserted estimate Proof of Proposition 4.1. Now the proof follows as direct consequence of Lemma 4.3 and Lemma 4.4.

Free boundary properties for the projected problem
In order to show that (z 1 , z 2 ) → ω ε is C 1 it is essential to show that the free boundaries {u ε = 1} and {u ε = −γ} have two-dimensional Lebesgue measure equal to zero, see the Step 2 in the proof of Proposition 6.2 below. The aim of this section is to establish such a property, via an argument involving the Faber-Krahn inequality. Throughout this section, for any S ⊂ R 2 , we denote by mS the two-dimensional Lebesgue measure of S. The main result in this section is the following. Before proving the previous result, we observe that arguing exactly as in Lemma 3.4, in view of the decay estimate (4.8) for ω ε , we can prove the following useful level set estimates.
The remaining part of the statement is obtained similarly.
6. The reduced functional and the proof of Theorem 2.1 We recall that the Euler-Lagrange functional for (1.1) is given by For every z 1 , z 2 ∈ Ω satisfying (3.6) we define the "reduced functional" where ω ε = ω ε (Z) is the error function obtained in Proposition 4.1. Then, we have: is a critical point for K ε , then u ε defined by u ε = P U ε,a ε,1 (Z),z 1 − P U ε,a ε,2 (Z),z 2 + ω ε is a solution for problem (1.1).
Hence, in order to conclude the proof of Theorem 2.1 we are left to obtain a critical point for K ε . For a 1 , a 2 > 0 let H a 1 ,a 2 be the Kirchhoff-Routh type Hamiltonian defined by H a 1 ,a 2 (z 1 , z 2 ) = a 2 1 h(z 1 ) + 2a 1 a 2Ḡ (z 1 , z 2 ) + a 2 2 h(z 2 ). Recall that The following proposition clarifies the relation between K ε and H a 1 ,a 2 : Proposition 6.2. The following expansions hold true as ε → 0: uniformly on compact subsets of M. Furthermore, where A 1 , A 2,ε , A 3,ε > 0 are the uniformly bounded constants defined by Bs(0) ϕ 2 1 .

(6.4)
In order to prove Proposition 6.2 we will use the following lemmata. Lemma 6.3. Let (z, ζ) ∈ M. For any a, b > 0 and for any sufficiently small ε > 0 the following expansions hold: Proof. Proof of (i). We compute, recalling (3.2) and Lemma 3.1: Proof of (ii). Similarly, we compute: Lemma 6.4. The following expansions hold: The convergences are uniform on compact subsets of M.
Proof. We write: In view of Lemma 3.2 we compute: On the other hand, for sufficiently small ε > 0 we have The remaining estimates are derived similarly.

(6.5)
We conclude the proof if the first part of (6.2) in view of Lemma 6.4.
We are left to prove the second part of (6.2), namely the C 1 -approximation property. The proof closely follows [10, Section 3] therefore we only outline the main steps.
The proof is based on a differential quotients techniques and strongly relies on Proposition 5.1. More precisely, let e ∈ R 4 be a unit vector and let s = 0, for any function v Z we denote the s-difference quotient of v Z in the direction e at Z with For any a ∈ R, let K a s := {y : d(y, {u = a}) < s}. Then, ∆ s (u − a) + = 1 {u>a} ∆ s u + O(1 K a s |∆ s u|). Now the crucial consequence of Proposition 5.1 is that mK 1 s → 0 and mK −γ s → 0 as s → 0. A careful analysis as in the proof of [10, Lemma 3.7] together with elliptic regularity applied to ∆ s ω ε yields the desired property D e ω ε,Z ∞ ≤ C ε 1−σ | ln ε| .
Proof of Theorem 2.1. We argue as in the proof of Theorem 1.3 in [15]. we equivalently seek critical points for the functional F ε defined by where A 1 , A 2,ε and A 3,ε are the constants defined in (6.4). In view of Proposition 6.2, F ε is approximated by H a ε,1 ,a ε,2 , uniformly in C 1 on compact subsets of M. Moreover, H a ε,1 ,a ε,2 (z 1 , z 2 ) → +∞ as (z 1 , z 2 ) → ∂M. Let C ⊂ M be a compact set such that catC = catM. Let U ⊂ M be an open set such that C ⊂ U and inf ∂U H a ε,1 ,a ε,2 > max C H a ε,1 ,a ε,2 .