Spreading equilibria under mildly singular potentials: pancakes versus droplets

We study global minimizers of a functional modeling the free energy of thin liquid layers over a solid substrate under the combined effect of surface, gravitational, and intermolecular potentials. When the latter ones have a mild repulsive singularity at short ranges, global minimizers are compactly supported and display a microscopic contact angle of $\pi/2$. Depending on the form of the potential, the macroscopic shape can either be droplet-like or pancake-like, with a transition profile between the two at zero spreading coefficient. These results generalize, complete, and give mathematical rigor to de Gennes' formal discussion of spreading equilibria. Uniqueness and non-uniqueness phenomena are also discussed.


Introduction and results
1.1. The problem. We consider a class of singular energy functionals of the form where the potential Q(u) satisfies the following structural assumptions: spreading coefficient: where γ SG and γ SL are the solid-gas and solid-liquid tensions, respectively. There is, however, a caveat to be made at this point.
In thermodynamic equilibrium of the solid with the vapor phase (the so-called "moist" case, which concerns for instance a surface which has been pre-exposed to vapor), γ SG is usually denoted by γ SV , and its value can never exceed γ SL + γ. Indeed, otherwise the free energy of a solid/vapor interface could be lowered by inserting a liquid film in between: the equilibrium solid/vapor interface would then comprise such film, leading to γ SV = γ SL + γ. Therefore, S ≤ 0 in the "moist" case. On the other hand, when the solid and the gaseous phase are not in equilibrium (the so-called "dry" case), there is no constraint on the sign of S. The cases S < 0, resp. S ≥ 0, are commonly referred to as partial wetting, resp. complete wetting: indeed, when Q ≡ −S, the global minimizer's support is compact if −S > 0, whereas if −S ≤ 0 the global minimizer does not exist and the final spreading equilibrium is a zero-thickness unbounded film (see e.g. [83, §19.4], where the complete form of the surface energy is considered instead of its lubrication approximation).
We are interested in non-negative global minimizers (hereafter simply called minimizers) of E under the constraint of fixed mass; that is, in the set (we shall omit the subscript M when unnecessary). with A > 0, B ∈ R, and m, n > 1. Since A > 0, the singularity of Q at u = 0 disfavors small heights of the droplet and corresponds to short-range repulsive forces. When the strength of the singularity is sufficiently high, namely when m ≥ 3, the very existence of a minimizer is precluded, since E[u] ≡ +∞ for any u ∈ D (see Lemma 2.10 below). However, this is not the case when the singularity is milder (m < 3), which is the focus of this manuscript. At long ranges, B < 0 corresponds to considering the effect of repulsive forces only (cf. the discussion in [31, II.D.1] and references therein), whereas B > 0 corresponds to considering short-range repulsive, long-range attractive, forces (cf. [92, II.E] and references therein). We anticipate that the long-range decay exponent n is not essential: it enters the analysis only in critical cases. Though our results cover a wide range of potentials, it will be convenient to introduce a few prototypical cases ( Fig. 1.A). The first one, Q a , is repulsive-attractive for B > 0 and purely repulsive for B ≤ 0: Q a (u) = Au 1−m − Bu 1−n − S for u > 0, B ∈ R, 1 < n < m. (1.5) For purely repulsive potentials (B < 0), a long-range decay exponent n larger than the short-range growth exponent m is often considered. A prototype which is suited to this situation is Q b (u) = A|B|u |B|u m + Au n − S for u > 0, B < 0, m < n, (1.6a) for which we only consider the convex case, corresponding to the constraint 1 + 2m + m 2 + 2n − 6mn + n 2 ≤ 0. (1.6b) Finally, when gravity is taken into account, the potential G has to be added: 1.3. The framework. An enormous amount of work has been done on the fundamentals of wetting phenomena, from different perspectives: referenced discussions may be found in [31,41,92,15,83,98]. Concerning the analysis of energy functionals E of the form (1.1) with a singular potential Q, the focus has mainly been on two aspects.
• Positive minimizers with Q ≡ +∞ for u ≤ 0 and/or m ≥ 3. In this case, short range repulsion is so strong that compactly supported minimizers do not exist, and energy minimization forces the creation of a tiny liquid layer fully separating gas and solid. In this framework, interesting qualitative properties of minimizers, such as (in)stability of the flat film, bifurcation, concentration, and asymptotic scaling laws with respect to the potential's parameters, have been successfully investigated, also in relation to dynamic phenomena such as coarsening and dewetting; see [4,9,55,19,20,68,77,78,79,80,82,94,54], the references therein, and [102] for a recent overview. • Potentials with A < 0. In this case, minimizers and critical points also have a rich structure: we refer to [69] and again to [77,78,79,80] for a thorough study, including classification, stability, and other qualitative properties. On the other hand, in the case of mildly singular potentials, Q(u) ≡ 0 for u ≤ 0 and 1 < m < 3, (1.8) the minimization problem (1.1)-(1.3) does not seem to have been explored so far. We are only aware of two very recent and interesting works [32,33], where the model case Q(u) = u 1−m χ {u>0} is considered on a bounded domain Ω with Dirichlet boundary condition (and no mass constraint). There, existence and regularity of minimizers is discussed, together with the regularity of the free boundary ∂{u > 0} ∩ Ω and the Γ-limit as m → 3 − .
The case (1.8) is the focus of the present manuscript. Given the vastity of the potentials which have been introduced and considered through the years, we prefer to study generic potentials rather than concentrating on model cases.
1.4. Existence and basic properties of minimizers. Solely under (1.2) and (1.8), the existence of a minimizer of E in D is guaranteed by standard direct methods and symmetry arguments (see Theorem 2.1). The assumption m < 3 is crucial, since E[u] ≡ +∞ on D if m ≥ 3 (see Lemma 2.10). It turns out that the minimizer we obtain is: (a) compactly supported; (b) radially symmetric (up to a translation of x); (c-) non-increasing along radii.
In the rest of this introduction we assume in addition that Q ∈ C 1 ((0, +∞)). If either N = 1, or if Q (u) satisfies a very mild additional condition for u 1 (see (3.2) below), then (a), (b), and (c-) in fact hold for any minimizer; in addition, any minimizer is (see Theorem 3.1): (c) strictly decreasing along radii; (d) a smooth solution to the Euler-Lagrange equation for some λ ∈ R: (1.9) 1.5. The one-dimensional case. For N = 1 we are able to obtain much more detailed information, such as uniqueness and asymptotic results, which are discussed in the next paragraphs. The key to both of them is the identification of λ, which we prove via a combination of ODE and variational arguments (Theorem 4.5): (1.10) Not surprisingly, the function R plays a crucial role in the analysis. First of all, it follows from (1.10) that a constant function u s ∈ (0, +∞) is a stationary solution to (1.9) if and only if Q (u s ) = λ = R(u s ) = Q(u s )/u s ; since u 2 R (u) = uQ (u) − Q(u), in fact u s is a stationary solution to (1.9) if and only if R (u s ) = 0.
We assume, as in the model cases (1.5), (1.6), and (1.7), that these stationary solutions do not accumulate at 0 or +∞: Of crucial importance is the smallest among the absolute minimum points of R, provided they exist: In the model cases, e * coincides with the unique global minimum point of R, whenever such point exists ( Fig. 1.B).
1.6. Uniqueness. As is often the case, uniqueness is related to convexity. If Q is convex in (0, e * ), by comparison arguments we show that the minimizer is unique (see Theorem 4.13). In terms of the model cases, this translates into (see Section 8): • uniqueness for Q a if B ≤ 0, or if B > 0 and −S ≤ 0, or if −S ≥ 0 and B ≥ c 1 (A, S), where • uniqueness for Q b and Q b,g . Interestingly, however, potentials Q exist such that the minimizer is not unique for at least one value of the mass M . Generally speaking, this occurs when R is not injective in (0, e * ) (see Theorem 7.4): this is the case, for instance, in model Q a with −S > 0 and  Figure 2. A prototypical droplet (N = 1). On the left, the macroscopic profile and the macroscopic contact angle θ mac ; on the right, a zoom into the contact-liner and the microscopic contact angle θ mic . All minimizers of E in D have θ mic = π/2. 1.7. Micro-macro relations and the regime M 1. When continuum models are considered, wetting phenomena are characterized by the presence of two interfaces of codimension-one (liquid-solid and liquid-gas) and an (unknown) contact line, i.e. a codimension-two interface where the solid, the liquid, and the surrounding gas or vapour meet (Fig. 2). Among the main topics of interest to the physics and applied math communities, are the modelling of the "microscopic" physics near these interfaces -e.g. in terms of intermolecular potentials, substrate's corrugation and, in a dynamic context, slippage, contact-line free boundary conditions, and rheological properties-and the analysis of how such microscopic laws affect the "macroscopic" behaviour of droplets. See the reviews [31,92,15,98] and also [39,44,95] for referenced discussions. In this context, of particular importance are the microscopic contact angle θ mic , identified with the arctangent of the droplet's slope at the contact line, and (various notions of) macroscopic, or effective, or apparent contact angle θ mac : generally speaking, this is the arctangent of the slope, near the contact line, of the profile that the droplet assumes in the bulk of the wetted region, see Fig. 2. For droplet's dynamics, after the pioneering works [67,37,101,100], the relation of θ mac and macroscopic profile with θ mic , microscopic modelling, and speed of the contact line has been extensively studied via both formal asymptotic methods (see e.g. [26,65,66,63,12,38,3,23] and the references therein) and rigorous arguments [50,46,34], especially in the case θ mic = 0. More details may be found e.g. in [38] and [15, §C]. In the framework of this paper, which is concerned with statics, the "microscopic" physics are encoded in the intermolecular part P of the potential Q. In order to associate to P a microscopic length-scale ε, for a given reference potential P 0 one could set Then the macroscopic profile of minimizers could be identified by taking the limit as ε → 0. However, due to the lack of scaling invariance of E for general Q, it is more convenient to look at the limit as M → +∞. The two regimes are equivalent when E has a scaling invariance. This is the case, for instance, when P has the form (1.12) and G = 0 (no gravity): indeed, with the scaling x = εx, u = εû, one obtains Hence, we will analyze the limit M → +∞: the goal is to identify a macroscopic profile, whence a macroscopic contact angle (if it exists), as the limit of (suitably rescaled) minimizers u M of E in D M .
1.8. Microscopic behavior. The microscopic behavior of minimizers of E in D is universally determined by the short-range form of the potential. Indeed, we show in Theorem 4.7 that wherer denotes the right boundary of the minimizer's support. This shows that mildly singular potentials produce steady states with θ mic = π/2 (Fig. 2).
1.9. Macroscopic behavior: Pancakes versus droplets. Let u M ba a minimizer of E in D M . By translation invariance, we may assume that supp u M = [−r M ,r M ] and that the maximal height is u 0M = u M (0). The behavior of u M for M 1 is essentially influenced by two quantities: the constant e * defined in (1.11), which is always finite in presence of gravity (i.e. D > 0), and the non-dimensionalized spreading coefficient S, which for a generic potential Q is defined by (−∞, +∞) −S := lim u→+∞ Q(u) when the limit exists and is finite.
We will prove in Section 5 that there are two generic behavior of u M as M → +∞. .
In Fig. 3 we report numerical solutions to the minimization problem in a prototypical case in which uniqueness holds.
• e * = +∞ Table 1. Synopsis of the main results. 7 1.10. Profiles of minimizers: macroscopic contact angles and thickness. Combining the information in Paragraphs 1.8 and 1.9, we can characterize minimizers as follows.
• Droplet: we have Fig. 4), wherer In this case, it is natural to define the macroscopic contact angle θ mac as the arctangent of the slope of the macroscopic profile at the boundary of its support: This analysis also identifies the transitional thickness as the height 2|S|δ at which the cross-over takes place: • Pancake: when e * < +∞, we have The pancake's thickness e * < +∞ is defined in (1.11): it satisfies R (e * ) = 0, that is, wherer =r M and u 0 = u 0M are as in (1.15).
1.11. Repulsive potentials: comparison with de Gennes' results. In part II.D of his milestone review [31], where final spreading equilibria are discussed, de Gennes considers two model cases. The first one ("van der Waals forces"), on which we focus, is of the generic form (1.4) with m = 3 and n = 4, which corresponds to Q b with m = 3 and n = 4. Now, we know from Lemma 2.10 that minimizers of E in D do not exist if m ≥ 3. However, de Gennes confines his analysis to scales not below 30Å, where "a continuum picture is still applicable". In any event, our results show that, replacing m = 3 by a generic exponent m ∈ (1, 3), most of his formal predictions can be rigorously justified down to u = 0. To proceed further, we distinguish three cases.
Partial wetting (−S > 0 in Q b ). When −S > 0, the macroscopic shape is of droplet type (see (1.13)). Our results confirm, in the case of negligible gravitational effects, both the relation between S and the macroscopic contact angle and, in the limiting case m = 3, the estimate for the transitional thickness (compare (1.17) and (1.18) with [31, (2.54) and the discussion below it]).
Limiting case (−S = 0 in Q b ). In the limiting case −S = 0, the macroscopic shape is given by f −1 p (see (1.15)). In particular, in the limiting case m = 3 and for p = 4, we recover the same scaling exponents for the microscopic and intermediate regimes in [31, (2.55)-(2.56)], the only difference being in the multiplicative constants, which turn out to depend on f p (0) and are therefore expressed in terms of Γ functions.
"Dry" complete wetting (−S < 0 in Q b , and Q b,g ). In [31], only the case Q b,g (with gravity) with −S < 0 is discussed. However, we see from Table 1 that the same qualitative result (pancake shape) holds for two other cases which do not seem to have been discussed there: The characterization of e * in (1.19) coincides with that in [31, (2.63)] whenever e * is uniquely defined. In particular, one easily checks that if D is relatively small and S is relatively large, namely which coincides with [31, (2.72)] in the critical case m = 3. However, the reader can easily realize that there are various other possibilities, depending on the relation between the four parameters S, A, B, D. Remark 1.1. As Table 1 shows, the above conclusions holds not only for model Q b , but also for model Q a when B ≤ 0 (which, in this case, is also purely repulsive). In addition, the qualitative aspects of our results remain true for the second model potential considered by de Gennes ("double-layer forces"): However, quantitative information need be modified in this case, taking into account that a log singularity of the potential corresponds to the limiting case "m = 1". We refrain from doing that for the sake of brevity.
1.12. Repulsive/attractive potentials. As we mentioned in Section 1.3, potentials which are short-range repulsive and long-range attractive, such as model case Q a with B > 0, have been widely discussed in the thin-film literature, using various forms of them, especially in order to model and analyze coarsening dynamics and dewetting phenomena; however, qualitative studies of mild singularities (Q(u) ≡ 0 for u ≤ 0 and 1 < m < 3) seem to have been missing so far. When gravity is present, the minimizer for M 1 is invariably a pancake. Hence we focus on Q a with B > 0. The different possible behaviors are summarized in Figure  5, where B ≤ 0 is also shown for completeness. If −S ≤ 0 (complete wetting), a unique minimizer exists with pancake shape. However, as opposed to purely repulsive potentials, a unique pancake-shaped minimizer may exist in the partial wetting regime (−S > 0), too, provided B is sufficiently large. In addition, as we mentioned already in Section 1.6, droplet-shaped minimizers can fail to be unique for moderate values of B.   Table 1). In particular, we expect non-uniqueness phenomena to occur whenever Q is not convex in (0, e * ), without the additional assumption that R is not injective (Theorem 7.4); e.g., model Q a with −S > 0 and 0 < B < c 2 (A, S). Higher dimension. Our qualitative study is one-dimensional. In higher dimensions, it is still possible to characterize the eigenvalue λ. However, the relation between u and λ becomes nonlocal, involving integrals of functions of u rather than u(0) alone (cf. 1.10). In addition, the Euler-Lagrange equation (in radial variable) becomes non-autonomous. The combination of these two features so far prevented us from developing an analogous qualitative study for N > 1.
Critical points and their stability. This manuscript is concerned with global minimizers of E in D. However, we expect that E also has critical points in D, consisting of two or more radially decreasing solutions to (1.9)-(1.10), suitably translated so to have disjoint positivity sets, with a possibly different λ for each of them. It would be very interesting to prove that such configurations are indeed critical points of E in D and to study their (in)stability in either variational and/or dynamical sense, see e.g. [16,21,22,25,70,79,87]. Full curvature problem. The gradient part of the functional in (1.1) may be derived from Stokes system on the basis of the main assumption in lubrication approximation: the vertical lengthscale is much smaller than the horizontal one [49,51]. If not for the full Stokes system, it would be interesting to perform an analogous study at least for the functional E with 1 2 |∇u| 2 replaced by 1 + |∇u| 2 ; in other words, the full curvature effect is retained, though the droplet is yet assumed to be a subgraph. In this case, we are only aware of the studies in [88,89,85,86], which concern existence and uniqueness for convex potentials in the one-dimensional case. Dynamics. Since the nineties [8], a lot of work has been done on existence [60,5,11,28,61] and qualitative properties (such as finite speed of propagation, waiting time, longtime behavior) [6,7,13,29,47,42,43] of the spreading dynamics associated to E, as modeled by thin-film equations, which in one space dimension formally read as with f depending on the slip condition adopted at the liquid-solid interface (f (u) = u 3 + bu 2 , b > 0, for Navier slip). When the potential is sufficiently singular (Q(u) ≡ +∞ for u ≤ 0 and/or m ≥ 3), existence and uniqueness are rather simple, since the datum is to be positive and the solution will as well [62,9]. On the other hand, for mildly or nonsingular potentials, (1.20) turns into a genuine free boundary problem. Concerning weak solutions, most efforts in its study concentrated on existence and qualitative properties of "zero contact-angle" solutions, which satisfy u x = 0 at ∂{u > 0} for a.e. t > 0. We mention in particular [10,30,90], where the case of power-law potentials is discussed. However, these zero contact-angle solutions have the property of converging to their mean for the Neumann problem on a bounded domain [10], regardless of their initial mass. It would be very interesting to see whether different classes of solutions to (1.20) exist, which instead satisfy a right-angle condition at ∂{u > 0} and converge to a stable critical point of E for long times. Such achievements would be analogous to the ones regarding weak solutions with Q ≡ 0 and finite non-zero microscopic contact-angle [93,14,84,24]. First results in this direction are contained in [35]: there, formal arguments support the existence of generic (both advancing and receding) traveling wave solutions of (1.20) for any speed and any m ∈ (1, 3), with a contact angle of π/2 at ∂{u > 0}. Notably, such waves exist even without slip conditions (i.e. for b = 0): hence mildly singular potentials may be seen as an alternative solution to the contact-line paradox. More recently, a well-posedness theory of "classical" solutions has been developed for both zero [48,45,56,57,58,97] and fixed non-zero [73,74,75,76] contact-angle. As a further step, it would also be interesting to develop a theory of "classical" solutions for the singular potentials Q addressed here. Another interesting question concerns, in the pancake case with e * 1, intermediate scaling laws for macroscopic droplets spreading over a microscopic pancake, in the spirit of [50]. of functions with compact support. We denote by |Ω| the Lebesgue measure of a Lebesgue measurable subset Ω of R N . We denote by B R (x) the ball of radius R and center x in R N and by ω N −1 the (N − 1)-dimensional measure of the unit sphere S N −1 = ∂B 1 (0). The Sobolev conjugate exponent of 2 is denoted by 2 * = 2N N −2 . For a measurable function f , we define We omit the domain of integration when it coincides with R N , and (when no ambiguity occurs) we also omit the differential dx when x is (a rescaling of) the spatial independent variable. If not otherwise specified, we will denote by C several constants whose value may change from line to line. These values will only depend on the data (for instance, C may depend on N ). We say that a function is radially strictly decreasing, resp. radially non-increasing (with respect to x 0 ∈ R N ), if it is radially symmetric (with respect to x 0 ∈ R N ) and the corresponding radial function is strictly decreasing, resp. non-increasing.

Existence of a minimizer
Note that (1.2) implies that (2.1) In this section we discuss existence and basic properties of minimizers.
Theorem 2.1 (Existence and basic properties of minimizers). Assume (1.2), m < 3, and M > 0. Then there exists a minimizer u of E in D. Moreover, u is radially nonincreasing w.r.to a certain x 0 ∈ R N , u is compactly supported, and u ∈ C . We divide the proof into lemmas.
Note that the range of α is not empty since 1 < m < 3. Straightforward computations show that for any u ∈ D. In particular, E is bounded from below in D.
Proof. The lower bound of E is immediate from (2.2). Recall that Q > 0 in (0, s 1 ), with s 1 defined in (2.1). If s 1 = +∞ then Q is non-negative and (2.2) is obvious with C = 0. Otherwise, we have There exists a minimizer u of E in D. Moreover, u is radially nonincreasing.
Proof. It follows from Lemma 2.2 and Lemma 2.4 that there exists a minimizing sequence In particular, for all k ∈ N, we have that In addition, by Nash inequality [17], In order to show that Q(ũ k ) is uniformly bounded in L 1 (R N ), we estimate and Combining (2.9) and (2.10) we conclude that In particular, for all k in N. This implies that {u k } is another minimizing sequence in D. In view of (2.13), there exists a non-negative, radially non-increasing function u k → u a.e. in R N and in L p (B R (0)) for any R > 0 and any 1 ≤ p < 2 * . (2.14) We now show that u = M , hence u ∈ D. For any R ≥ 1, we have In order to estimate the second integral on the right-hand side of (2.16), we note that, in view of (1.2) and (2.15), Passing to the limit in (2.16) as R → +∞ using (2.18) and Beppo Levi's theorem, we conclude that u = M , hence u ∈ D. Now we prove that u is a minimizer of E in D. The passage to the limit in the Dirichlet energy is straightforward by lower semi-continuity: Let's focus on the potential energy. Let 0 < δ < s 1 be such that |{u = δ}| = 0 (note that this is the case for a.e. δ > 0, see Lemma A.1). Choosing R δ −1/N and using (2.15), we deduce that u k ≤ CR −N < δ in R N \ B R (0) for all k sufficiently large. Therefore It follows from (2.14) that u k → u a.e. in R N . Hence, since |{u = δ}| = 0, χ {u k >δ} → χ {u>δ} a.e.. Moreover, by (1.2), Q − ≤ C. Hence by Fatou lemma, applied to the first 14 term on the righ-hand side of (2.20), and Lebesgue theorem, applied to the second term on the right-hand side of (2.20), we obtain Remark 2.6. If N = 1, the compact embedding H 1 (R) C 1 2 (R) implies that any minimizer belongs to C 1 2 (R).
. Indeed, with the same argument just used in the proof of Lemma 2.5 (cf. (2.9)-(2.11)), we have The next Lemma implies that the minimizer given by Lemma 2.5 has compact support.
Lemma 2.8. Any radially non-increasing function u : has compact support.
Proof. We can assume without loss of generality that u is radially symmetric with respect to with v non-increasing in (0, +∞). Since u ∈ L 1 (Ω), arguing as in the proof of (2.15) we see that v(R) ≤ CR −N for all R ≥ 1. Consequently, arguing as in (2.18) we obtain for R 1 Since v is non-increasing, this implies that v(r) = 0 for all r sufficiently large and completes the proof.
Now we recall a simple property of radially symmetric functions in H 1 (R N ).
We conclude the section by proving the result anticipated in Remark 2.3.
Remark 2.11. Lemma 2.10 is related to Theorem 2 of [81] (see also [103]): there, in the model case Q(s) = As 1−m , it is proved that a solution to the Euler-Lagrange equation associated to E (cf. This implies that there exists R ∈ [r/2,r) such that and we have obtained a contradiction. 16

The Euler-Lagrange equation
In this section we assume that m < 3 and Q ∈ C 1 ((0, +∞)) is such that (1.2) holds. (3.1) In higher dimension we will also need additional information on the behavior of Q (s) for large s (more precisely on Q − and Q + , the negative and positive parts of Q ): for some C > 0, with q < +∞ if N = 2 and q ≤ 2 * if N ≥ 3. We will show: . Then any minimizer of E in D has compact support, is radially strictly decreasing w.r.to some x 0 ∈ R N in {u > 0}, and is a classical solution to Remark 3.2. Starting from the pioneering works [27,81], an enormous interest has been given to the homogeneous Dirichlet problem for elliptic equations with a source term which is singular with respect to u (see e.g. the Introduction of [64]). Of course, in this case both the domain and λ are fixed. Concerning the case Q (u) ≈ −u −m with 1 < m < 3, we refer in particular to [52,91,59,18,36].
We begin by proving that (3.3) is satisfied by any radially non-increasing minimizer. Proof. Let x 0 be the symmetry center of u. It follows from Lemma 2.8 and Lemma 2.9 that u ∈ C(R N \ {x 0 }) and that supp u = Br(x 0 ) for somer ∈ (0, +∞). Let ϕ ∈ I, where Since u ∈ C(R N \ {x 0 }) and u is radially non-increasing, ε > 0 (depending on ϕ) exists such that u ≥ ε in supp ϕ. Choosing |t| < ε ϕ ∞ , we have that Passing to the limit as t → 0 on the first two terms on the right-hand side of (3.5) is trivial.
Next, we give a boundedness result for solutions to (3.3) if N ≥ 2. Proof. Since u is a distributional solution to (3.3) with supp u = Br(x 0 ), we have Hence, taking a non-negative ϕ and dropping the non-negative term concerning Q + , we obtain By Lemma 2.9, u is continuous outside x 0 , which, since u is radially non-increasing, implies that u is bounded away from zero in any compact subset of Br(x 0 ). This fact, together with (3.2), allow us to extend the class of test functions in (3.13) to non-negative functions in H 1 c (Br(x 0 )). Let assume that sup u > 1 (otherwise there is nothing to prove). For k ≥ 1, let G k (u) = (u − k) + . Note that G k (u) ∈ H 1 0 (Br(x 0 )) and supp G k (u) = {u ≥ k} ⊆ {u ≥ 1} Br(x 0 ) (again using continuity and monotonicity of u). Hence we can take G k (u) as test function in (3.13), obtaining (3.14) Now fix By the Sobolev embedding theorem, applied to the left-hand side of (3.14), and Hölder inequality, applied to the right-hand side of (3.14), we have for a constant C depending on N andr. Hence, for h > k, Starting from inequality (3.15) and applying Lemma 4.1 of [99], it is standard to conclude that u ∈ L ∞ (Br(x 0 )).
We are now ready to prove Theorem 3.1.
Proof of Theorem 3.1. Letũ be a minimizer of E in D and let u be the Schwarz symmetrization ofũ. Arguing as in the proof of Lemma 2.5 we deduce that , which together with (3.16) implies that Since u is a radially non-increasing minimizer, by Lemma 2.8 it has compact support.
Let v be the non-increasing function defined by u(x) = v(r) with r = |x − x 0 | and let supp v = [0,r]. By Proposition 3.3 and Corollary 3.5, v is a classical solution of the following one-dimensional problem: We claim that dv dr < 0 in (0,r). (3.18) We argue by contradiction assuming that there exists a first r 0 ∈ (0,r) such that dv dr (r 0 ) = 0. Since dv dr ≤ 0, d 2 v dr 2 (r 0 ) = 0: but then Q (v(r 0 )) = λ, whence v(r) ≡ v(r 0 ) > 0 for all r ≥ r 0 , in contradiction with v(r) = 0. In view of (3.17) and

The one-dimensional case
The rest of the manuscript is concerned with the case N = 1. In the next statement we summarize, for N = 1, the results contained in Section 3: Corollary 4.1. Assume (3.1) and N = 1. Any minimizer u of E in D is even with respect to some x 0 ∈ R, which up to a translation we may assume to be zero: x 0 = 0. Moreover supp u = [−r,r] for somer ∈ (0, +∞), u < 0 in (0,r), u ∈ C 2 ((−r,r)) ∩ C([−r,r]), and λ > 0 exists such that u is a classical solution to In the rest of the manuscript, we will always assume (up to a translation) that the symmetry point of a minimizer is located at x 0 = 0. In one space dimension, we will be able to obtain uniqueness (up to a translation) and qualitative properties of minimizers of E in D. The key additional information is a characterization of the eigenvalue λ, which we now discuss under a mild additional information on the behavior of Q (s) for s 1. We assume that C > 0 exists such that Integrating by parts the first term and using (4.6), we obtain (4.4). We now prove (4.5). For α > 0, we consider the mass-preserving rescaling u α (x) = αu(αx) ∈ D. Performing the change of variable x = αx, we obtain We show that E[u α ] is differentiable with respect to α and The only non-trivial limit is Hereafter in the proof, C denotes a generic constant which may depend on α, but not on t and τ . Take |t| < α 2 , so that α 2 < α + τ < 3α 2 for |τ | < |t|. In view of (1.2) and (4.2), δ > 0 exists such that (4.10) Using (4.10) 1 twice, we see that if u ≥ δ 0 and the L 1 -bound follows from (4.9) since Q(u) ∈ L 1 (R) (cf. Remark 2.7). Thus (4.8), whence (4.7), hold. Since u = u 1 is a minimizer of E in D, it follows that d dα E[u α ]| α=1 = 0: hence (4.7) coincides with (4.5). Now we are ready to characterize λ. In view of (4.1) and (4.13), any minimizer of E in D is a solution to −u + Q (u) = R(u 0 ) in {u > 0}, u(0) = u 0 > 0, u (0) = 0 (P u 0 ) whose solutions we now discuss. First of all, any solution to (P u 0 ) is even and In addition, multiplying (P u 0 ) by u and integrating from x = 0, we obtain In the next lemma we give a necessary and sufficient condition on u 0 for a solution of (P u 0 ) to have compact support (supp u = [−r,r]) and negative derivative in (0,r). This will identify an admissible set A to which the maximal height of minimizers must belong. We also list a few properties of such solutions, which will be used in the sequel.  In particular, max u ∈ A for any minimizer u of E in D. If (a) or (b) hold, then

R(s)=Q(s)/
and for all x ∈ [0,r) it holds that Note that Z is well defined since R (u 0 ) = 0.
Proof of (b) =⇒ (a). We now assume that u 0 ∈ A, which implies R (u 0 ) < 0. Because of (4.16), u is strictly decreasing in a right neighborhood of x = 0. Assume by contradiction that x > 0 exists such that u (x) = 0 and u < 0 in (0, x). In particular, u(x) < u 0 and, by (4.17), R(u(x)) = R(u 0 ). This contradicts the definition of A. Therefore u < 0 as long as u is defined, and (4.19) follows from (4.17). Integrating (4.19) with respect to x we obtain (4.20). Integrating (4.19) with respect to u, we obtain (4.21). In particular, Since R (u 0 ) = 0 and s(R(s) − R(u 0 )) ∼ Q(s) → +∞ as s → 0 + , the right-hand side is finite: therefore u has compact support and the proof is complete.

4.3.
Asymptotics near the interface. Now we investigate the asymptotic behaviour near ∂{u > 0} = {−r,r} of solutions to (P u 0 ) with u 0 ∈ A; in particular, for minimizers of E. Since any solution of (P u 0 ) is even, it suffices to study the behaviour of u as x →r − .
In the following lemma we give a characterization of e * in terms of A and some restrictions on the maximal height. Proof. It is obvious that sup A ≤ e * . If by contradiction sup A < e * , then for all s ∈ (sup A, e * ) we would have either R (s) ≥ 0 or R(t) ≤ R(s) for some t < s, in contradiction with the definition of e * . Therefore sup A = e * . If e * = +∞ and z 0 < +∞, let u 0 ∈ A. By its definition, R(e min ) is a global minimum for R in (0, e max ], which implies that R (e min ) = 0 and R(s) ≥ R(e min ) for s ∈ [e min , e max ]. The former implies that u 0 = e min and the latter implies that u 0 / ∈ (e min , e max ].

4.5.
Uniqueness. We will now prove comparison and uniqueness results for minimizers of E in D M under the following additional assumption on Q: Q (s) is non-decreasing for s ∈ (0, e * ).   Proof. The proof of (iii) ⇒ (i) is obvious. 25 Proof of (ii) ⇒ (iii). It follows from Lemma 4.6 that u 0i ∈ A. Since u i are strictly decreasing, it follows from Lemma 4.10 that u 1 , u 2 < e * . Subtracting the corresponding equations, we obtain Since u 02 ∈ A, R(u 02 ) < R(t) for all t ∈ (0, u 02 ): in particular, R(u 02 ) < R(u 01 ). As long as u 1 ≤ u 2 , by (4.30), we have Q (u 1 ) ≤ Q (u 2 ). Hence (u 1 − u 2 ) < 0 as long as u 1 ≤ u 2 .
Proof of (i) ⇒ (ii). First we note that u 01 = u 02 : otherwise, u 1 and u 2 would solve the same equation (P u 0 ), whence u 1 = u 2 by Picard-Lindelöf theorem, in contradiction with On the other hand, using (ii) ⇒ (iii), u 01 > u 02 implies u 1 > u 2 in supp u 2 , in contradiction with M 1 < M 2 .
As a by-product of Lemma 4.11 we obtain the uniqueness result: Proof. Let u 1 and u 2 be two minimizers of E in D M (both of them symmetric with respect to x 0 = 0). Since u 1 and u 2 have the same mass, it follows from Lemma 4.11 that u 1 (0) = u 2 (0). This implies u 1 = u 2 by Picard-Lindelöf theorem.

Pancakes versus droplets
We assume throughout the section that We claim that µ ∈ C(A).
Since A is open, s ∈ A in a neighbourhood of u 0 . By Lemma 4.6, we know that u s is even, has compact support, say [−r s ,r s ], and is strictly decreasing in (0,r s ). By Lemma A.3, u s → u u 0 in C 2 loc ((−r u 0 ,r u 0 )) andr s →r u 0 as s → u 0 , hence a.e. in (−r u 0 ,r u 0 ). Therefore, by dominated convergence, µ(s) → µ(u 0 ). We will also need the following a-priori estimate, in the spirit of Theorem 4.7.
Proof. For u 0 as in the statement, the properties of Q imply that ε K < K −1 exist such that Q(s) − R(u 0 )s ≤ 2As 1−m for all s ∈ (0, ε K ), We preliminarily work out an upper bound on u. It follows from (4.19) that as long as u < ε K .
Integrating it in (x,r), we see that as long as u < ε K .
Proof. We recall that e * = +∞ implies Q > 0 (thus R > 0) in (0, +∞) (Remark 4.8), and we note for later reference that only −S ≥ 0 is used in Step 1 of this proof.

Lemma 5.3 (the droplet's energy).
Under the assumptions of Lemma 5.2, Proof. We let u = u k andr =r k for notational convenience. Recalling that Q, thus R, is positive in (0, +∞) (Remark 4.8), we note that Note that (5.9) implies that R(+∞) = 0. Therefore we may apply Lemma 5.1 with K = 1, leading to which by monotonicity implies that
We are now ready to conclude the analysis of generic macroscopic shapes of minimizers. Proof. The droplet's case. By Lemma 5.7, a > 0 exists such that (a, +∞) ⊂ A. By (5.3), µ ∈ C((a, +∞)). Hence Lemma 5.2 is applicable and yields µ(s) → +∞ as s → +∞. Therefore, for any sequence M k → +∞ there exists a sequence A u 0k → +∞ such that µ(u 0k ) = M k . Moreover, by Theorem 4.7, u k (the solution to (P u 0k )) belongs to D M k . It follows from Lemma 5.3 that u k is such that E[u k ] √ M k . To conclude, assume by contradiction that u 0M does not converge to +∞. Then a subsequence M k exists such that u 0M k → α ∈ [0, +∞). By Lemma 4.6, u 0M k ∈ A for all k. By Lemma 5.6, since R > 0 in (0, +∞) (cf. The pancake's case. By Lemma 4.10, A ⊆ (0, e * ). By Lemma 5.7, I = (e * − δ, e * ) ⊆ A for some δ ≤ e * . By (5.3), µ ∈ C(I). By Lemma 5.4, µ(s) → +∞ as s → e − * . Therefore, for any sequence M k → +∞ there exists a sequence A u 0k → e − * such that µ(u 0k ) = M k . By Theorem 4.7 the solution u k to (P u 0k ) belongs to D M k and, by Lemma 5.5,u . Assume by contradiction that u 0M does not converge to e * . Then a subsequence M k exists such that u 0M k → α ∈ [0, e * ). By Lemma 4.6, u 0M k ∈ A for all k. By Lemma 5.6, we have lim inf for all y ∈ (0, 1), that is, for all x ∈ (0, 1), a contradiction. Assume that α ∈ (0, +∞). By continuous dependence (Theorem 8.40 of [71]), u M → u in C 2 loc ({u > 0}), where u is the solution of (P α ). By Remark 5.10 we have α ∈ A, hence it follows from Lemma 4.6 that u has compact support and, therefore, finite mass µ(α).
In this limiting case, the macroscopic shape turns out to depend on the behavior of Q and its derivatives as s → +∞: we assume that K > 0 and p > 1 exist such that Lemma 7.1. Assume (5.1) and (7.2). Let {u 0k } ⊂ A be such that u 0k → 0 as k → +∞. Then µ(u 0k ) → 0 as k → +∞.
for all x ∈ (r k −δ K ,r k ) and k sufficiently large. In terms of w k , this means that for all y ∈ r − δ K 2 ,r (7.5) for k sufficiently large. It follows from the monotonicity of w k and the locally uniform convergence w k → u that |Q(w k )| Since 2(1−m) m+1 > −1 if m < 3, the right-hand side of (7.6) belongs to L 1 ((0,r)): therefore an application of Lebesgue theorem in (7.3) implies the result.
A straightforward application of dominated convergence theorem then yields . We are now ready to conclude. We know that P(M ) → 0 as M → 0 (Lemma 6.1), P(M ) → e * as M → +∞ (Theorem 5.8).
These two information, together with the assumption A = (0, e * ), contradict the continuity of P in (0, +∞) and complete the proof.

Model cases
Here we take a closer look at the four model cases referred to in the Introduction. We recall that large-mass asymptotic results depend on two quantities: e * , which is the smallest among the global minimum points of R(s) = Q(s)/s, if any, and +∞ otherwise; and the sign of S. Uniqueness also depends on the sign of Q (s) for s ∈ (0, e * ). Throughout the section, N = 1 and u M is a minimizer of E in D M . Any Q (and R) in this section obviously satisfies (5.1), (5.33), (7.2), (5.9) if −S > 0 and D = 0, and (5.34)-(5.35) if −S = 0, B ≤ 0 and D = 0. Therefore, for the macroscopic shape we will only need to check whether e * < +∞ or not (cf. Theorem 5.8 and Theorem 5.12), and for uniqueness we will only need to discuss convexity of Q in (0, e * ) (cf. Theorem 4.13).  If R never changes sign, then e * = +∞. If R changes sign once, then e * < +∞ and R(e * ) < 0 = R(+∞). If R changes sign twice, then -If −S > 0 and B ≥ c 1 (A, S), we have seen that e * < +∞ and that R changes sign twice. Since G (s) = −sQ (s), the unique zero of Q is located at the above defined point s 2 = g. Above, we have also seen that G(g) < 0, hence R (g) > 0. Since R changes sign twice and recalling the definition of e * , this implies that e * ≤ g. Therefore Q ≥ 0 in (0, e * ).
In the case −S > 0 and B > c 3 (A, D), we do not know how to ascertain the sign of Q in (0, e * ).
Proof. Thanks to Theorem 8.39 and Theorem 8.40 of [71], u k → u in C 2 loc ((−r, r)). It follows that for all x ∈ (−r, r), u k (x) is positive for k sufficiently large; hence r ≤ lim inf k→+∞ r k . It remains to prove that r ≥ lim sup k→+∞ r k =: R. If r = +∞, nothing is to be proved. If r < +∞, assume by contradiction that r < R. Multiplying the equation in (A.1) by −u k , integrating in (0, x) and using the initial conditions, we have (u k (x)) 2 = 2F k (u k (x)). (A.3) Since u is continuous at x = r and u(r) = 0, for every ε > 0 there exists δ > 0 such that u(r − δ) < ε/2. By locally uniform convergence and recalling that u k is non-increasing, u k < ε in (r − δ, R − δ) for all k sufficiently large. Fix x ∈ (r − δ, R − δ). Since u k is non-increasing, it follows from (A.3) that u k (x) = − √ 2 F k (u k (x)). Hence, (A.2) implies that for every M > 0 we can choose ε sufficiently small and k sufficiently large such that u k (y) < −M ∀y ∈ (r − δ, x), (A.4) whence we deduce that for every k sufficiently large Choosing M sufficiently large and recalling that u k is non-negative, we obtain a contradiction.