Sharp concentration estimates near criticality for radial sign-changing solutions of Dirichlet and Neumann problems
We consider radial solutions of the slightly subcritical problem either on () or in a ball satisfying Dirichlet or Neumann boundary conditions. In particular, we provide sharp rates and constants describing the asymptotic behavior (as ) of all local minima and maxima of as well as its derivative at roots. Our proof is done by induction and uses energy estimates, blow-up/normalization techniques, a radial pointwise Pohozaev identity, and some ODE arguments. As corollaries, we complement a known asymptotic approximation of the Dirichlet nodal solution in terms of a tower of bubbles and present a similar formula for the Neumann problem.
Let , , and consider the subcritical problem in the whole space
or in a unitary ball centered at the origin
with either Dirichlet boundary conditions
or Neumann boundary conditions
We are interested in a precise description of the asymptotic behavior of classical radially symmetric solutions as .
The study of this type of results for the Dirichlet problem originated in , where using ODE arguments (Emden-Fowler theory) it is proved that the unique positive radial solution of (1.2), (1.3) satisfies
In particular, (1.5), (1.6) show that concentrates at the origin and uniformly in compact subsets of as . This behavior is consistent with the well-known fact that (1.2), (1.3) has no nontrivial solutions (in any star-shaped domain) for , due to the Pohozaev identity .
In  the question of the asymptotic behavior of is revisited (considering also the operator for ) for , and the estimates (1.5), (1.6) are shown using PDE methods based on the Pohozaev identity, Green functions, and elliptic regularity theory.
The authors in  conjectured that a similar asymptotic behavior as in (1.5), (1.6) should also occur for positive nonradial solutions in general domains (note that, by a moving-plane argument, all positive solutions of (1.2),(1.3) are radial ). This conjecture was proved independently in  and , where it is shown that least-energy solutions possess the same limiting behavior with constants depending on the associated Green function. Furthermore, in [19, Proposition 1] a -convergence of the solution at the boundary of the domain is shown; in particular in the case of a ball, by (1.6), this implies that
After these seminal works, many extensions for related problems have been studied, for example, for Dirichlet positive solutions of operators of the form ; the number of interesting papers for this problem is too large to give here a complete list of references, so just to give a glimpse of the results in this direction we refer to  and the references therein.
The first paper to consider Dirichlet sign-changing subcritical solutions with annular-shaped nodal domains seems to be , where the authors use a Lyapunov-Schmidt reduction scheme in domains with symmetries (general domains were considered afterwards in ). See also  where the particular case of a ball is considered using a similar strategy as in . Particularly important for our approach is the recent paper , where the Morse index of radial solutions of the slightly subcritical problem is studied using energy methods. We describe in more detail the results of [24, 15, 16] and the relationship with ours after Theorem 1.3.
Concerning nodal solutions of variants of (1.2), the literature is again very extensive. The following is an incomplete list of references whose only aim is to show the diversity of results and techniques used in this direction, see [4, 5, 6, 21, 24, 1, 20] and the references therein.
From a variational point of view, least-energy solutions have been considered in the sublinear () case , in the superlinear-subcritical () case , and in the critical () case  (see also ). In contrast to the Dirichlet problem, the Neumann b.c. allow the existence of nontrivial solutions if and, due to a symmetry-breaking phenomenon, minimal-energy solutions are not radially symmetric [23, 28, 13], and they do not blow up as (we study in detail the qualitative properties of this kind of solutions in a forthcoming paper). For the are no radial solutions to (1.2), (1.4), since the change of sign would force the existence of an interior nodal sphere; nevertheless, for , radial solutions of (1.2), (1.4) do exist and therefore it is natural to study the asymptotic behavior of Neumann solutions as , and we do this in Theorem 1.2 below, where we detail its concentration rates.
In the literature one can also find an ample study of Neumann positive solutions of with ; here again we only mention the survey paper  and the references therein to have an idea of this topic, but we emphasize that many other interesting papers studying this problem are available.
Equation (1.1) is not only a mathematical paradigm in nonlinear analysis of PDEs, but also it has a physical motivation, since, for , radial solutions of (1.1) solve the Lane-Emden equation of index , namely,
(usually only positive values of are considered). Equation (1.8) is used in astrophysics to model self-gravitating spheres of plasma, such as stars or self-consistent stellar systems in polytropic-convective equilibrium, where the pressure and the density () satisfy a nonlinear relationship , see . In this setting, a solution is often called a polytrope, and it contains (up to constants) important physical information, such as the radius of the star (the first root of ), the total mass (), the pressure (), and (for an ideal gas) the temperature is proportional to .
The aim of this paper is to extend and generalize the results from  to any radial solution of (1.1) and of (1.2) satisfying either (1.3) or (1.4) with an arbitrary number of interior nodal spheres. In particular, we show similar estimates to (1.5), (1.6) and describe in a precise way the asymptotic behavior of all critical points, of all roots, and the values of and at these points respectively. Here, a very delicate and precise analysis is needed to obtain the explicit rates and the associated constants. Our results seem to be the first to consider explicit constants regarding sign-changing solutions and the exact asymptotic behavior of the Neumann problem. As shown below, it turns out that the asymptotic study of the Dirichlet and the Neumann problem is closely intertwined and intimately related to the subcritical problem in the whole space .
To state our main results let us introduce some notations. For a radial function we use a common abuse of notation and identify with for . Then, we say that has -interior zeros if has interior zeros in . Given , it is known (see for example ) that there is a unique (up to a sign) radial solution of the Dirichlet problem (1.2), (1.3) with exactly interior zeros and, for , a unique (up to a sign) radial solution of the Neumann problem (1.2), (1.4) with exactly interior zeros. Moreover, between two consecutive zeros there is exactly one critical point, which is either a local minimum or maximum. For radial solutions of (1.2), throughout this paper we use and to denote decreasing sequences of critical points and zeros in , respectively, see Figure 1 (Dirichlet case) and Figure 2 (Neumann case).
Let denote the standard Gamma function (which satisfies if ) and set
Our main results are the following.
Theorem 1.1 (Dirichlet case).
Theorem 1.2 (Neumann case).
The proof of Theorems 1.1 and 1.2 is done by induction, and relies on a radial Pohozaev identity, a blow-up/normalization procedure, energy estimates, direct computations, and some ODE arguments. A more detailed discussion of this strategy can be found after Corollary 1.6 below.
Observe that, in the Neumann case, the behavior of (that is, of the derivative of the solution at the largest interior zero) is particularly interesting, since its behavior changes drastically depending on the dimension (as in many situations in critical problems, see for example , dimension 4 is a threshold). Indeed we have that
Another difference between the Neumann and the Dirichlet case is the behavior of the solution at the largest critical point , where in the Dirichlet case but for Neumann b.c; in fact, the Dirichlet solution is unbounded in the nodal set that touches the boundary as , whereas the Neumann solution goes uniformly to zero as in this region. Actually, our approach also yields information on the asymptotic behavior of for fixed ; the next result can be seen as an extension of (1.6).
As mentioned above, in [24, 15] the authors use a Lyapunov-Schmidt reduction scheme to study the asymptotic profile of radial sign-changing solutions of (1.2), (1.3). This approach uses the family of all positive solutions in of the critical problem (see [3, 29])
where is a function which is uniformly bounded in , are some positive constants, and is given by (1.11). Formula (1.12) is sometimes called a superposition of bubbles or a tower of bubbles. We can use now Theorems 1.1 and 1.3 to complement (1.12).
Corollary 1.4 (Dirichlet tower of bubbles).
We remark that, up to some calculations and substitutions, the constants (1.13) can also be deduced from the proofs in [24, 15], which rely on different arguments than ours. Note that the Lyapunov-Schmidt approach provides a general shape of the solution; however, it seems difficult to obtain precise information regarding rates as in Theorem 1.1 using only (1.12). In this regard, the Lyapunov-Schmidt scheme and our approach complement each other.
Corollary 1.5 (Neumann tower of bubbles).
For our last result, we present a consequence of Theorems 1.1 and 1.2 regarding radial solutions of (1.1). These solutions have infinitely many oscillations and between two consecutive roots there is only one local maximum or minimum (see [22, page 294]). It is easily seen that solutions of (1.17) and (1.2) satisfying (1.3) or (1.4) are all connected via suitable rescalings. As a consequence, we have the following asymptotic profiles.
Let , , , and be the radial solution of
Moreover, let and be respectively the (divergent) increasing sequences of all zeros and critical points of , such that
We now discuss in more detail the proofs of our main results and the relationship with Corollary 1.6. As mentioned earlier, the proof of Theorems 1.1 and 1.2 is intertwined and holds a close relationship with the solution of (1.17), since we base our induction on rescalings using the sequence
To explain our approach, consider first
|the positive solution of the Dirichlet problem (1.2), (1.3).|
Here and would be the coefficients, and the unknown variables, while and are suitable right-hand sides (note that is not explicit and may depend on , but these terms vanish after taking the limit as ). Consider now
|the solution of the Neumann problem (1.2), (1.4) with one interior zero and .|
We wish to establish the behavior of the sequences
where is the unique zero of , and , are the unique critical points. To determine (1.18), we require a suitable system of four equations. Since and are related (by uniqueness) via the rescaling
we can use the known information on to obtain two equations (Lemma 3.2 for ). The other two equations (Lemma 3.7) are obtained by some computations using the equation on (see Lemma 3.3) together with energy estimates (Lemma 3.4) and a normalization argument (Lemma 3.6). The resulting system of equations is nonlinear, but a solution can be found directly by substitution.
|the solution of the Dirichlet problem (1.2), (1.3) with one interior zero and .|
In this case, the solution has two zeros, and two critical points, , and we study the behavior of the sequences
Therefore, we need a suitable system of 6 equations. As before, using a rescaling we can relate and to obtain 4 equations (Lemma 4.2 for ); the other two (equations (4.11) and (4.14)) are obtained using a radial Pohozaev identity (see (4) in the proof of Lemma 4.5) and a blow-up procedure in the set (Lemma 4.4). Here the bounds obtained in  are crucial, see Theorem 4.3.
Finally, we can consider in a similar fashion and show Theorems 1.1 and 1.2 by induction. In the proof that we present below in Sections 3 and 4, we follow this inductive strategy starting from explicit formulas for the constants , , , , , , , and . To deduce these formulas in the first place, we argued exactly as described above but with unknown coefficients, and we obtain recurrence identities relating , , , , , , , and that unequivocally define them. These relations were then developed for and , from which general formulas can be deduced. This approach requires hundreds—if not thousands—of algebraic manipulations and, to avoid any calculation mistake, we used a symbolic calculus software444Mathematica 220.127.116.11, Wolfram Research Inc., 2017.. Although this implementation was in itself a nontrivial computational challenge, to keep this paper short, we only present the rigorous proof by induction starting from known formulas for the coefficients.
To close this introduction, we refer to  for a broader perspective on the problem (1.2), (1.3); in particular, using a dynamical-system approach, the authors in  provide a full classification of the set of all positive and nodal (regular and singular) radial solutions of the equation
without any restriction on and for all .
The paper is organized as follows. In Section 2 we recall the rates for positive solutions of the Dirichlet problem (this serves as the inductive base in our argument). In Section 3 we study the radial Neumann problem whereas Section 4 is devoted to the Dirichlet problem. The proof of Theorems 1.1 and 1.2 can be found at the end of Section 4. Section 5 contains the proof of Theorem 1.3 and Corollaries 1.4, 1.5, and 1.6.
2 Positive Dirichlet solution
In this section we establish the induction base of our argument to show Theorems 1.1 and 1.2. Let be the unique positive radial solution of (1.2), (1.3). The function has a unique critical point and a unique zero .
3 The radial Neumann problem
As explained in the introduction, the proofs of Theorems 1.1 and 1.2 are intertwined and performed together by induction. In the previous section we proved the starting point of the induction procedure: Theorem 1.1 for . The purpose of this section is to prove the following.
Let , , , and let be a solution of (1.2), (1.4) with interior zeros. Let be the decreasing sequence of all the critical points of in and the decreasing sequence of all the zeros of in . The constants , , , , , , , and are explicit constants given by Theorems 1.1 and 1.2. The constant is given in (1.9). Observe that, in virtue of the Neumann boundary conditions and the uniqueness of the Cauchy problem for radial solutions of (1.2) (or by Hopf’s Lemma), we have
To ease notation we omit the dependency of . Observe that given by is a Dirichlet solution with interior zeros. If and are the critical points and roots of then, since Theorem 1.1 holds for solutions with interior zeros, we have that