p–Laplacian diffusion coupled to logistic reaction: asymptotic behavior as p goes to 1

This work discusses the limit as p goes to 1 of solutions to problem P-Δpu=λ|u|p-2u-|u|q-2u,x∈Ω,u=0x∈∂Ω,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\left\{ \begin{array}{ll} -\varDelta _p u = \lambda |{u}|^{{p}-2}{u} -|{u}|^{{q}-2}{u},&{} \qquad x\in \varOmega ,\\ \ u=0 &{}\qquad x\in \partial \varOmega , \end{array}\right. } \end{aligned}$$\end{document}where Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varOmega$$\end{document} is a bounded smooth domain of RN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^N$$\end{document}, λ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda >0$$\end{document} is a parameter and the exponents p, q satisfy 1<p<q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1< p < q$$\end{document}. Our interest is focused on the radially symmetric case. We prove in this radial setting that solutions up\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_p$$\end{document} to (P) converge to a limit u as p→1+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\rightarrow 1+$$\end{document}. Moreover, the limit function u defines a solution to the natural ‘limit problem’ which involves the 1–Laplacian operator. In addition, a precise description of the structure of the set of all possible solutions to such a problem is achieved. This is accomplished by means of the introduction of a suitable energy condition. Furthermore, a detailed analysis of the profiles of all these solutions is also performed.


Introduction
Since the late seventies, reaction-diffusion systems have been one of the more active areas in nonlinear analysis [17,7,12,36,42]. The so-called logistic problem is a reference model in the field where a wide variety of techniques have been tested (sub-and super-solutions, degree and bifurcation theory, critical point theory). Under such a term it is understood the nonlinear eigenvalue problem, where ⊂ ℝ N is a bounded smooth domain and the diffusion is governed by the p-Laplace operator p u = div (|∇u| p−2 ∇u) . Exponents p, q are assumed to satisfy The number > 0 plays the rôle of a bifurcation parameter. In fact, a well-extended insight in the theory from the very beginning is just observing (1.1) as a crude perturbation of the "pure" eigenvalue problem, The main objective of the present work is to analyze the fine aspects of the asymptotic behavior of problem (1.1) as p → 1+ . In the first place, this involves discussing the existence of the limit u = lim p→1+ u p of a given family u p of solutions to (1.1). In the second place, it should be decided whether such possible limits u solve in some weak sense the natural "limit problem." In other words, that one obtained by directly inserting p = 1 in (1.1), where 1 = div ∇u |∇u| is the one-Laplacian operator. To complete the analysis, a third task to be faced is that of describing all of the possible nontrivial solutions to (1.3). Previous experiences on the "natural" associated eigenvalue problem, strongly suggests that characterizing the solutions to (1.3) requires imposing certain restrictions. As a matter of fact, the higher eigenvalues to (1.4) have not been studied until few years ago [8,31,33,41]. It was just discovered in [8] that infinitely many anomalous eigenpairs to (1.4) arise if the corresponding Euler-Lagrange inclusion is not suitably constrained. The very same phenomenon occurs in the 1D-version of our problems (1.1), (1.3) as recently remarked in [40]. On the other hand, our analysis in the present work is an extended nontrivial continuation of [41]. The radial spectrum of (1.4) is analyzed for the first time in this work. As for applications, the linear diffusion case p = 2 of (1.1) arises in population dynamics, where it describes the equilibrium regime of a species subject to logistic self-regulation and spatial migration [7,34,35]. In reaction dynamics, a solution to (1.1) furnishes the stationary concentration u of a chemical substance, which diffuses throughout a reactor ⊂ ℝ N and is subject to parallel competing reactions [18]. That is why major emphasis has been put on studying its positive solutions (see [7,32] for a comprehensive overview (1.1) − p u = |u| p−2 u − |u| q−2 u x∈ , u = 0 x ∈ , in population dynamics). The nonlinear diffusion case p ≠ 2 is comparatively less understood. Most of the results have to do with positive solutions to (1.1) which has been analyzed in a series of works [13,25,22,23,26]. On the other hand, problems involving the 1-Laplacian are deserving a growing interest in the literature. This is mainly due to the pioneering works [3,4,10]. The issue of formulating a proper notion of solution to problems as (1.3) counts among the challenges achieved in these references (see Sect. 2.3). From the very beginning, the applications of 1 range from image processing [5,38] to elasticity [27].
However, the structure of the whole set of nontrivial solutions to (1.1) still remains unknown in many concerns, with the exception of the case N = 1 [25,40]. The problem in a general N-dimensional domain is plagued of obstacles. To quote only a few, there are not any kind of bifurcation results available from the higher eigenvalues n,p of − p (bifurcation at the first eigenvalue 1,p has been studied in [9,15]). The only exception is the radially symmetric case where is a ball [24,39]. What is worse, the complete spectrum of − p remains nowadays undetermined [16,30]. That is why there hardly exist results providing the existence of two signed solutions to (1.1) when grows (see [20] where such a kind of existence issues are addressed in a problem with the same structure).
After these considerations, it seems reasonable that an analysis of problems (1.1) and (1.3) can only be undertaken in the radially symmetric case. In a first step, a detailed account of the set of all possible nontrivial radial solutions to (1.1) is presented in this work. Solutions to this problem in a ball B R ⊂ ℝ N are shown to be organized in continuous curves emanating from the radial eigenvalues ̃n ,p to (2.17). More importantly, it is shown that the interval >̃n ,p is the precise existence domain for each of these curves. In this regard, global existence results in [24] (valid in the case p > 2 ) are substantially sharpened for the particular case of (1.1).
Once the nontrivial solutions to (1.1) are known, two main objectives are pursued in this work. First, to analyze the limit of these solutions as p → 1+ . Second, to characterize such limits as properly defined solutions to (1.3). It turns out that both problems are deeply connected. On one hand, a compactness type result permits us extracting limits u of families of solutions u p to (1.1) as p → 1+ . Moreover, every such a limit u defines a solution to (1.3) and so this statement actually constitutes a true existence tool. In fact, the result is also valid in a general smooth domain ⊂ ℝ N . On the other hand, an uniqueness result allows us concluding the validity of the full limit u = lim p→1 u p . In addition, it furnishes a quite detailed description of the profile of the limit u. This stage of the analysis heavily rests upon the radial requirement. It is worth to point out that solutions comprised under the uniqueness result must satisfy suitable symmetry and energy conditions which are revealed in this work. In fact, without restrictions, problem (1.3) could exhibit an uncontrolled number of solutions (Sect. 5.4).
As a final conclusion, we are able to furnish a rather complete picture of the nontrivial solutions to (1.3) in a ball B R . It is shown that its radial solutions satisfying an energy condition are organized in continuous curves. Every such a curve emanates from a radial eigenvalue ̄n to − 1 . Moreover, the structure of solutions lying in the same curve is explicitly described. In particular, solutions belonging to the same curve undergo the same number of jumps. Of course, this feature is reminiscent of the nodal properties exhibited by the solutions to (1.1) lying in a fixed branch.
This work is organized as follows. Next section deals with the preliminaries. Sect. 2.2 discusses the basic properties of problem (1.1), while the concept of solution to (1.3) together with the compactness principle satisfied for this problem (Theorem 3) is presented in Sect. 2.3. It is remarked that the material in these subsections is valid on a general domain ⊂ ℝ N . The main features reported here were firstly tested in the one-dimensional case ( [40]). Due to its intrinsic interest for our purposes in the present work, a partial overview of the later paper is contained in Subsection 2.4. Theorem 4 describing the nontrivial radial solutions to (1.1) is shown in Sect. 3. It includes important Lemma 1 which introduces and studies the zeros n of the solutions to the initial value problem associated to (1.1). The analysis of the asymptotic behavior of problem (1.1) as p → 1 is launched in Sect. 4. Two preliminary results stating the finiteness of the limits lim p→1 n , lim p→1 n (Theorem 6) and proving the validity of the strict inequality lim p→1 n < lim p→1 n+1 (Theorem 7) are introduced in this section. Proving the main result of this work, Theorem 8, is the objective of Sect. 5. This task is performed in two steps. The first one discusses the existence of solutions to the initial value problem connected to (1.3) (Theorem 9). Relevant Theorem 10 is the keystone on which the uniqueness feature is built. This second step permits us obtaining the proof of our main statement.

Notation
In what follows, we assume N ≥ 2 and denote H N−1 the (N − 1)-dimensional Hausdorff measure in ℝ N . Bounded domains ⊂ ℝ N are supposed to be of class C 1, . Thus, an outward unit normal (x) is defined for all x ∈ . Lebesgue and Sobolev spaces are denoted by L q ( ) and W 1,p 0 ( ) , respectively. The space of functions of bounded variation is denoted by BV( ) . It consists of those L 1 -functions whose distributional gradient is a Radon measure with finite total variation. Even though derivatives of members in BV( ) are not functions, they exhibit traces in L 1 ( ) , while this space enjoys the same ranges of continuous and compact embeddings than W 1,1 ( ) . We regard BV( ) endowed with the norm and refer to [1] for a comprehensive account on the theory of functions of bounded variation.
A substantial part of this work is focused on radial solutions. So we deal with a ball in ℝ N centered at the origin and of radius R > 0 , it will be denoted by B R . Observe that a radial function u ∈ W 1,p x |x| (see further details in Sect. 3). In the same vein, a radial function u ∈ BV(B R ) satisfies where v ∈ L 1 ((0, R), r N−1 dr) . However, v ′ is now a Radon measure in (0, R) with total variation |v ′ | so that the measure r N−1 |v � | is finite. Moreover, the identity where N N = H N−1 ( B(0, 1)) , holds true for all radial test functions (|x|) in C ∞ 0 (B R ) (precise details are omitted for brevity).
The space of continuous functions C(J) on an interval J is regarded with the uniform convergence on compacta (a similar remark applies to C 1 (J)).
Finally, for a given measurable function u in , the notation will be used to mean that v ∈ L ∞ ( ) satisfies ‖v‖ ∞ ≤ 1 and v(x)u(x) = |u(x)| a. e. in . Accordingly, infinitely many v's can be found whenever u vanishes in a positive measure set.

Logistic p-Laplacian problems
Although we are mainly interested in the radial case, the introduction of some general properties of the nonlinear problem is quite convenient for later reference. Henceforth, exponents p, q fall in the range, For its use in this section, we introduce the notion of weak solution to (2.2).
The requirement u ∈ L q ( ) is natural if one thinks of the variational formulation of (2.2). In addition, since elements v ∈ W 1,p 0 ( ) ∩ L q ( ) can be approximated in this space by functions of C 1 0 ( ) then test functions in W 1,p 0 ( ) ∩ L q ( ) can be also inserted in (2.4). Finally, we are next showing that weak solutions lie on L ∞ ( ) and so we can test in (2.2) with arbitrary v ∈ W 1,p 0 ( ). Some important features of (2.2) are the goal of the following result.  v ∈ sign (u) (iv) For every > 1,p there exists a unique positive solution u to (2.2). Family u is smooth and increasing in while uniformly on compact sets of .
The assertion of the C 1, smoothness of solutions follows from the estimate (2.5) and the classical results in [14,43].
The existence of a positive solution when > 1,p is obtained by using, say the method of sub-and super-solutions. See for instance [11] and [23] (see also [12] provided that p ≥ 2 ). It is sufficient to choose u − = 1 (⋅) , > 0 small enough, 1 a first positive eigenfunction, as a sub-solution and u + = 1 q−p as a super-solution. Uniqueness of a positive solution is a consequence of [13]. The family u is increasing in . Indeed, it is implicit in the fact that u 0 becomes a sub-solution of (2.2) for > 0 . Finally, asymptotic estimate (2.6) and further features on (2.2) are addressed in [22]. ◻

Remark 1
Only the regime 1 < p ≤ 2 is our main concern in this work. However, the complementary range p > 2 enjoys especial phenomena, the most relevant being that the flat core O = {u (x) = 1 q−p } becomes nonempty and converges to as → ∞ [22,26].

Remark 2
By means of variational methods, one can show the existence of further nontrivial (two-signed) solutions to (2.2), for as large as desired. In fact, the number of these solutions grows beyond any bound as → ∞ . See for instance [20] for this kind of results.

The 1-Laplacian limit problem
The main objective of this work is to let p go to 1 in problem (2.2) and obtaining limits of solutions. Accordingly, an important part of our endeavor will be to analyze the resulting Dirichlet problem deduced from (2.2) as p → 1 . Namely: The concept of solution to this problem relies on Anzellotti's theory (see [6]), which we next recall. Given ∈ L ∞ ( , ℝ N ) and u ∈ BV( ) , it was introduced a distribution in [6] which resembles the dot product ⋅ Du for pairs ( , u) satisfying certain compatibility conditions. For instance, div ∈ L N ( ) and u ∈ BV( ) or div ∈ L r ( ) and A further feature of the theory in [6] is the notion of weak trace on of the normal component, denoted [ , ] , of a field ∈ L ∞ ( , ℝ N ) . In fact, under the assumption that div is a finite Radon measure, the trace is appropriately defined, satisfies [ , ] ∈ L ∞ ( ) and ‖[ , ]‖ L ∞ ( ) ≤ ‖ ‖ L ∞ ( ,ℝ N ) . Most importantly, a Green formula connecting the measure ( , Du) and the weak trace [ , ] is established in [6]. Namely: for those pairs ( , u) satisfying the conditions already mentioned (see [6]).
We are now ready to introduce the notion of solution to (2.7) which is based on that introduced in [4].
there exist ∈ L ∞ ( , ℝ N ) and ∈ L ∞ ( ) satisfying: is a set of measure zero since then ‖ ‖ ∞ ≤ 1 and ⋅ ∇u = |∇u| implies = ∇u |∇u| . For a general u ∈ BV( ) , Du |Du| cannot belong to L ∞ ( , ℝ N ) . A similar observation applies to which plays the rôle of u |u| and they have the same value when {u = 0} is a null set.
Remark 4 We point out that the Radon measure ( , Du) is well defined since div ∈ L q � ( ) and u ∈ BV( ) ∩ L q ( ) . Moreover, ( , Dv) is defined too whenever v ∈ BV( ) ∩ L q ( ) and so equation in 2) together with (2.10) implies that the equality holds for all these test functions v in BV( ) ∩ L q ( ) . For the moment, we are not allowed to consider ( , Dv) for an arbitrary v ∈ BV( ) . Nevertheless, the next result implies that actually div ∈ L ∞ ( ) , so that ( , Dv) has always a meaning for every v ∈ BV( ).

Proof
Observe that |u| ≥ 1 q−1 implies 1 − |u| q−1 ≤ 0 ; in this case, the right hand side becomes nonpositive, while the left hand side is always nonnegative. So it is enough with choosing k = 1 q−1 to conclude that ‖G k (u)‖ BV( ) = 0 what entails the desired estimate.
(ii) Let u be a nontrivial solution to (2.7). By using u as a test function in Green's formula (2.11), it yields Resorting to conditions 3) and 4) of Definition 2, we get Thus, we infer from (2.13) that and the result follows. ◻ We are next stating that solutions ( p , u p ) to (2.2) converge as p → 1 and up to subsequences, to a solution ( , u) to (2.7), provided that p → .
Furthermore, u defines a solution to problem (2.7) by choosing and as the functions referred to in Definition 2. Remark 5 It is worth remarking that the above theorem could yield the trivial solution. This occurs, for instance, when lim p→1 p = 1 . Notice that lim p→1 1,p = 1 ( [28, Corollary 6]). Accordingly, obtaining a nontrivial solution u requires some extra computations. Indeed, it can be shown that for every > 1 , the limit as p → 1 of the family of positive solutions u to (2.2) defines a nonnegative and nontrivial solution u to (2.7). Details are omitted for brevity.
Proof By setting v = u p in (2.4) and taking into account (2.5), we achieve a uniform estimate of the form for a no depending on p positive constant M. This implies that u p is bounded in BV( ) and modulus a subsequence we find u ∈ BV( ) such that u p → u both a. e. and in L r ( ) as p → 1 , provided that r < N N−1 . However, since u p is uniformly bounded in L ∞ ( ), such a convergence is upgraded to L s ( ) for all s ≥ 1.
The remaining assertions of the theorem are shown by employing similar arguments as in [4] (see also [41,Theorem 6]). Accordingly, their proof is omitted. ◻

Review of the one-dimensional case
For future reference as auxiliary tools, some features of the one-dimensional version of problem (2.2), are next reported (see [40] for a detailed account and [21] for related one-dimensional problems).
To begin with the one-dimensional version, the eigenvalue problem is Its full set of eigenvalues consists in the sequence {̂n ,p }: Notice that lim p→1 t 1 (p) = 2 , hence lim p→1̂n,p = 2n for every n ∈ ℕ.
To study (2.18), it is quite convenient to consider the following initial value problem: where 0 < < 1 plays the rôle of ‖v‖ ∞ and is regarded as a parameter. It can be shown that to every in this range corresponds a unique solution v 0 (t) . Such a solution is described in terms of the function: As key properties, v 0 (t) decreases from to − when , is symmetric with respect to t = T and becomes periodic with period 2T. Going back to (2.18), all the relevant information concerning this problem can be now expressed in terms of v 0 (t) . In this regard, notice that so that v 0 vanishes exactly at the points t = − T( ) 2 + nT( ) . Hence, solutions to problem (2.18) can be viewed as a shift of v 0 . It should be remarked that this solution v 0 depends on , which plays the rôle of the amplitude of v 0 . Taking these facts into account, one deduces the following features.  (2) Zeros of v are exactly t = kT( ) , 0 ≤ k ≤ n , v attains its maximum at and is expressed in this interval by The left hand side can be alternatively written as Property 1) asserts that solving (2.18) amounts to discuss the solutions to (2.19). Next result is just introduced for this and further purposes of the present paper (see [40,Lemma1]).
Proposition 1 Assume that 1 < p ≤ 2 . Then, function T ∶ (0, 1) → ℝ is continuous and increasing. In addition, It should be remarked that eigenvalues ̂n ,p to (2.16) can be expressed as ̂n ,p = (nT(0)) p . These are just the values referred to in the next statement where the solvability of equivalent problems (2.15) and (2.18) is completely described. Its proof reduces to analyze the solutions to (2.19) and is a direct consequence of Proposition 1.
The following auxiliary result addresses the limit behavior as p goes to 1 (see [40, Lemma 2] for a proof). It will be instrumental in the arguments of Sects. 4 and 5.
(a) Function T introduced in (2.20) verifies: where the convergence holds in

Radial solutions
In this section, we study (2.2) in a ball B R = B(0, R) ⊂ ℝ N : As was pointed out in Theorem 1, problem (2.2) exhibits a unique positive solution. Thus, it must be radial if = B R . In fact, uniqueness is in principle necessary since the validity of Gidas-Ni-Nirenberg symmetry for equations − p u = f (u) requires suitable conditions on the nonlinear term f ( [19]). Nevertheless, we are further interested in solutions with both signs and therefore we focus our attention on radial solutions. Assume that ũ ∈ W 1,p 0 (B) is a radially symmetric solution to (3.1), then ũ can be a. e. identified with a function u(r), r = |x| , such that u, |u r | p−2 u r ∈ C 1 [0, R] , u r (0) = u(R) = 0 and pointwise solves, Moreover, we are only concerned with the parameter range 1 < p ≤ 2 . In this case, On the other hand, nontrivial solutions u satisfy the estimate ‖u‖ ∞ ≤ 1 q−p (Theorem 1). Hence, by introducing the scaling nontrivial solutions are sought in the range ‖v‖ ∞ ≤ 1 . In addition, it should be remarked that the decreasing character of the energy E(v, v t ) below (see (3.10) and (3.9)) implies that solutions u to (3.1) satisfying u(0) > 0 achieve their maximum at r = 0 . Accordingly, = v(0) is a natural parameter to describe normalized solutions (3.3). Observe that unlike the one-dimensional case (problems (2.18) and (2.19)), a shift is not necessary now.
So, to handle (3.1) and (3.2), we are led to the initial value problem with 0 < < 1 . Notice that when = 1, the solution to (3.4) is given by v(t) = 1. Main features on (3.4) are next depicted. The sequence of radial eigenvalues =̃n ,p to the Dirichlet problem in the ball B R (see [2,9,44]), is involved in the next and forthcoming statements. Observe that ̃n , Due to our purposes here, exponent p is restricted to the range 1 < p ≤ 2.
(ii) Solution v is oscillatory, i. e., it exhibits a sequence of infinitely many simple zeros, The asymptotic estimate holds true, where n = n − n−1 and t 1 (p) is the value introduced in (2.17). In particular, (iv) Every n defines a continuous function of and, Proof The existence and uniqueness of a local solution v to this problem have been largely discussed in [37] and [23]. That such a solution can be extended to all t > 0 is a consequence of the relation, which express the decaying along solutions of the total energy E defined by We next describe the oscillatory character of v. From the equation, we get, ) must be positive for t small enough, wherewith v ′ < 0 and v decreases in the same interval. Next, we are showing that v must vanish at finite time.
This is again not possible. Finally, by doing v → −v , the conditions on −v for t ≥ 1 are just the same as those for v at the beginning of the reasoning at t = 0 . This shows that v exhibits infinitely many simple positive zeros n (notice that v � ( n ) ≠ 0 ). But v cannot accumulate zeros in (0, ∞) since the only solution to (3.4) Thus, n → ∞ . Moreover, a careful review of the proof permits us concluding the existence of a unique critical point n ∈ ( n , n+1 ) of v for every n. Additionally, the continuous dependence of v on ( [23,37]) entails that every n depends continuously on . To prove (3.6), assume on the contrary that inf ℝ + E > 0 . Then, Let us point out that Ascoli-Arzelà's Theorem implies that Hence, By taking into account both (−1) n v t ( n ) → v � ∞ and n → ∞ , together with the uniform convergence of functions v n and their derivatives, it follows that v = v ∞ (t) solves the problem, We next observe that and, by proceeding as in telescoping series, it leads to Performing a change of variable, we deduce and ∑ ∞ n=1 a n converges. However, while by Cesàro's Theorem Thus the series ∑ ∞ n=1 a n diverges. The contradiction has arisen from assuming that > 0 . Therefore, inf n = 0.
To show (3.7) set n = (−1) n v t ( n ) . Then, due to the fact that together with n → 0 and V(v) ∼ 1 p |v| p as v → 0 , we find that the sequence of functions, On the other hand, v =ṽ n (t) solves the problem, A compactness argument again permits us ensuring that This implies that as desired. The fact 1 ( ) → ∞ as → 1− follows from the continuous dependence of v(⋅, ) on the parameter (see [23]). On the other hand, that 1 increases with is a consequence of the uniqueness of a positive solution to the Dirichlet problem, where c ≥ 0 is constant and B an arbitrary ball (see [13]). Finally and arguing as above, v (t) ∶= 1 v(t, ) solves, and in the limit as → 0+ , v converges in It is well known that exhibits a sequence of positive zeros n → ∞ and that the sequence ̃n ,p = p n just defines the eigenvalues of − p in B 1 [9,41]. On the other hand, the convergence v → in C 1 together with the simplicity of all of the zeros involved entail that n ( ) → n as → 0+ for all n ∈ ℕ .
where 0 < < 1 and for some n ∈ ℕ . Eqs. (3.14), (3.15) define a continuous curve of solutions ( n ( ), u n (⋅, )) parameterized in ∈ (0, 1) . This proves the first assertion in v), while (3.13) is a consequence of inequality (4.10) to be shown in next section. Notice that this curve can be alternatively represented as a (possibly multivalued) family u n, when is regarded as the governing parameter. From (3.15), one finds that u n, is defined for 1 p R > n while the asymptotic behaviors in either (3.11) or (3.12) are a consequence of iv) in Lemma 1. In addition, every solution in u n, vanishes at The uniqueness of a positive solution to (3.1) was already established in Theorem 1.
The characterization of nontrivial solutions asserted in vi) is achieved when such solutions are observed as solving the initial value problem (3.4). ◻

Remark 6
First limits in (3.11) and (3.12) assert that the n-th family bifurcates from u = 0 at =̃n ,p . It was stated in [24] (see also [39]) that such a bifurcation locally occurs in the direction >̃n ,p . However, inequality (3.13) substantially improves this result since it implies that u n, is only defined when >̃n ,p .

Remark 7
In the regime p > 2 , radial solutions u to (3.1) may develop a central core {u = ± 1 q−p } as is large.

Limit as p → 1 : direct approach
It this section, the more subtle question of finding the limit profiles as p → 1+ of the branches of solutions u n, of Theorem 4 is addressed. Our first results provide some partial answers to this problem. In the forthcoming statements, a reference to p is incorporated to the notation whenever it is necessary. For instance, v p (t, ) stands for the solutions to (3.4), while n,p ( ) designates its n-th zero. They are just new names for the former v(t, ) and n ( ) , respectively.

Lemma 2
Let v p (t, ) be the solution to (3.4) and let 1,p ( ) designate its first zero. Then, Moreover, (4.1) for all n ∈ ℕ . This is a quite delicate question. Its proof relies upon the following result, one of the featured achievements in [41]. Notice that relevant quantities, e.g., the radial eigenvalues ̃n ,p , are labeled with subindex p to stress its dependence on p. We now prove that limits in (4.5) are finite. the weight q is defined by and = 0 . Notice now that u vanishes exactly at n − 1 points in the interval (0, 1) and that problem (4.8) has a unique eigenvalue exhibiting an eigenfunction with that property ( [44]). Namely, the n-th eigenvalue n (q) . Therefore,

Now,
But n (q) is increasing in the weight q ( [44]). Thus: The first inequality implies that Thus, and so, The second inequality entails whence, To achieve (4.1) in Lemma 2, observe that ̄1 = N ( [41]). ◻ We now analyze the gap between the values ̄n ( ) ± and its behavior as n becomes large.        = (a, b) is a finite interval, m, q ∈ C(J) . It is well known that 1,p (m, q) is increasing in m and q.
Let v(t) be the solution to (3.4) (subscript p will be omitted whenever possible) and consider the particular case of problem (4.14) where m = t N−1 , q = −t N−1 (1 − |v| q−p ) and J = J n ∶= ( n−1 , n ) . Then, it holds that its main eigenvalue is and has u = v |J n as an associated main eigenfunction. Setting the estimates hold true.
As for (4.11) suppose that v > 0 in J n (otherwise replace v → −v ), set as above n−1 = max J n v and n−1 the critical point in J n . From the fact that v decreases in [ n−1 , n ] an that (4.14) −(m(t)|u � | p−2 u � ) � + q(t)|u| p−2 u = |u| p−2 u t∈ J u | J = 0, 1,p (m, q) = 0, we obtain that for n−1 < t < n . In particular, whence (4.11) follows by taking limits as p → 1 . Similarly, it follows from (4.17) that for fixed n, with n−1 = lim p→1 n−1 . We stress that in Sect. 5, a sharpened version of all of the previous estimates will be stated. Fig. 1 and 2 strongly suggest that all of numbers n,p ( ) and n,p stabilize to single values as p → 1+ . In addition, solution v p (t, ) develops flat patterns between consecutive values of the limits of n,p . This issue is addressed in detail in the next section.  As it is the case when the operator − 1 is involved in the equations, problem (5.1) has a tendency to exhibit an uncontrolled amount of solutions. See for instance [8] dealing with eigenvalue problems, [40] on the one-dimensional case (2.15) or Remark 14 below. To identify proper solutions, we handle an energy condition (see (5.7) below) similar to that introduced in [41].

Remark 9 Numerical simulations in
In order to formulate a uniqueness result, we also require suitable symmetry restrictions on the solutions.

Definition 3
A solution u ∈ BV(B R ) to (5.1) is said to be radial if aside from u, function and field referred to in Definition 2 are also radial. In the latter case, this means that, where w ∈ L ∞ (0, R).

Remark 10 Condition (5.3) is reminiscent of the fact that for a radial
In the next statement, solutions to (5.1) are understood to be radial in the sense of Definition 3. The continuity mentioned in the point iii) below is regarded in the sense of the strict topology of the space BV(B R ) ( [1]).

An initial value problem
We are mimicking the existence analysis in Sect. 3. Our reference initial value problem (3.4) there: is more conveniently written now in the equivalent form ( , u) = (̄n( ),ū n (⋅, )) ∈ ℝ × BV(B R ), ( n ( ), u n (⋅, )) = (̄n( ),ū n (⋅, )), In addition, notation � = d dt will be often used with the meaning v � = v t . A formal expression for the limit problem of (5.8) as p → 1 reads as follows, where v, w vary in suitable spaces of functions defined in (0, ∞) and equations are understood in distributional sense. Precise details to clarify the meaning of a solution to (5.9) are next explained. Of course, we are keeping in mind Definition 2.
As it turns out from the results below, a convenient space for the solutions (v, w) to (5.9) is where we denote, According to Sect. 2.3, a function u belongs to BV(I) with I = (0, b) if u ∈ L 1 (I) and its distributional derivative u ′ is a Radon measure with finite total variation |u � |(I) . As customary, W 1,∞ (I) denotes the space of functions w ∈ L ∞ (I) with a weak derivative w � ∈ L ∞ (I).
It can be shown that every function u ∈ BV(I) can be identified a. e. with a function ũ which is of bounded variation in the classical sense in I (see [1]). The identification of u with ũ is henceforth assumed without further comments. In particular BV(I) ⊂ L ∞ (I).
On the other hand, we point out that the first equation in (5.9) will be satisfied in the sense that the total variation |v ′ | is equal to the product wv ′ . When v ∈ BV(I) and w ∈ W 1,∞ (I) such a product is naturally defined as ⟨wv � , ⟩ = ⟨v � , w ⟩ , ∈ C ∞ 0 (I) , since w is a Lipschitz function. Moreover, by suitably approximating w, it follows from the definition of v ′ in the sense of distributions that Hence, wv ′ coincides with the definition of the pairing (w, v � ) introduced in Sect. 2.3 (see (2.8)). It should be also recalled that (w, v � ) is a Radon measure in I such that, for all open interval J ⊂ I , where | ⋅ | means the total variation of the corresponding measure.
plays a substantial rôle in the forthcoming considerations. More properly the formal limit equation of (3.9) as p → 1 will play such a rôle. This formal limit is given by For a function v ∈ BV loc (0, ∞) Eq. (5.10) is understood in distributional sense. Notice that the power term is well defined as v ∈ L ∞ (0, b) for each b > 0. The next definition is an adaptation of a corresponding one in [41] where it was proposed for the study of the limit of the eigenvalue problem (3.5) as p → 1.

Definition 4 A couple of functions
loc (0, ∞) defines a solution to (5.9) provided that the following conditions hold.

Existence results
Our next statement furnishes the existence of a solution to (5.9).

Proof of Theorem 9
We begin by observing that ̄+ n ( ) → ∞ as n → ∞ (see Theorem 6). Thus, if we prove the desired claims on each interval (0,̄+ n ( )) , then it will hold on every Fix n ∈ ℕ and set: r = |x| . Then, u p defines a solution to (3.1) with = p . Take a suitable subsequence as p → 1 (denoted with the same index) to get lim p→1 n,p ( ) =̄+ n ( ) and define = lim p→1 p . Now observe that hypotheses of Theorem 3 hold, so that (2.14) implies the estimate from where, by Young's inequality, we deduce an estimate of {v p } in BV( , n,p ) for all > 0 . On the other hand, applying Theorem 3, we may choose a further subsequence and find radial functions u ∈ BV(B R ) , ∈ L ∞ (B R ) and a field ∈ L ∞ (B R , ℝ N ) satisfying ‖ ‖ ∞ ≤ 1 , ‖ ‖ ∞ ≤ 1 so that assertions 1) to 4) in the theorem are satisfied. Thus, by extracting again a subsequence if necessary, we infer that u p (x) → u(x) a.e. in B R . Summarizing, a sequence p m , no depending on n, can be found so that all of the previous limits hold true as p m → 1+ (subindex m will be omitted).
In the sequel and by abuse of notation, u p (r) and u(r) are replacing u p (x) and u(x) when necessary. The same criterium will be applied to other possible radial functions.
We now set, Assertions i) to v) are next to be verified. Explicit reference to will be avoided whenever possible.
We also need to connect test functions on (0,̄+ n ( )) and test functions on B R . Given ∈ C ∞ 0 (0,̄+ n ( )) , consider (x) = ( |x|) . Owing to Theorem 3, Property 2), we obtain Passing to polar coordinates, we get and so, by scaling separately each integral we arrive at We next observe that both A p → 1 and p → 1 while n,p →̄+ n . In addition, ( p t) → (t) for each t. Thus, The desired convergence follows by directly employing ∈ L s � (0,̄+ n ) in the previous argument.
(iii) First observe that (x) ⋅ x |x| is a weak limit of radial functions and so defines a radial function w(r) . If w(t) =w( −1 t), then w ∈ L ∞ (0,̄+ n ) with ‖w‖ ∞ ≤ 1 . A similar procedure than that developed above gives the weak convergence. To check that equation (5.13) holds, take ∈ C ∞ 0 (0,̄+ n ) and consider As u solves (2.7), we get Passing to polar coordinates leads to By setting t = r, it is found that w solves (5.13). Observe that then and the right hand side is bounded on any interval (a,̄+ n ) with a > 0 . Moreover, we deduce from Lemma 2 that whose solution satisfying w(0) = 0 is given by Thus, w ′ is bounded on (0,̄+ n ) and so w ∈ W 1,∞ (0,̄+ n ) . Actually, w ∈ W 1,∞ (0, +∞) since bounds do not depend on the interval.
iv) Choose ∈ C ∞ 0 (0,̄+ n ) and define ∈ C ∞ 0 (B R ) as in (5.17). It follows from the identity |Du| = ( , Du) as measures that Performing the same manipulations as above and employing (2.1), we obtain, and we are done.
v) For a nonnegative ∈ C ∞ 0 (0,̄+ n ( )) choose now the variant, of the test function defined in (5.17). By Theorem 3, Property 4), and taking once again a subsequence, Passing to polar coordinates, performing separate scalings in the integrals and multiplying by N − 1, we deduce where: Hence, Now recalling (5.15) and taking into account that p → in C ∞ 0 (0,̄+ n ) as p → 1 and in particular, that its support is bounded away from zero, we find that the first limit in (5.18) vanishes. On the other hand, by Lebesgue's theorem, we obtain Thus, we conclude from (5.18) that, and the energy identity (5.10) is proved.
Finally, the other assertion of v) follows immediately from the fact that ‖v p ‖ ∞ = for all 1 < p ≤ 2 . ◻ Figure 1 shows the profiles of v p (t, ) corresponding to = 0.5 , q = 2.5 , N = 2 and decreasing values of p ∈ (1,2] . Flat plateaus arise when p becomes close to one. In strong difference with the 1D case (problem (2.19)), a decaying in the amplitude of the solutions to (3.4) is observed and this feature is transmitted to the limit as p → 1+.

A uniqueness result
The next one is a sort of uniqueness statement for the initial value problem (5.13).
iv) n satisfies the following asymptotic estimate,

Proof of Theorem 10
As a first remark, let (v, w) be any possible solution to (5.9) where 0 < < 1 .
Since v satisfies the energy equation (5.10), it follows that g(v) = |v| − 1 q |v| q is non increasing along the solution. As the function g is increasing in (0, 1), we deduce that |v| is nonincreasing; in particular, |v(t)| ≤ for all t ≥ 0.
We are now following the argument of the proof of [41,Theorem 19]. (2) Nature of (v, w) in the initial component of C . Since |w| < 1 near t = 0, there exists a first component (0, b) in C . From v(0+) = , it follows from 1) that v = in (0, b) while direct integration of (5.12) yields for t ∈ (0, b) , the last equality being implied by the relation w(b) = −1 . Thus, we set 1 = b . Notice that b > N since h( ) < 1 . We next use (5.12) to observe that, for t ≥ 1 . This together with (5.23) implies that,   in (a, b).
(c) The following relation holds true: The finiteness of (a, b) is consequence of the representation which holds in (a, b) and the fact that |w| < 1 . Furthermore, (5.24)  Assume that w(t) = 1 . Then, from (w, v � ) = |v � |, one learns that v � = |v � | and so v is nondecreasing in I. However, (5.12) implies that If v(t 0 ) ≠ 0 at some t 0 ∈ I , then (near t 0 ) |v| q−2 v would be strictly decreasing and this is not possible. The case w(t) = −1 is similarly handled.
(6) Components of C are contiguous in the sense that the upper end of one component coincides with the lower end of another. More precisely, beyond every component exists a further component (b, d) where v(t) = c � and cc ′ < 0 holds. The assertion is a consequence of 3), 4) and 5) (see [41]).

Proof of i), ii), iii)-(5.19).
Starting at the first interval (0, 1 ) with 0 = and by employing step 6), we are attaching successive components, named I n ∶= ( n−1 , n ) . Function v attains the constant value (−1) n n in I n , with n > 0 since signs on these intervals are alternated. Energy condition (5.10) implies that n is not increasing. By (5.25), it is found that n follows the recursive law (5.19). Observe that this law gives 1 = N h( ) as expected. Proof of iii)- (5.20), dependence n ( ) , n ( ) and v). We first discuss equation (5.19) to show that every n can actually be computed. Given n−1 and n−1 , the new term x = n must be found by solving

such an equation is transformed into
It is rather clear that this equation possesses a unique root y =̂n > N which is a smooth function of ̃n −1 . This implies that is the next term in the sequence. Moreover, it also defines a smooth function of both n−1 , n−1 . We emphasize that it follows from n ≥ 1 + n−1 (see 3) a)) that lim n→∞ n = ∞ . Thus, v is defined in (0, ∞).
Function v exhibits a jump at every n . Thus, the energy relation (5.10) implies that which can be written as (5.20): We now check that this recursive relation certainly produces a decreasing sequence 0 < n < . Proceeding by induction, assume that both 0 < n−1 < and n > 1 have already been found. Then, In fact this inequality amounts to , so that as q > 1.
Next, it can be checked that the function x ↦ g(x) + N−1 n x is increasing in the interval has a unique solution x in the range 0 < x < 1 , and such a root must be x = n . Moreover, we deduce This means that n is a smooth function of both n , n−1 . In addition, since n lies in the range where g(⋅) + N−1 n ⋅ is increasing, it follows from (5.20) that 0 < n < n−1 . We now proceed recursively and use the dependence of n on n−1 and n−1 shown above, to conclude that n and n are smooth functions of . Moreover, in the particular case n = 1, both functions are increasing in .
Estimate (5.22) in assertion v) is shown by direct substitution and the help of [41,Theorem 19].
Proof of n → 0 and estimate (5.21). By setting, then (5.19) leads to whence lim a n = 1 . On the other hand, as n → ∞ where ̄= lim n = inf n . We are next showing that ̄= 0 so the proof of estimate (5.21) is attained. Accordingly, let us verify that ̄= 0 . For n fixed choose 0 < a < b so that,

3
The decaying character of the energy E: and equation ( for a certain constant C > 0 . Therefore, ̄ must be zero. ◻

Remark 13
For 0 < < 1 the sequence n of values obtained in Theorem 10 are denoted in the sequel as ̄n ( ) . This is done to highlight on the one hand their dependence on , and on the other its rôle as a limit when p → 1 . Next statement clarifies this last remark. Notations v p (⋅, ) and n,p ( ) (beginning of Section 4) are going to be employed. Proof Convergence assertion (5.29) is a consequence of the uniqueness shown in Theorem 10.

Proof of Theorem 8
Let (v(t), w(t)) be the solution to (5.9) introduced in Theorem 10. By setting, we are checking that the assertions in Theorem 8 hold true.
Regarding the property of being a solution to (5.1), we choose where w(r) = w( r) and 1 (t) is just the function, On the other hand, distributions div , ( , Du) and |Du| are invariant under rotations in ℝ N (a detailed checking of this and forthcoming similar assertions is omitted to brief). Accordingly, they are equal provided that take the same values when acting on radial test functions ∈ C ∞ 0 (B R ) , (x) = (|x|) . Thus, to check that (5.1) holds, we observe that both v and w are smooth enough up to t = 0 and that equality is satisfied. It is equivalent to Multiplying by a test function ∈ C 1 [0, R] which vanishes near r = R and integrating by parts we obtain which is the weak version of −div = − |u| q−2 u in polar coordinates. Regarding the identity ( , Du) = |Du|, it suffices with checking it in D( ) ∶= B R ⧵ B for 0 < < R small since it is clearly true near zero. Thus, define as in (5.17) where ∈ C ∞ 0 ( ,̄n( )) . Then, ∈ C ∞ 0 (D( )), while some computations show that Taking the same test functions and , it can be seen that Since (v, w) is the solution to (5.9), it follows that (w, v t ) = |v t | and so ( , Du) = |Du| as measures in D( ) . It yields ( , Du) = |Du| as measures in B R , so that the required coupling between and |Du| is verified. The validity of the energy relation (5.7) is proven by a direct scaling argument based on (5.10).
Regarding the boundary condition, it follows from [1, Theorem 3.87] that the trace of u at R is given by Hence, since [ , ] =w(R) = w(̄n( )) = (−1) n . Parametrization (5.5) for ū n, together with its continuity in is provided by the expression for v(t, ) and the smoothness of ̄n and n with respect to stated in Theorem 10. In addition, crucial relation (5.6) was the objective of Corollary 1.
We next address the uniqueness issue in v). So, let u ∈ BV(B R ) be a radial solution in the sense of Definition 3, with associated function (r) and field =w(r) x r . It is also supposed that u satisfies the energy relation (5.7).
We start with the equation, By testing with radial functions (|x|) ∈ C ∞ 0 (B R ), we obtain By using the limit condition in (5.3), we arrive at Since u ∈ L ∞ (B R ) (Theorem 2), it follows that w ∈ W 1,∞ (0, R) . Moreover, equation is satisfied in weak sense. Define now, Such a limit exists because u is chosen to be of bounded variation in classical sense. In addition, | | < 1 due to (2.12) (Theorem 2) and no generality is lost if we assume that ≥ 0 . We now observe that (5.7) implies that the group |u| − |u| q q is nonincreasing. Therefore, This in particular rules out the option = 0.
Let us introduce now the scalings,  Then, it is found that the pair (v, w) fulfills the properties i), ii) and iii) in Definition 4, where 1 assumes the rôle of in iii), being (0, R ) the reference interval. In addition, a scaling computation ensures us that the energy relation (5.10) holds. Finally, the boundary condition: is satisfied at the endpoint b = R . We now come back to the proof of Theorem 10 and observe that dispose of enough conditions to conclude that v(t) exactly matches, in the interval (0, R ) , the solution obtained in this theorem. Since the boundary condition (5.32) is only fulfilled at the points ̄k ( ), there must exist some n so that, Thus, we have shown that solution u =ū n, with = R −1̄n ( ) . This finishes the proof of Theorem 8.

Remark 14
If we drop condition (5.7), then further families of solutions than those in Theorem 8 can be found. The most simple example is extracted from the solution (v, w) to problem (5.9) defined by together with Then, defines a solution to (5.1) in every ball whose radius is greater than R provided that