Abstract
This work discusses the limit as p goes to 1 of solutions to problem
where \(\varOmega\) is a bounded smooth domain of \({\mathbb {R}}^N\), \(\lambda >0\) is a parameter and the exponents p, q satisfy \(1< p < q\). Our interest is focused on the radially symmetric case. We prove in this radial setting that solutions \(u_p\) to (P) converge to a limit u as \(p\rightarrow 1+\). 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.
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1 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 \(\varOmega \subset {\mathbb {R}}^N\) is a bounded smooth domain and the diffusion is governed by the p–Laplace operator \(\varDelta _p u = \hbox \mathrm{div\,}(|\nabla u|^{p-2}\nabla u)\). Exponents p, q are assumed to satisfy
The number \(\lambda >0\) 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 \(p\rightarrow 1+\). In the first place, this involves discussing the existence of the limit \(u=\lim _{p\rightarrow 1+} u_p\) 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 \(\varDelta _1 = \hbox \mathrm{div\,}\left(\dfrac{\nabla u}{|\nabla u|}\right)\) 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 \(\varOmega \subset {\mathbb {R}}^N\) 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 in population dynamics). The nonlinear diffusion case \(p\ne 2\) 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 \(\varDelta _1\) 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 \(\varOmega\) is plagued of obstacles. To quote only a few, there are not any kind of bifurcation results available from the higher eigenvalues \(\lambda _{n,p}\) of \(-\varDelta _p\) (bifurcation at the first eigenvalue \(\lambda _{1,p}\) has been studied in [9, 15]). The only exception is the radially symmetric case where \(\varOmega\) is a ball [24, 39]. What is worse, the complete spectrum of \(-\varDelta _p\) remains nowadays undetermined [16, 30]. That is why there hardly exist results providing the existence of two signed solutions to (1.1) when \(\lambda\) 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 \(B_R\subset {\mathbb {R}}^N\) are shown to be organized in continuous curves emanating from the radial eigenvalues \({{\tilde{\lambda }}}_{n,p}\) to (2.17). More importantly, it is shown that the interval \(\lambda >{{\tilde{\lambda }}}_{n,p}\) 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 \(p\rightarrow 1+\). 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 \(p\rightarrow 1+\). 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 \(\varOmega \subset {\mathbb {R}}^N\). On the other hand, an uniqueness result allows us concluding the validity of the full limit \(u = \lim _{p\rightarrow 1}u_p\). 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 \({{\bar{\lambda }}}_n\) to \(-\varDelta _1\). 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 \(\varOmega \subset {\mathbb {R}}^N\). 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 \(\theta _n\) of the solutions to the initial value problem associated to (1.1). The analysis of the asymptotic behavior of problem (1.1) as \(p\rightarrow 1\) is launched in Sect. 4. Two preliminary results stating the finiteness of the limits \(\varliminf _{p \rightarrow 1}\theta _n\), \(\varlimsup _{p\rightarrow 1} \theta _n\) (Theorem 6) and proving the validity of the strict inequality \(\varlimsup _{p\rightarrow 1} \theta _n < \varliminf _{p \rightarrow 1}\theta _{n+1}\) (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.
2 Preliminary facts
2.1 Notation
In what follows, we assume \(N\ge 2\) and denote \({\mathscr{H}}^{N-1}\) the \((N - 1)\)-dimensional Hausdorff measure in \({\mathbb {R}}^N\). Bounded domains \(\varOmega \subset {\mathbb {R}}^N\) are supposed to be of class \(C^{1,\alpha }\). Thus, an outward unit normal \(\nu (x)\) is defined for all \(x\in \partial \varOmega\).
Lebesgue and Sobolev spaces are denoted by \(L^q(\varOmega )\) and \(W_0^{1,p}(\varOmega )\), respectively. The space of functions of bounded variation is denoted by \(BV(\varOmega )\). It consists of those \(L^1\)–functions whose distributional gradient is a Radon measure with finite total variation. Even though derivatives of members in \(BV(\varOmega )\) are not functions, they exhibit traces in \(L^1(\partial \varOmega )\), while this space enjoys the same ranges of continuous and compact embeddings than \(W^{1,1}(\varOmega )\). We regard \(BV(\varOmega )\) 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 \({\mathbb {R}}^N\) centered at the origin and of radius \(R>0\), it will be denoted by \(B_R\). Observe that a radial function \(u\in W_0^{1,p}(B_R)\) can be represented as \(u(x)=v(|x|)\) where \(v,v'\in L^p((0,R),r^{N-1}dr)\), \(v'\) being the weak derivative of v, while \(\nabla u(x)=v'(|x|)\dfrac{x}{|x|}\) (see further details in Sect. 3). In the same vein, a radial function \(u\in BV(B_R)\) satisfies \(u(x)=v(|x|)\) where \(v\in L^1((0,R),r^{N-1}dr)\). 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 \(N\omega _N = {\mathscr{H}}^{N-1}(\partial B(0,1))\), holds true for all radial test functions \(\varphi (|x|)\) in \(C_0^\infty (B_R)\) (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 \(\varOmega\), the notation
will be used to mean that \(v\in L^\infty (\varOmega )\) satisfies \(\Vert v\Vert _\infty \le 1\) and \(v(x)u(x)= |u(x)|\) a. e. in \(\varOmega\). Accordingly, infinitely many v’s can be found whenever u vanishes in a positive measure set.
2.2 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).
Definition 1
A weak solution to (2.2) is defined as a function \(u\in W_0^{1,p}(\varOmega )\cap L^q(\varOmega )\) such that equality
is satisfied for all functions \(v\in C_0^1(\varOmega )\).
The requirement \(u\in L^q(\varOmega )\) is natural if one thinks of the variational formulation of (2.2). In addition, since elements \(v\in W_0^{1,p}(\varOmega )\cap L^q(\varOmega )\) can be approximated in this space by functions of \(C_0^1(\varOmega )\) then test functions in \(W_0^{1,p}(\varOmega )\cap L^q(\varOmega )\) can be also inserted in (2.4). Finally, we are next showing that weak solutions lie on \(L^\infty (\varOmega )\) and so we can test in (2.2) with arbitrary \(v\in W_0^{1,p}(\varOmega )\).
Some important features of (2.2) are the goal of the following result.
Theorem 1
Problem (2.2) exhibits the next features.
-
(i)
All possible solutions u belong to \(L^\infty (\varOmega )\) and satisfy the estimate
$$\begin{aligned} \Vert u\Vert _\infty \le \lambda ^{\frac{1}{q-p}}. \end{aligned}$$(2.5) -
(ii)
Nontrivial solutions are only possible for \(\lambda >\lambda _{1,p}\), \(\lambda _{1,p}\) being the first Dirichlet eigenvalue of \(-\varDelta _p\).
-
(iii)
For fixed \(\lambda > \lambda _{1,p}\) there exists \(0< \beta < 1\) not depending on \(\lambda\) varying in bounded intervals such that the whole set of nontrivial solutions to (2.2) constitutes a compact set in \(C^{1,\beta }(\overline{\varOmega })\).
-
(iv)
For every \(\lambda >\lambda _{1,p}\) there exists a unique positive solution \({u_\lambda }\) to (2.2). Family \({u_\lambda }\) is smooth and increasing in \(\lambda\) while
$$\begin{aligned} \lim _{\lambda \rightarrow \lambda _{1,p}}\Vert {u_\lambda }\Vert _\infty = 0,\qquad \lambda ^{-\frac{1}{q-p}}{u_\lambda }\rightarrow 1\quad \lambda \rightarrow \infty , \end{aligned}$$(2.6)uniformly on compact sets of \(\varOmega\).
Proof
We first observe that \(v = (|u|-\lambda ^{\frac{1}{q-p}})^+\mathrm{sign\ }u\in W_0^{1,p}(\varOmega )\cap L^q(\varOmega )\). Then, it can be inserted in (2.4) as a test function leading to:
Thus, \((|u|-\lambda ^{\frac{1}{q-p}})^+=0\) which amounts to \(|u|\le \lambda ^{\frac{1}{q-p}}\).
By choosing \(v=u\) in (2.4) we obtain:
and so we deduce
Hence, \(\lambda >\lambda _{1,p}\).
The assertion of the \(C^{1,\beta }\) smoothness of solutions follows from the estimate (2.5) and the classical results in [14, 43].
The existence of a positive solution when \(\lambda >\lambda _{1,p}\) is obtained by using, say the method of sub- and super-solutions. See for instance [11] and [23] (see also [12] provided that \(p\ge 2\)). It is sufficient to choose \(u^- = \varepsilon \phi _1(\cdot )\), \(\varepsilon > 0\) small enough, \(\phi _1\) a first positive eigenfunction, as a sub-solution and \(u^+ = \lambda ^{\frac{1}{q-p}}\) as a super-solution.
Uniqueness of a positive solution is a consequence of [13]. The family \({u_\lambda }\) is increasing in \(\lambda\). Indeed, it is implicit in the fact that \(u_{\lambda _0}\) becomes a sub-solution of (2.2) for \(\lambda >\lambda _0\). Finally, asymptotic estimate (2.6) and further features on (2.2) are addressed in [22]. \(\square\)
Remark 1
Only the regime \(1<p\le 2\) is our main concern in this work. However, the complementary range \(p>2\) enjoys especial phenomena, the most relevant being that the flat core \({\mathcal {O}}_\lambda = \{{u_\lambda }(x)=\lambda ^{\frac{1}{q-p}}\}\) becomes nonempty and converges to \(\varOmega\) as \(\lambda \rightarrow \infty\) [22, 26].
Remark 2
By means of variational methods, one can show the existence of further nontrivial (two–signed) solutions to (2.2), for \(\lambda\) as large as desired. In fact, the number of these solutions grows beyond any bound as \(\lambda \rightarrow \infty\). See for instance [20] for this kind of results.
2.3 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 \(p\rightarrow 1\). Namely:
The concept of solution to this problem relies on Anzellotti’s theory (see [6]), which we next recall. Given \(\mathbf{z} \in L^\infty (\varOmega , {\mathbb {R}}^N)\) and \(u\in BV(\varOmega )\), it was introduced a distribution in [6] which resembles the dot product \(\mathbf{z}\cdot Du\) for pairs \((\mathbf{z}, u)\) satisfying certain compatibility conditions. For instance, \(\hbox \mathrm{div\,}\mathbf{z}\in L^N(\varOmega )\) and \(u\in BV(\varOmega )\) or \(\hbox \mathrm{div\,}\mathbf{z}\in L^r(\varOmega )\) and \(u\in BV(\varOmega )\cap L^{r'}(\varOmega )\). The distribution \((\mathbf{z},Du):{C}_0^\infty (\varOmega )\rightarrow {\mathbb {R}}\) is defined by
When \(\mathbf{z}\) and u are compatible, every integral in (2.8) is well defined. It is proved in [6] that \((\mathbf{z}, Du)\) is a Radon measure with finite total variation. More precisely, it is shown that for every Borel B set with \(B\subseteq U\subseteq \varOmega\) (U open), it holds
A further feature of the theory in [6] is the notion of weak trace on \(\partial \varOmega\) of the normal component, denoted \(\left[ \mathbf{z}, \nu \right]\), of a field \(\mathbf{z}\in L^\infty (\varOmega ,{\mathbb {R}}^N)\). In fact, under the assumption that \(\hbox \mathrm{div\,}\mathbf{z}\) is a finite Radon measure, the trace is appropriately defined, satisfies \(\left[ \mathbf{z}, \nu \right] \in L^\infty (\partial \varOmega )\) and \(\Vert \left[ \mathbf{z}, \nu \right] \Vert _{L^\infty (\partial \varOmega )} \le \Vert \mathbf{z} \Vert _{L^\infty (\varOmega ,{\mathbb {R}}^N)}\). Most importantly, a Green formula connecting the measure \(\left( \mathbf{z}, Du \right)\) and the weak trace \(\left[ \mathbf{z}, \nu \right]\) is established in [6]. Namely:
for those pairs \((\mathbf{z}, u)\) 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].
Definition 2
A function \(u\in BV(\varOmega )\cap L^q(\varOmega )\) is said to be a solution to problem (2.7) if there exist \(\mathbf{z}\in L^\infty (\varOmega , {\mathbb {R}}^N)\) and \(\beta \in L^\infty (\varOmega )\) satisfying:
-
(1)
\(\Vert \mathbf{z}\Vert _\infty \le 1\) and \(\Vert \beta \Vert _\infty \le 1\),
-
(2)
\(-\hbox \mathrm{div\,}\mathbf{z}=\lambda \beta -|u|^{q-2}u\) in \({{\mathscr{D}}}'(\varOmega )\),
-
(3)
\((\mathbf{z}, Du)=|Du|\) as measures and \(\beta u=|u|\) a.e. in \(\varOmega\),
-
(4)
\([\mathbf{z},\nu ]\in \mathrm{sign\ }(-u)\) \({\mathscr{H}}^{N-1}\)–a.e. on \(\partial \varOmega\).
Remark 3
Conditions \(\mathbf{z}\in L^\infty (\varOmega , {\mathbb {R}}^N)\), with \(\Vert \mathbf{z}\Vert _\infty \le 1\), and \((\mathbf{z}, Du)=|Du|\) indicate that the vector field \(\mathbf{z}\) plays the rôle of \(\displaystyle \frac{Du}{|Du|}\). In fact, they are equal when \(u\in W^{1,1}(\varOmega )\) and \(\{\nabla u=0\}\) is a set of measure zero since then \(\Vert \mathbf{z}\Vert _\infty \le 1\) and \(\mathbf{z}\cdot \nabla u=|\nabla u|\) implies \(\displaystyle \mathbf{z}=\frac{\nabla u}{|\nabla u|}\). For a general \(u\in BV(\varOmega )\), \(\displaystyle \frac{Du}{|Du|}\) cannot belong to \(L^\infty (\varOmega , {\mathbb {R}}^N)\). A similar observation applies to \(\beta\) which plays the rôle of \(\displaystyle \frac{u}{|u|}\) and they have the same value when \(\{u=0\}\) is a null set.
Remark 4
We point out that the Radon measure \((\mathbf{z}, Du)\) is well defined since \(\hbox \mathrm{div\,}\mathbf{z}\in L^{q'}(\varOmega )\) and \(u\in BV(\varOmega )\cap L^q(\varOmega )\). Moreover, \((\mathbf{z}, Dv)\) is defined too whenever \(v\in BV(\varOmega )\cap L^q(\varOmega )\) and so equation in 2) together with (2.10) implies that the equality
holds for all these test functions v in \(BV(\varOmega )\cap L^q(\varOmega )\). For the moment, we are not allowed to consider \((\mathbf{z}, Dv)\) for an arbitrary \(v\in BV(\varOmega )\). Nevertheless, the next result implies that actually \(\hbox \mathrm{div\,}\mathbf{z}\in L^\infty (\varOmega )\), so that \((\mathbf{z}, Dv)\) has always a meaning for every \(v\in BV(\varOmega )\).
In the next statement, \(\lambda _1\) denotes the first Dirichlet eigenvalue of \(-\varDelta _1\) in \(\varOmega\) [8, 28, 41]. As shown in [28], \(\lambda _1\) coincides with the Cheeger constant of \(\varOmega\) and is variationally expressed by
Theorem 2
Let \(q>1\). Then problem (2.7) exhibits the following features.
-
(i)
All possible solutions u belong to \(L^\infty (\varOmega )\) and satisfy the estimate
$$\begin{aligned} \Vert u\Vert _\infty \le \lambda ^{\frac{1}{q-1}}. \end{aligned}$$(2.12) -
(ii)
Nontrivial solutions are only possible for \(\lambda >\lambda _1\).
Proof
-
(i)
Set \(G_k(t) = (|t|-k)^+\mathrm{sign\ }(t)\), \(k>0\), and choose \(v = G_k(u)\in BV(\varOmega )\cap L^q(\varOmega )\) as a test function in (2.11). Then,
Now, it follows as a consequence of [29, Proposition 2.7] (see also [6, Proposition 2.8]) that equality \((\mathbf{z}, Du)=|Du|\) as measures implies that \(({\mathbf {z}},DG_k(u)) = |DG_k(u)|\) and so,
Observe that \(|u|\ge \lambda ^{\frac{1}{q-1}}\) implies \(1 -\frac{\ |u|^{q-1}}{\lambda }\le 0\); in this case, the right hand side becomes nonpositive, while the left hand side is always nonnegative. So it is enough with choosing \(k = \lambda ^{\frac{1}{q-1}}\) to conclude that \(\Vert G_k(u)\Vert _{BV(\varOmega )}=0\) 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. \(\square\)
We are next stating that solutions \((\lambda _p,u_p)\) to (2.2) converge as \(p\rightarrow 1\) and up to subsequences, to a solution \((\lambda ,u)\) to (2.7), provided that \(\lambda _p\rightarrow \lambda\).
Theorem 3
Let \(\{(\lambda _p,u_p)\}_{p>1}\) be a family of nontrivial solutions to (2.2) with \(\lambda _p > \lambda _{1,p}\), the first Dirichlet eigenvalue of \(-\varDelta _p\), such that \(\lim _{p\rightarrow 1+}\lambda _p=\lambda\). Then, up to a subsequence, there exist \(u\in BV(\varOmega )\), \(\mathbf{z}\in L^\infty (\varOmega , {\mathbb {R}}^N)\) and \(\beta \in L^\infty (\varOmega )\) with \(\Vert \mathbf{z}\Vert _\infty \le 1\), \(\Vert \beta \Vert _\infty \le 1\) such that the following properties hold.
-
(1)
\(u_p\rightarrow u\) strongly in \(L^s(\varOmega )\) for all \(1\le s<\infty\).
-
(2)
\(|u_p|^{p-2}u_p\rightharpoonup \beta\) weakly in \(L^s(\varOmega )\) for all \(1\le s<\infty\). Moreover \(\beta u = |u|\) a. e. in \(\varOmega\).
-
(3)
\(|\nabla u_p|^{p-2}\nabla u_p\rightharpoonup \mathbf{z}\) weakly in \(L^s(\varOmega , {\mathbb {R}}^N)\) for all \(1\le s<\infty\).
-
(4)
\(\lim _{p\rightarrow 1+}\int _\varOmega \varphi |\nabla u_p|^p=\int _\varOmega \varphi |Du|\) for every nonnegative \(\varphi \in C_0^\infty (\varOmega )\).
Furthermore, u defines a solution to problem (2.7) by choosing \(\mathbf{z}\) and \(\beta\) 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 \(\lim _{p\rightarrow 1}\lambda _p=\lambda _1\). Notice that \(\lim _{p\rightarrow 1}\lambda _{1,p}=\lambda _1\) ( [28, Corollary 6]). Accordingly, obtaining a nontrivial solution u requires some extra computations. Indeed, it can be shown that for every \(\lambda >\lambda _1,\) the limit as \(p\rightarrow 1\) of the family of positive solutions \({u_\lambda }\) 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 \(BV(\varOmega )\) and modulus a subsequence we find \(u\in BV(\varOmega )\) such that \(u_p\rightarrow u\) both a. e. and in \(L^r(\varOmega )\) as \(p\rightarrow 1\), provided that \(r< \frac{N}{N-1}\). However, since \(u_p\) is uniformly bounded in \(L^\infty (\varOmega ),\) such a convergence is upgraded to \(L^s(\varOmega )\) for all \(s\ge 1\).
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. \(\square\)
2.4 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 \(\{\hat{\lambda }_{n,p}\}\):
Notice that \(\lim _{p\rightarrow 1}t_1(p)=2\), hence \(\lim _{p\rightarrow 1}\hat{\lambda }_{n,p} = 2n\) for every \(n\in {\mathbb {N}}\).
As for (2.15), the scaling \(u(x)=\lambda ^{\frac{1}{q-p}}v(t)\), \(t=\lambda ^{\frac{1}{p}}x\) leads to the equivalent form,
Solutions to (2.15) satisfy the estimate \(\Vert u\Vert _\infty <\lambda ^{\frac{1}{q-p}}\) and hence corresponding solutions to (2.18) verify \(\Vert v\Vert _\infty <1\).
To study (2.18), it is quite convenient to consider the following initial value problem:
where \(0< \alpha < 1\) plays the rôle of \(\Vert v\Vert _\infty\) and is regarded as a parameter. It can be shown that to every \(\alpha\) in this range corresponds a unique solution \(v_0(t)\). Such a solution is described in terms of the function:
where \(V(v) = \frac{1}{p} {|{v}|}^p -\frac{1}{q}{|{v}|}^q\). As key properties, \(v_0(t)\) decreases from \(\alpha\) to \(-\alpha\) when \(0\le t \le T\), vanishes at \(t = \dfrac{T}{2}\), 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 \(t= -\frac{T(\alpha )}{2}+n T(\alpha )\). 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 \(\alpha\), which plays the rôle of the amplitude of \(v_0\). Taking these facts into account, one deduces the following features.
-
(1)
Function
solves (2.18) if and only if there exists \(n\in {\mathbb {N}}\) such that \(\alpha\) solves the equation:
Moreover, (2.21) is the unique solution to (2.18) normalized so as \(v_t(0)>0\), fulfilling \(\max v = \alpha\) and vanishing \(n-1\) times in \((0,\lambda ^{\frac{1}{p}})\).
-
(2)
Zeros of v are exactly \(t=kT(\alpha )\), \(0\le k\le n\), v attains its maximum \(\alpha\) at \(t = \left( \frac{1}{2}+2k\right) T(\alpha )\), \(0\le k \le \left[ \frac{1}{2}(n-\frac{1}{2})\right]\) and its minimum \(-\alpha\) at \(t = \left( \frac{3}{2}+2k\right) T(\alpha )\), \(0\le k\le \left[ \frac{1}{2}(n-\frac{3}{2})\right]\) (here \([\cdot ]\) denotes the integer part).
-
(3)
Function v is increasing in \(\left[ 0,\frac{T(\alpha )}{2}\right]\) and is expressed in this interval by
The left hand side can be alternatively written as \(\psi _0(v(t))\) where \(\psi _0\>:\>[0,\alpha ]\rightarrow \left[ 0,\frac{T(\alpha )}{2}\right]\) is the inverse of v.
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 \(1< p \le 2\). Then, function \(T:(0,1)\rightarrow {\mathbb {R}}\) is continuous and increasing. In addition,
\(t_1(p)\) being the value in (2.17), while
It should be remarked that eigenvalues \({\hat{\lambda }}_{n,p}\) to (2.16) can be expressed as \({\hat{\lambda }}_{n,p} = (nT(0))^p\). 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.
Proposition 2
Problems (2.15) and (2.18) admit a nontrivial solution if and only if,
Moreover, to every
there corresponds exactly n solutions u (respectively, v) to (2.15) (r. (2.18)) satisfying the normalizing condition \(u_x(0)>0\) (r. \(v_t(0)>0\)).
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.
Proposition 3
Assume that \(1< p \le 2\). Then, the following properties hold.
-
(a)
Function T introduced in (2.20) verifies:
$$\begin{aligned} {{\overline{T}}}(\alpha ):= \lim _{p\rightarrow 1}T(\alpha )=\frac{2}{1-\alpha ^{q-1}}\qquad \hbox { for all}\ 0<\alpha <1. \end{aligned}$$(2.24) -
(b)
For \(0< \alpha < 1,\) the function v defined by (2.23), alternatively \(v = \psi _0^{-1}(t)\), satisfies:
$$\begin{aligned} \lim _{p\rightarrow 1}v(t)=\alpha , \end{aligned}$$where the convergence holds in \(C^1\left(0,\frac{1}{1-\alpha ^{q-1}}\right)\).
3 Radial solutions
In this section, we study (2.2) in a ball \(B_R=B(0,R)\subset {\mathbb {R}}^N\):
As was pointed out in Theorem 1, problem (2.2) exhibits a unique positive solution. Thus, it must be radial if \(\varOmega = B_R\). In fact, uniqueness is in principle necessary since the validity of Gidas–Ni–Nirenberg symmetry for equations \(-\varDelta _p u = f(u)\) 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 \({{\tilde{u}}}\in W_0^{1,p}(B)\) is a radially symmetric solution to (3.1), then \({{\tilde{u}}}\) can be a. e. identified with a function u(r), \(r=|x|\), such that \(u,|{u_r}|^{{p}-2}{u_r}\in C^1[0,R]\), \(u_r(0)=u(R)=0\) and pointwise solves,
Moreover, we are only concerned with the parameter range \(1 < p \le 2\). In this case, \(u_r = |{w}|^{{p'}-2}{w}\) where \(p'=\frac{p}{p-1}\) and \(w = |{u_r}|^{{p}-2}{u_r}\). Thus, \(u\in C^2[0,R]\).
On the other hand, nontrivial solutions u satisfy the estimate \({\Vert {u}\Vert }_\infty \le \lambda ^{\frac{1}{q-p}}\) (Theorem 1). Hence, by introducing the scaling
nontrivial solutions are sought in the range \({\Vert {v}\Vert }_\infty \le 1\). 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, \(\alpha = v(0)\) 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 \(0<\alpha <1\). Notice that when \(\alpha =1,\) the solution to (3.4) is given by \(v(t)=1\).
Main features on (3.4) are next depicted. The sequence of radial eigenvalues \(\lambda = {{\tilde{\lambda }}}_{n,p}\) to the Dirichlet problem in the ball \(B_R\) (see [2, 9, 44]),
is involved in the next and forthcoming statements. Observe that \({{\tilde{\lambda }}}_{n,p} = R^{-p}{{\tilde{\lambda }}}_{n,p}(B_1)\) where \({{\tilde{\lambda }}}_{n,p}(B_1)\) are the Dirichlet eigenvalues of \(-\varDelta _p\) in the unit ball \(B_1\).
Due to our purposes here, exponent p is restricted to the range \(1< p\le 2\).
Lemma 1
Assume that p, q satisfy (2.3) while \(1 < p \le 2\). Then, for every \(0< \alpha < 1,\) problem (3.4) satisfies the following properties.
-
(i)
It admits a unique solution \(v=v(\cdot ,\alpha )\) which is defined and \(C^2\) in \([0,\infty )\). Moreover,
$$\begin{aligned} \lim _{t\rightarrow \infty }(v(t),v_t(t)) = (0,0). \end{aligned}$$(3.6) -
(ii)
Solution v is oscillatory, i. e., it exhibits a sequence of infinitely many simple zeros,
$$\begin{aligned} 0< \theta _1< \theta _2 <\cdots , \end{aligned}$$such that \(\theta _n\rightarrow \infty\).
-
(iii)
The asymptotic estimate
$$\begin{aligned} \lim _{n\rightarrow \infty } \varDelta \theta _n = t_1(p) \end{aligned}$$(3.7)holds true, where \(\varDelta \theta _n=\theta _n-\theta _{n-1}\) and \(t_1(p)\) is the value introduced in (2.17). In particular,
$$\begin{array}{*{20}c} {\theta _{n} \sim nt_{1} (p)} & {{\text{as n}} \to \infty .} \\ \end{array} {\text{ }}$$ -
(iv)
Every \(\theta _n\) defines a continuous function of \(\alpha\) and,
$$\begin{aligned} \lim _{\alpha \rightarrow 0+}\theta _n(\alpha )=\omega _n \qquad \& \qquad \lim _{\alpha \rightarrow 1-}\theta _n(\alpha ) = \infty , \end{aligned}$$(3.8)where \(\omega _n = {{\tilde{\lambda }}_{n,p}(B_1)}^{\frac{1}{p}}\) and \(\tilde{\lambda }_{n,p}(B_1)\) is the n–th radial eigenvalue of \(-\varDelta _p\) in \(B_1\). In addition, function \(\theta _1\) is increasing in \(\alpha\).
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,
Here \(v' = v_t\), \(\varphi _r(t) = |{t}|^{{r}-2}{t}\) and \(f(v) = |{v}|^{{p}-2}{v} -|{v}|^{{q}-2}{v}\).
Observe that \(f(v(t))>0\) implies \(v'(t)<0\). Hence, it follows from \(0<\alpha <1\) that f(v(t)) 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. Otherwise, \(0< l< v(t)<\alpha\) and so \(f(v)\ge \delta >0\) for all \(t>0\). Then, one finds \(v'(t)\le -\varphi _{p'}\left(\frac{\delta }{N}\right)t^{p'-1}\) and consequently
which is not possible. Thus, a first zero \(t=:\theta _1\) arises. In addition, a first value \(t=:\tau _1>\theta _1\) exists such that \(v'(\tau _1)=0\). Otherwise, \(v'< 0\) for all \(t\ge \theta _1\) and \(f(v(t))\le -\eta\) for \(t\ge t_1:= \theta _1+\varepsilon\). Then,
This is again not possible. Finally, by doing \(v\rightarrow -v\), the conditions on \(-v\) for \(t\ge \tau _1\) 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 \(\theta _n\) (notice that \(v'(\theta _n)\ne 0\)). But v cannot accumulate zeros in \((0,\infty )\) since the only solution to (3.4) with initial data \(v(t_0)=v'(t_0)=0\) is \(v=0\). Thus, \(\theta _n\rightarrow \infty\). Moreover, a careful review of the proof permits us concluding the existence of a unique critical point \(\tau _n\in (\theta _n,\theta _{n+1})\) of v for every n. Additionally, the continuous dependence of v on \(\alpha\) ( [23, 37]) entails that every \(\theta _n\) depends continuously on \(\alpha\).
To prove (3.6), assume on the contrary that \(\inf _{{\mathbb {R}}^+} E>0\). Then,
Moreover, \(\inf _{{\mathbb {R}}^+} E = V(\underline{\alpha })\). Define
Sequence \(\{v_n\}\) is bounded in \(C^1[0,b]\) for all \(b>0\). In addition, \(v=v_n(t)\) solves
Let us point out that Ascoli–Arzelà’s Theorem implies that
in \(C^{1}[0,b]\), for all \(b>0\). On the other hand, inequalities \(E(\theta _n)\ge E(\tau _n)\ge E(\theta _{n+1})\) yield
Hence,
By taking into account both \((-1)^nv_t(\theta _n)\rightarrow v_\infty '\) and \(\theta _n\rightarrow \infty\), together with the uniform convergence of functions \(v_n\) and their derivatives, it follows that \(v = v_\infty (t)\) solves the problem,
By choosing \(b>T(\underline{\alpha })\), \(T(\cdot )\) being the function defined in (2.20), we obtain
We next observe that
and, by proceeding as in telescoping series, it leads to
Performing a change of variable, we deduce
and \(\sum _{n=1}^\infty a_n\) converges. However,
while by Cesàro’s Theorem
Thus the series \(\sum _{n=1}^\infty a_n\) diverges. The contradiction has arisen from assuming that \(\underline{\alpha }>0\). Therefore, \(\inf \alpha _n = 0\).
To show (3.7) set \(\beta _n = (-1)^nv_t(\theta _n)\). Then, due to the fact that
together with \(\beta _n\rightarrow 0\) and \(V(v)\sim \frac{1}{p}|v|^p\) as \(v\rightarrow 0\), we find that the sequence of functions,
is bounded in \(C^1[0,b]\) for all \(b>0\). On the other hand, \(v= {{\tilde{v}}}_n(t)\) solves the problem,
A compactness argument again permits us ensuring that
in \(C^{1}[0,b]\) where \(v={{\tilde{v}}}(t)\) is the solution to problem
This implies that
as desired.
The fact \(\theta _1(\alpha )\rightarrow \infty\) as \(\alpha \rightarrow 1-\) follows from the continuous dependence of \(v(\cdot ,\alpha )\) on the parameter \(\alpha\) (see [23]). On the other hand, that \(\theta _1\) increases with \(\alpha\) is a consequence of the uniqueness of a positive solution to the Dirichlet problem,
where \(c\ge 0\) is constant and B an arbitrary ball (see [13]).
Finally and arguing as above, \(v_\alpha (t) := \frac{1}{\alpha }v(t,\alpha )\) solves,
and in the limit as \(\alpha \rightarrow 0+\), \(v_\alpha\) converges in \(C^1[0,b]\) for all \(b>0\) to the solution \(\phi (t)\) to
It is well known that \(\phi\) exhibits a sequence of positive zeros \(\omega _n\rightarrow \infty\) and that the sequence \({\tilde{\lambda }}_{n,p} = \omega _n^p\) just defines the eigenvalues of \(-\varDelta _p\) in \(B_1\) [9, 41]. On the other hand, the convergence \(v_\alpha \rightarrow \phi\) in \(C^1\) together with the simplicity of all of the zeros involved entail that \(\theta _n(\alpha )\rightarrow \omega _n\) as \(\alpha \rightarrow 0+\) for all \(n\in {\mathbb {N}}\). \(\square\)
Theorem 4
Assume that \(1 < p \le 2\). Then, problem (3.1) exhibits the following properties.
-
(i)
[Range and amplitude] Nontrivial solutions u are only possible when \(\lambda >{{\tilde{\lambda }}}_{1,p}\) while their normalized amplitude,
$$\begin{aligned} \alpha := \lambda ^{-\frac{1}{q-p}}{\Vert {u}\Vert }_\infty , \end{aligned}$$satisfies \(0< \alpha < 1\).
-
(ii)
[Positive solutions] There exists a unique positive (radial) solution \(u_{{1},\lambda }\) for all \(\lambda >{{\tilde{\lambda }}}_{1,p}\). Moreover,
$$\left\| {u_{{1,\lambda }} } \right\|_{\infty } \to 0 \quad {\text{as}}\;{\mkern 1mu} \lambda \to \tilde{\lambda }_{{1,p}} \qquad {{ \& }}\qquad \lambda ^{{ - {\frac{1}{{q - p}}}}} \left\| {u_{{1,\lambda }} } \right\|_{\infty } \to 1\quad {\text{ as}}\; {\mkern 1mu} \lambda \to \infty$$(3.11) -
(iii)
[Existence of branches] For every \(n\ge 2,\) two symmetric families \(\pm u_{{n},\lambda }(r)\) of nontrivial radial solutions exist which are exactly defined for all \(\lambda > {{\tilde{\lambda }}}_{n,p}\) and satisfy,
$$\left\| {u_{{n,\lambda }} } \right\|_{\infty } \to 0\quad {\text{as}}\;\lambda \to \tilde{\lambda }_{{{\text{n}},{\text{p}}}} \qquad {{ \& }}\qquad \lambda ^{{{\text{ - }}{\frac{{\text{1}}}{{{\text{q - p}}}}}}} \left\| {{\text{u}}_{{{\text{n}},\lambda }} } \right\|_{\infty } \to {\text{1}}\quad {\text{ as}}\;\lambda \to \infty .$$(3.12) -
(iv)
[Nodal properties] Every solution \(u_{n,\lambda }(r)\) satisfies \(u_{{n},\lambda }(0)>0\) and vanishes exactly at \(n-1\) values \(r_k\in (0,R)\).
-
(v)
[Continuity] The n–th family \(u_{{n},\lambda }\) can be globally parameterized, in terms of the normalized amplitude \(\alpha \in [0,1)\), as a continuous curve
$$\begin{aligned} (\lambda ,u) = (\lambda _n(\alpha ),u_n(\cdot ,\alpha )) \end{aligned}$$in \({\mathbb {R}}\times C^2[0,R]\), that is, \(u_{n,\lambda }=u_n(\cdot ,\alpha )\) when \(\lambda = \lambda _n(\alpha )\). Moreover,
$$\begin{aligned} \lambda _n(\alpha )>{\tilde{\lambda }}_{n,p},\qquad \hbox { for all}\ 0< \alpha < 1. \end{aligned}$$(3.13) -
(vi)
[Uniqueness] Let u be a nontrivial solution to (3.1). Then, u belongs to some of the families \(\pm u_{{n},\lambda }\), \(n\in {\mathbb {N}}\), introduced in ii) and iii).
Proof
According to the change (3.3), a nontrivial radial solution u to (3.1) is represented as
where \(0< \alpha < 1\) and
for some \(n\in {\mathbb {N}}\). Eqs. (3.14), (3.15) define a continuous curve of solutions \((\lambda _n(\alpha ),u_n(\cdot ,\alpha ))\) parameterized in \(\alpha \in (0,1)\). 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 \(u_{{n},\lambda }\) when \(\lambda\) is regarded as the governing parameter.
From (3.15), one finds that \(u_{{n},\lambda }\) is defined for \(\lambda ^{\frac{1}{p}} R > \omega _n\) while the asymptotic behaviors in either (3.11) or (3.12) are a consequence of iv) in Lemma 1. In addition, every solution in \(u_{{n},\lambda }\) 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). \(\square\)
Remark 6
First limits in (3.11) and (3.12) assert that the n–th family bifurcates from \(u=0\) at \(\lambda = {{\tilde{\lambda }}}_{n,p}\). It was stated in [24] (see also [39]) that such a bifurcation locally occurs in the direction \(\lambda >{\tilde{\lambda }}_{n,p}\). However, inequality (3.13) substantially improves this result since it implies that \(u_{{n},\lambda }\) is only defined when \(\lambda >{\tilde{\lambda }}_{n,p}\).
Remark 7
In the regime \(p>2\), radial solutions u to (3.1) may develop a central core \(\{u = \pm \lambda ^{\frac{1}{q-p}}\}\) as \(\lambda\) is large.
4 Limit as \(p\rightarrow 1\): direct approach
It this section, the more subtle question of finding the limit profiles as \(p\rightarrow 1+\) of the branches of solutions \(u_{{n},\lambda }\) 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, \(v_p(t,\alpha )\) stands for the solutions to (3.4), while \(\theta _{n,p}(\alpha )\) designates its n–th zero. They are just new names for the former \(v(t,\alpha )\) and \(\theta _n(\alpha )\), respectively.
Lemma 2
Let \(v_p(t,\alpha )\) be the solution to (3.4) and let \({\theta _{1,p}}(\alpha )\) designate its first zero. Then,
Moreover,
the convergence being in the topology of \(C^1\left[0, \frac{{{{\overline{T}}}}(\alpha )}{2}\right)\) where \({{\overline{T}}}(\alpha )\) is the value introduced in (2.24).
Proof
The energy E in (3.10) is decreasing along \(v(t) =v_p(t,\alpha )\) while relation
where \(f(v) = |{v}|^{{p}-2}{v} - |{v}|^{{q}-2}{v}\), reveals that v decreases up to \(t = {\theta _{1,p}}\) (\(\alpha\) is removed to brief). In fact, v decreases until its first critical point \(t= \tau _1 \in ({\theta _{1,p}},{\theta _{2,p}})\). Thus,
which implies that,
In particular, by setting \(t = {\theta _{1,p}}\) we get
Hence, the first inequality in (4.1) follows by taking limits and observing that
Set now,
Function \(v = \psi _0^{-1}(t)\), \(t \in \left[0, \frac{T(\alpha )}{2}\right]\), defines the solution to equation in (2.18) having \(v'(0)>0\) and \(\max v = \alpha >0\) (Sect. 2.4). On the other hand, (4.4) implies that
while Proposition 3 asserts \(\psi _0^{-1}\rightarrow \alpha\) as \(p\rightarrow 1+\).
In addition, Equation (4.3) yields
for \(0< t < \frac{{{{\overline{T}}}}(\alpha )}{2}\). All these facts put together entail (4.2).
The complementary upper estimate in (4.1) is a consequence of Theorem 6 below. \(\square\)
Our next result states the finiteness of the limits,
for all \(n\in {\mathbb {N}}\). 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 \({\tilde{\lambda }}_{n,p}\), are labeled with subindex p to stress its dependence on p.
Theorem 5
Let
be the sequence of Dirichlet radial eigenvalues of the p–Laplacian in the unit ball \(B_1\subset {\mathbb {R}}^N\). Then, the limits
exist for all n. Moreover, \({\bar{\omega }}_n\) is increasing, \(\bar{\omega }_n\rightarrow \infty\) and,
where \(\varDelta {\bar{\omega }}_n = {{\bar{\omega }}}_{n}-{{\bar{\omega }}}_{n-1}\).
We now prove that limits in (4.5) are finite.
Theorem 6
For all \(n\in {\mathbb {N}}\) and \(0\le \alpha < 1\), limits \({{\bar{\theta }}}_n^{\pm }(\alpha )\) in (4.5) are finite. Moreover,
In particular \({{\bar{\theta }}}_n^\pm (\alpha )\rightarrow \infty\) as \(n\rightarrow \infty\).
Proof
Write again \(\theta _n = {\theta _{n,p}}(\alpha )\) for short. Define,
Then, u solves the eigenvalue problem
where operator \({\mathscr{L}}_p\) is defined by (the radial p–Laplacian):
the weight q is defined by
and \(\lambda = 0\). 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 \(\lambda _n(q)\). Therefore,
Now,
But \(\lambda _n(q)\) 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 \({\bar{\omega }}_1 =N\) ( [41]). \(\square\)
We now analyze the gap between the values \({{\bar{\theta }}}_n(\alpha )^\pm\) and its behavior as n becomes large.
Theorem 7
Limits \({{\bar{\theta }}}_n^\pm (\alpha )\) in (4.5) satisfy,
for all \(\in {\mathbb {N}}\). Moreover, for n fixed
where \(\varDelta {\theta _{n,p}}(\alpha ) = {\theta _{n,p}}(\alpha )-{\theta _{n-1,p}}(\alpha )\), while
Furthermore,
Proof
A variant of the argument in the proof of Theorem 6 is going to be employed. Define \(\lambda _1(m,q)\) the first eigenvalue of the problem,
where \(J = (a,b)\) is a finite interval, \(m,q\in C({{\overline{J}}})\). It is well known that \(\lambda _{1,p}(m,q)\) 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 \(J = J_n:= (\theta _{n-1},\theta _n)\). Then, it holds that its main eigenvalue is
and has \(u=v_{|J_n}\) as an associated main eigenfunction. Setting
the estimates
hold true.
The monotonicity of \(\lambda _1\) in (m, q) then implies
where \(\lambda _{1,p}(J_n)=\lambda _{1,p}(m,q)\) for the choices \(m=1\), \(q=0\). Thus,
\(t_1(p)\) being the value provided in (2.17).
The second inequality in (4.16) says that
By taking \(\limsup\) as \(p\rightarrow 1\) we find
which proves (4.12).
Estimate (4.13) follows from (4.12) by noticing (Theorem 5) that \({{\bar{\omega }}}_n-{{\bar{\omega }}}_{n-1}\rightarrow 2\) as \(n\rightarrow \infty\).
As for (4.11) suppose that \(v>0\) in \(J_n\) (otherwise replace \(v\rightarrow -v\)), set as above \(\alpha _{n-1} = \max _{J_n}v\) and \(\tau _{n-1}\) the critical point in \(J_n\). From the fact that v decreases in \([\tau _{n-1},\theta _n]\) an that
we obtain that
for \(\tau _{n-1}<t<\theta _n\). In particular,
whence (4.11) follows by taking limits as \(p\rightarrow 1\). \(\square\)
Remark 8
Upper estimate in (4.7) and the corresponding ones in (4.11) and (4.12) can slightly be refined. By observing that the upper estimate in (4.15) may be replaced by
we obtain the sharper one
where \({\bar{\alpha }}_{n-1} = \varlimsup _{p\rightarrow 1}\alpha _{n-1}\). This in turn implies,
a better alternative than (4.12). Moreover, since \(\lim _{n\rightarrow \infty }\alpha _n = 0,\) it should be expected that \(\lim _{n\rightarrow \infty }\bar{\alpha }_n = 0\). This together with (4.13) would lead to
Similarly, it follows from (4.17) that for fixed n,
with \({\underline{\alpha }}_{n-1} = \varliminf _{p\rightarrow 1}\alpha _{n-1}\).
We stress that in Sect. 5, a sharpened version of all of the previous estimates will be stated.
Remark 9
Numerical simulations in Fig. 1 and 2 strongly suggest that all of numbers \({\theta _{n,p}}(\alpha )\) and \(\alpha _{n,p}\) stabilize to single values as \(p\rightarrow 1+\). In addition, solution \(v_p(t,\alpha )\) develops flat patterns between consecutive values of the limits of \({\theta _{n,p}}\). This issue is addressed in detail in the next section.
5 Limit as \(p\rightarrow 1\): the BV framework
Two main objectives of this work are now to be accomplished. First, to show the existence of the limit as \(p\rightarrow 1\) of the solutions \(u_{{n},\lambda }\) to (3.1) obtained in Theorem 4. Second, to prove that these limits define solutions \({\bar{u}}_{{n},\lambda }\) to the limit problem,
Resulting families \({\bar{u}}_{{n},\lambda }\) give rise to continuous curves bifurcating from the eigenvalues to \(-\varDelta _1\). In fact, radial eigenvalues \(\lambda = {{\bar{\lambda }}}_n\) to
have been recently introduced in the form ( [41]):
\({{\bar{\omega }}}_n\) being the values referred to in Theorem 5.
As it is the case when the operator \(-\varDelta _1\) 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].
In order to formulate a uniqueness result, we also require suitable symmetry restrictions on the solutions.
Definition 3
A solution \(u\in BV(B_R)\) to (5.1) is said to be radial if aside from u, function \(\beta\) and field \({\mathbf {z}}\) referred to in Definition 2 are also radial. In the latter case, this means that,
where \({{\tilde{w}}}\in L^\infty (0,R)\).
Remark 10
Condition (5.3) is reminiscent of the fact that for a radial \(C^1\) function \(u(x) = v(r)\) one has \(\nabla u = v'(r) \dfrac{x}{r}\) where \(v'(0)=0\).
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]).
Theorem 8
The problem (5.1) exhibits the following properties.
-
(i)
[Range of existence and amplitude estimate] Nontrivial solutions \(u\in BV(B_R)\cap L^q(B_R)\) are only possible if \(\lambda >{{\bar{\lambda }}}_1\). Normalized amplitude \(\alpha = \lambda ^{-\frac{1}{q-1}}\Vert u\Vert _\infty\) of solutions satisfies \(0< \alpha < 1\).
-
(ii)
[Existence] To every eigenvalue \({{\bar{\lambda }}}_n,\) there corresponds a symmetric family \(\pm {{\bar{u}}}_{n,\lambda }\) of nontrivial solutions bifurcating from \(u=0\) at \({{\bar{\lambda }}}_n\) which is exactly defined for \(\lambda >{{\bar{\lambda }}}_n\). Moreover, the normalized amplitude of \({{\bar{u}}}_{n,\lambda }\) satisfies
$$\begin{aligned} \lim _{\lambda \rightarrow \infty } \lambda ^{-\frac{1}{q-1}}\Vert {{\bar{u}}}_{n,\lambda }\Vert _\infty = 1. \end{aligned}$$(5.4) -
(iii)
[Continuity of branches] Family \({{\bar{u}}}_{n,\lambda }\) can be represented as a continuous curve
$$\begin{aligned} (\lambda ,u) = ({{\bar{\lambda }}}_n(\alpha ),{{\bar{u}}}_n(\cdot ,\alpha ))\in {\mathbb {R}}\times BV(B_R), \end{aligned}$$when parameterized by the normalized amplitude \(0< \alpha < 1\). This means that \({{\bar{u}}}_{n,\lambda } = {{\bar{u}}}_n(\cdot ,\alpha )\) when \(\lambda = {{\bar{\lambda }}}_n(\alpha )\). More precisely,
$$\begin{aligned} {{\bar{\lambda }}}_n(\alpha ) = R^{-1}{{\bar{\theta }}}_n(\alpha ), \qquad {{\bar{u}}}_n(r,\alpha ) = \lambda ^{\frac{1}{q-1}}\sum _{k=1}^n (-1)^k\alpha _{k-1}\chi _{I_k}(\lambda r), \end{aligned}$$(5.5)\(\chi _{I_k}(t)\) being the characteristic function of the interval,
$$\begin{aligned} I_k = \left(R\frac{{{\bar{\theta }}}_{k-1}(\alpha )}{{{\bar{\theta }}}_n(\alpha )},R\frac{{{\bar{\theta }}}_{k}(\alpha )}{{{\bar{\theta }}}_n(\alpha )}\right), \end{aligned}$$and where \(\alpha _k\), \({{\bar{\theta }}}_k\) are smooth functions of the amplitude \(\alpha\).
-
(iv)
[Convergence of branches and zeros] For every n and \(0<\alpha < 1\)
$$\begin{aligned} \lim _{p\rightarrow 1}\theta _n(\alpha ) = {{\bar{\theta }}}_n(\alpha ), \end{aligned}$$(5.6)while the family \(u_{{n},\lambda }\) converges to the family \({\bar{u}}_{{n},\lambda }\) as \(p\rightarrow 1+\) in the sense:
$$\begin{aligned} \lim _{p\rightarrow 1}(\lambda _n(\alpha ),u_n(\cdot ,\alpha )) = ({{\bar{\lambda }}}_n(\alpha ),{{\bar{u}}}_n(\cdot ,\alpha )), \end{aligned}$$\(\lambda _n\), \(u_n\) being the functions introduced in v) of Theorem 4.
-
(v)
[Uniqueness] Every nontrivial solution u fulfilling the ‘energy’ condition,
$$\begin{aligned} \frac{d}{dr}\left(\lambda |u| - \frac{|u|^q}{q}\right) = -\frac{N-1}{r} |u_r| \qquad \hbox { in}\ {{\mathscr{D}}}'(0,R), \end{aligned}$$(5.7)necessarily belongs to some of the previous families \(\pm {{\bar{u}}}_{n,\lambda }\).
The remaining of this section is devoted to the proof of Theorem 8. Section 5.2 states a compactness result which entails the existence of solutions. The key uniqueness result is presented in Sect. 5.3.
5.1 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
where \(f(v) = |{v}|^{{p}-2}{v} - |{v}|^{{q}-2}{v}\), \(0< \alpha < 1\). In addition, notation \(' = \dfrac{d}{dt}\) will be often used with the meaning \(v'=v_t\).
A formal expression for the limit problem of (5.8) as \(p\rightarrow 1\) reads as follows,
where v, w vary in suitable spaces of functions defined in \((0,\infty )\) 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 \(u\in L^1(I)\) and its distributional derivative \(u'\) is a Radon measure with finite total variation \(|u'|(I)\). As customary, \(W^{1,\infty }(I)\) denotes the space of functions \(w\in L^\infty (I)\) with a weak derivative \(w'\in L^\infty (I)\).
It can be shown that every function \(u\in BV(I)\) can be identified a. e. with a function \({{{\tilde{u}}}}\) which is of bounded variation in the classical sense in I (see [1]). The identification of u with \({{{\tilde{u}}}}\) is henceforth assumed without further comments. In particular \(BV(I)\subset L^\infty (I)\).
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 \(v \in BV(I)\) and \(w\in W^{1,\infty }(I)\) such a product is naturally defined as \(\left\langle {wv^{\prime},\varphi } \right\rangle {\text{ }} = {\text{ }}\left\langle {v^{\prime},w\varphi } \right\rangle\), \(\varphi \in C_0^\infty (I)\), 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 \(J\subset I\), where \(|\cdot |\) means the total variation of the corresponding measure.
Equation (3.9) for the dissipation of the energy
plays a substantial rôle in the forthcoming considerations. More properly the formal limit equation of (3.9) as \(p\rightarrow 1\) will play such a rôle. This formal limit is given by
For a function \(v\in BV_{\text {loc}}(0,\infty )\) Eq. (5.10) is understood in distributional sense. Notice that the power term is well defined as \(v\in L^\infty (0,b)\) 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 \(p\rightarrow 1\).
Definition 4
A couple of functions \((v,w)\in BV_{\text {loc}}(0,\infty )\times W^{1,\infty }_{\text {loc}}(0,\infty )\) defines a solution to (5.9) provided that the following conditions hold.
-
(i)
Function \(\Vert w\Vert _\infty \le 1\) while,
$$(w,v^{\prime } ) = |v^{\prime } |\qquad {\text{in}}\,{\mathscr{D}}^{\prime } ({\text{0}},\infty ).{\text{ }}$$(5.11) -
(ii)
There exists \(\beta \in L^\infty (0,\infty )\) satisfying \(\Vert \beta \Vert _\infty \le 1\), \(\beta v = |v|\) and such that v solves the equation
$$\begin{aligned} w'+\frac{N-1}{t} w - |{v}|^{{q}-2}{v} = -\beta ,\qquad \hbox {in } {{\mathscr{D}}}'(0,\infty )\,. \end{aligned}$$(5.12) -
(iii)
Initial conditions are fulfilled in the following sense,
$$\begin{aligned} v(0+) = \alpha ,\qquad w(0) = 0. \end{aligned}$$
Remark 11
We are showing in Sect. 5.4 that any radial solution in the sense of Definition 3 gives rise, up to scaling, to a solution of problem (5.9).
5.2 Existence results
Our next statement furnishes the existence of a solution to (5.9).
Theorem 9
Fix \(0<\alpha < 1\) and for \(1< p \le 2\) let \((v_p,w_p)\in C^1([0,\infty ))^2\) be the solution to the initial value problem (5.8). Then, up to subsequences,
and (v, w) solves (5.9). More precisely, the following properties hold true.
-
(i)
For each \(b>0\), \(v_p\rightarrow v\) in \(L^1((0,b),t^{N-1}\ dt)\) where \(v\in BV_{\text {loc}}(0,\infty )\).
-
(ii)
There exists \(\beta _1 \in L^\infty (0,\infty )\) with \(\Vert \beta _1\Vert _\infty \le 1\) such that \(|{v_p}|^{{p}-2}{v_p}\rightharpoonup \beta _1\) weakly in \(L^s((0,b),t^{N-1}\ dt)\) for all \(1\le s < \infty\) and \(b >0\). Moreover,
$$\begin{aligned} \beta _1 v = |v|. \end{aligned}$$ -
(iii)
\(w_p\rightharpoonup w\) weakly in \(L^s((0,b),t^{N-1}\ dt)\) for all \(1\le s < \infty\) and all \(b >0\), where \(w \in L^\infty (0,\infty )\cap W^{1,\infty }_{\text {loc}}(0,\infty )\), \(\Vert w\Vert _\infty \le 1\) and solves the equation,
$$- w^{\prime} - \frac{{N - 1}}{t}w + |v|^{{q - 2}} v = \beta _{1} ,\qquad {\text{in}}\,{\mathscr{D}}^{\prime } ({\text{0}},\infty ).{\text{ }}$$(5.13) -
(iv)
Identity \((w,v') = |v'|\) is fulfilled in \({{\mathscr{D}}}'(0,\infty )\).
-
(v)
\(\Vert v\Vert _\infty = \alpha\) while the energy equation (5.10) is satisfied in the sense of \({{\mathscr{D}}}'(0,\infty )\).
Remark 12
Convergence in i) actually holds in \(L^s((0,b),t^{N-1}\ dt)\) for all \(1\le s<\infty\).
Proof of Theorem 9
We begin by observing that \({{\bar{\theta }}}_n^+(\alpha )\rightarrow \infty\) as \(n\rightarrow \infty\) (see Theorem 6). Thus, if we prove the desired claims on each interval \((0,{{\bar{\theta }}}_n^+(\alpha ))\), then it will hold on every (0, b) (\(b>0\)).
Fix \(n\in {\mathbb {N}}\) and set:
\(r = |x|\). Then, \(u_p\) defines a solution to (3.1) with \(\lambda =\lambda _p\). Take a suitable subsequence as \(p\rightarrow 1\) (denoted with the same index) to get \(\displaystyle \lim _{p\rightarrow 1}\theta _{n,p}(\alpha )={{\bar{\theta }}}_n^+(\alpha )\) and define \(\displaystyle \lambda =\lim _{p\rightarrow 1}\lambda _p\). 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 \(BV(\sigma , \theta _{n,p})\) for all \(\sigma >0\). On the other hand, applying Theorem 3, we may choose a further subsequence and find radial functions \(u\in BV(B_R)\), \(\beta \in L^\infty (B_R)\) and a field \(\mathbf{z}\in L^\infty (B_R,{\mathbb {R}}^N)\) satisfying \(\Vert \mathbf{z}\Vert _\infty \le 1\), \(\Vert \beta \Vert _\infty \le 1\) so that assertions 1) to 4) in the theorem are satisfied. Thus, by extracting again a subsequence if necessary, we infer that \(u_p(x)\rightarrow u(x)\) 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 \(p_m\rightarrow 1+\) (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 \(\alpha\) will be avoided whenever possible.
-
(i)
The \(L^1\)–convergence \(u_p\rightarrow u\) implies
and so,
as \(p\rightarrow 1\). Since \(\sigma _p\rightarrow 1\) and v(t) is continuous in \((0,{{\bar{\theta }}}_n^+)\) up to a numerable set we observe that,
From the estimate,
and (5.16) we obtain that,
As \(\theta _{n,p}\rightarrow {{\bar{\theta }}}_n^+\) then we find that \(v_p\rightarrow v\) in \(L^1((0,{{\bar{\theta }}}_n^+),t^{N-1}dt)\). This \(L^1\)–convergence jointly with our BV–estimate gives \(v\in BV(\sigma , {{\bar{\theta }}}_n^+)\) for all \(\sigma >0\). We recall that Lemma 2 yields \(v(t)=\alpha\) on \(\left(0, \frac{1}{1-\alpha ^{q-1}}\right)\). Therefore, \(v\in BV(0, {{\bar{\theta }}}_n^+(\alpha ))\). Finally, as \(p_m\rightarrow 1\) no depends on n, then v is actually defined on the whole interval \((0,+\infty )\) and \(v\in BV_{loc}(0,\infty )\).
-
(ii)
By putting \(\beta _1(t)=\beta (\lambda ^{-1}t),\) one finds that \(\beta _1\in L^\infty (0,{{\bar{\theta }}}_n^+(\alpha ))\), \(\Vert \beta _1\Vert _\infty \le 1\), while the identity \(\beta _1 v=|v|\) is a straightforward consequence of the identity \(\beta u=|u|\) a.e. in \(B_R . \)
We also need to connect test functions on \((0,{{\bar{\theta }}}_n^+(\alpha ))\) and test functions on \(B_R\). Given \(\psi \in C_0^\infty (0,{{\bar{\theta }}}_n^+(\alpha ))\), consider \(\varphi (x) = \psi (\lambda |x|)\). Owing to Theorem 3, Property 2), we obtain
Passing to polar coordinates, we get
and so, by scaling separately each integral we arrive at
where \(\sigma _p = \lambda \lambda _p^{-\frac{1}{p}}\), \(A_p = \lambda _p^{\frac{p-1}{q-p}}\sigma _p^N\). We next observe that both \(A_p\rightarrow 1\) and \(\sigma _p\rightarrow 1\) while \(\theta _{n,p}\rightarrow {{\bar{\theta }}}_n^+\). In addition, \(\psi (\sigma _pt)\rightarrow \psi (t)\) for each t. Thus,
The desired convergence follows by directly employing \(\psi \in L^{s'}(0,{{\bar{\theta }}}_n^+)\) in the previous argument.
-
(iii)
First observe that \(\mathbf{z}(x)\cdot \frac{x}{|x|}\) is a weak limit of radial functions and so defines a radial function \({{\tilde{w}}}(r)\). If \(w(t)={{\tilde{w}}}(\lambda ^{-1}t),\) then \(w\in L^\infty (0,{{\bar{\theta }}}_n^+)\) with \(\Vert w\Vert _\infty \le 1\). A similar procedure than that developed above gives the weak convergence. To check that equation (5.13) holds, take \(\psi \in C_0^\infty (0,{{\bar{\theta }}}_n^+)\) and consider
As u solves (2.7), we get
Passing to polar coordinates leads to
By setting \(t = \lambda r,\) it is found that w solves (5.13). Observe that then
and the right hand side is bounded on any interval \((a,{{\bar{\theta }}}_n^+)\) with \(a>0\). Moreover, we deduce from Lemma 2 that
whose solution satisfying \(w(0)=0\) is given by
Thus, \(w'\) is bounded on \((0,{{\bar{\theta }}}_n^+)\) and so \(w\in W^{1,\infty }(0,{{\bar{\theta }}}_n^+)\). Actually, \(w\in W^{1,\infty }(0,+\infty )\) since bounds do not depend on the interval.
iv) Choose \(\psi \in C_0^\infty (0,{{\bar{\theta }}}_n^+)\) and define \(\varphi \in C_0^\infty (B_R)\) as in (5.17). It follows from the identity \(|Du|=(\mathbf{z}, Du)\) as measures that
Performing the same manipulations as above and employing (2.1), we obtain,
and we are done.
v) For a nonnegative \(\psi \in C_0^\infty (0,{{\bar{\theta }}}_n^+(\alpha ))\) 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 \(\psi _p\rightarrow \psi\) in \(C_0^\infty (0,{{\bar{\theta }}}_n^+)\) as \(p\rightarrow 1\) 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 \({\Vert {v_p}\Vert }_\infty = \alpha\) for all \(1< p \le 2\). \(\square\)
Figure 1 shows the profiles of \(v_p(t,\alpha )\) corresponding to \(\alpha = 0.5\), \(q=2.5\), \(N=2\) and decreasing values of \(p\in (1,2]\). 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 \(p\rightarrow 1+\).
Profiles of \(v_p\) corresponding to \(N=2\), \(q = 2.5\), \(\alpha = 0.5\) and \(p=2\), \(p=1.5\) and \(p=1.1\)
5.3 A uniqueness result
The next one is a sort of uniqueness statement for the initial value problem (5.13).
Theorem 10
Let \(0< \alpha < 1\). Then, the initial value problem (5.9) admits a unique solution \((v,w)\in BV_{\text {loc}}(0,\infty )\times W^{1,\infty }_{\text {loc}}(0,\infty )\) satisfying the energy condition (5.10). Moreover, there exist positive monotone sequences \(\alpha _n\) and \(\theta _n\) which verify
such that the following properties are satisfied.
-
(i)
\(v(t) = (-1)^{n-1}\alpha _{n-1}\) on each interval \((\theta _{n-1},\theta _n)\) wherein \(\alpha _0 = \alpha\) and \(\theta _0 = 0\).
-
(ii)
\(w\in W^{1,\infty }(0,\infty )\), w is strictly monotone on each interval \((\theta _{n-1},\theta _n)\) while \(w(\theta _n) = (-1)^n\) for every \(n\ge 1\).
-
(iii)
Sequences \(\alpha _n\) and \(\theta _n\) satisfy the recurrence relations:
$$\begin{aligned} \frac{h(\alpha _{n-1})}{N} \theta _n^N-\theta _n^{N-1} = \frac{h(\alpha _{n-1})}{N} \theta _{n-1}^N+\theta _{n-1}^{N-1}\qquad n\ge 1, \end{aligned}$$(5.19)where \(\theta _0=0\), \(\alpha _0 = \alpha\), \(h(x) = \mathrm{sign\ }x -|{x}|^{{q}-2}{x}\) while
$$\begin{aligned} g(\alpha _n)+\frac{N-1}{\theta _n}\alpha _n = g(\alpha _{n-1})-\frac{N-1}{\theta _n}\alpha _{n-1} \qquad n\ge 1, \end{aligned}$$(5.20)with \(g(x) = |x|-\frac{1}{q} |x|^q\).
-
(iv)
\(\theta _n\) satisfies the following asymptotic estimate,
$$\begin{aligned} \lim _{n\rightarrow \infty }(\theta _n-\theta _{n-1}) = 2. \end{aligned}$$(5.21) -
(v)
For every \(n\ge 1,\) both \(\theta _n\) and \(\alpha _n\) are smooth functions of \(\alpha \in [0,1)\). Moreover,
$$\begin{aligned} \lim _{\alpha \rightarrow 0+}\theta _n = {{\bar{\omega }}}_n, \end{aligned}$$(5.22)\({{\bar{\omega }}}_n\) being the reference values introduced in Theorem 5.
Proof of Theorem 10
As a first remark, let (v, w) be any possible solution to (5.9) where \(0< \alpha < 1\). Since v satisfies the energy equation (5.10), it follows that \(g(v)= |v|-\frac{1}{q}|v|^q\) is non increasing along the solution. As the function g is increasing in (0, 1), we deduce that |v| is nonincreasing; in particular, \(|v(t)|\le \alpha\) for all \(t\ge 0\).
We are now following the argument of the proof of [41, Theorem 19].
-
(1)
Function v is constant in every component of the set \({\mathcal {C}}:=\{t:|w(t)|< 1\}\). In fact, let (a, b) be any of such components, \(J\subset (a,b)\) an arbitrary open interval. Then,
Since \(\Vert w\Vert _{\infty ,J}<1\) then \(|v'|(J)=0\). Thus, v is constant in J.
-
(2)
Nature of (v, w) in the initial component of \({\mathcal {C}}\). Since \(|w|<1\) near \(t=0,\) there exists a first component (0, b) in \({\mathcal {C}}\). From \(v(0+)=\alpha,\) it follows from 1) that \(v=\alpha\) in (0, b) while direct integration of (5.12) yields
for \(t\in (0,b)\), the last equality being implied by the relation \(w(b)=-1\). Thus, we set \(\theta _1 = b\). Notice that \(b>N\) since \(h(\alpha )<1\). We next use (5.12) to observe that,
for \(t\ge \theta _1\). This together with (5.23) implies that,
-
(3)
Let (a, b) be a component of \({\mathcal {C}}\) where \(v(t) =c\), \(c\ne 0\), is a constant. Then, we claim the validity of the following facts.
-
(a)
(a, b) is finite while \(b-a\ge 1\).
-
(b)
\(\mathrm{sign\ }c = \mathrm{sign\ }w(a)\), being \((\mathrm{sign\ }c)w(t)\) decreasing in (a, b).
-
(c)
The following relation holds true:
$$\begin{aligned} \frac{h(|c|)}{N} b^N -b^{N-1} = \frac{h(|c|)}{N} a^N +a^{N-1}. \end{aligned}$$(5.25)
The finiteness of (a, b) is consequence of the representation
which holds in (a, b) and the fact that \(|w|< 1\). Furthermore, (5.24) implies the second assertion in a) since
To check b), observe first that \(\mathrm{sign\ }c = \mathrm{sign\ }h(c)\) when \(|c|< 1\). If \(w(a)=1\) and \(c<0\), then
due to \(a\ge \theta _1\) and \(|c|\le \alpha\). This would imply that \(w>1\) near \(t= a\) which is not possible. A similar argument allows to deal with the case \(w(a)=-1\) and \(c>0\). Thus, \(w(a)=\mathrm{sign\ }c\). On the other hand,
which is decreasing. Since \((\mathrm{sign\ }c)w(t) = w(a)w(t)\), point b) is proven.
Finally, (5.25) follows by direct integration of (5.12).
-
(4)
Solution v can not undergo a discontinuity at \(\theta \ge \theta _1\) such that \(v(\theta -) = c\ne 0\) and \(v(\theta +)=0\). In fact, since v only has jump discontinuities, that fact and (5.10) would imply
and so
Hence,
which is not possible. We stress that condition (5.10) is essential in this step (see Remark 14 below).
-
(5)
If either \(w(t)=1\) or \(w(t)=-1\) in an whole interval \(I=(a,b)\) then \(v(t)=0\) there. 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 \(v(t_0)\ne 0\) at some \(t_0\in I\), 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 \({\mathcal {C}}\) are contiguous in the sense that the upper end of one component coincides with the lower end of another. More precisely, beyond every component (a, b) in \({\mathcal {C}}\) where \(v(t)= c\ne 0,\) there 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 \((0,\theta _1)\) with \(\alpha _0 = \alpha\) and by employing step 6), we are attaching successive components, named \(I_n:=(\theta _{n-1},\theta _n)\). Function v attains the constant value \((-1)^n\alpha _n\) in \(I_n\), with \(\alpha _n>0\) since signs on these intervals are alternated. Energy condition (5.10) implies that \(\alpha _n\) is not increasing. By (5.25), it is found that \(\theta _n\) follows the recursive law (5.19). Observe that this law gives \(\theta _1=\frac{N}{h(\alpha )}\) as expected.
Proof of iii)–(5.20), dependence \(\theta _n(\alpha )\), \(\alpha _n(\alpha )\) and v).
We first discuss equation (5.19) to show that every \(\theta _n\) can actually be computed. Given \(\alpha _{n-1}\) and \(\theta _{n-1}\), the new term \(x=\theta _n\) must be found by solving
By setting \(y = h(\alpha _{n-1})x\), \({{\tilde{\theta }}}_{n-1}=h(\alpha _{n-1})\theta _{n-1}\) such an equation is transformed into
It is rather clear that this equation possesses a unique root \(y = {\hat{\theta }}_n>N\) which is a smooth function of \({{\tilde{\theta }}}_{n-1}\). This implies that
is the next term in the sequence. Moreover, it also defines a smooth function of both \(\alpha _{n-1}\), \(\theta _{n-1}\).
We emphasize that it follows from \(\theta _n\ge 1+\theta _{n-1}\) (see 3) a)) that \(\lim _{n\rightarrow \infty }\theta _n=\infty\). Thus, v is defined in \((0,\infty )\).
Function v exhibits a jump at every \(\theta _n\). 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 \(0< \alpha _n< \alpha\). Proceeding by induction, assume that both \(0<\alpha _{n-1}< \alpha\) and \(\theta _n >\theta _1\) have already been found. Then,
In fact this inequality amounts to
But \(\theta _n>\frac{N}{h(\alpha _{n-1})}\), so that
as \(q>1\).
Next, it can be checked that the function \(x\mapsto g(x)+\frac{N-1}{\theta _n}x\) is increasing in the interval \(0\le x\le \left(1+ \frac{N-1}{\theta _n}\right)^{\frac{1}{q-1}}\). Thus, equation
has a unique solution x in the range \(0< x < 1\), and such a root must be \(x=\alpha _n\). Moreover, we deduce
This means that \(\alpha _n\) is a smooth function of both \(\theta _n, \alpha _{n-1}\). In addition, since \(\alpha _n\) lies in the range where \(g(\cdot )+\frac{N-1}{\theta _n}\cdot\) is increasing, it follows from (5.20) that \(0< \alpha _n< \alpha _{n-1}\).
We now proceed recursively and use the dependence of \(\theta _n\) on \(\theta _{n-1}\) and \(\alpha _{n-1}\) shown above, to conclude that \(\alpha _n\) and \(\theta _n\) are smooth functions of \(\alpha\). Moreover, in the particular case \(n=1,\) both functions are increasing in \(\alpha\).
Estimate (5.22) in assertion v) is shown by direct substitution and the help of [41, Theorem 19].
Proof of \(\alpha _n\rightarrow 0\) and estimate (5.21). By setting,
then (5.19) leads to
whence \(\lim a_n = 1\). On the other hand,
as \(n\rightarrow \infty\) where \({\bar{\alpha }} = \lim \alpha _n = \inf \alpha _n\). We are next showing that \({\bar{\alpha }} = 0\) so the proof of estimate (5.21) is attained.
Accordingly, let us verify that \({\bar{\alpha }} = 0\). For n fixed choose \(0< a < b\) so that,
The decaying character of the energy E:
and equation (5.10) imply that,
and so,
Thus, the series
converges. On the other hand, it follows from (5.28) and Cesàro’s Theorem that \(\lim _{n\rightarrow \infty }\frac{\theta _n}{n}=\frac{2}{h({\bar{\alpha }})}\). Hence,
for a certain constant \(C>0\). Therefore, \({\bar{\alpha }}\) must be zero.
\(\square\)
Remark 13
For \(0< \alpha < 1\) the sequence \(\theta _n\) of values obtained in Theorem 10 are denoted in the sequel as \({{\bar{\theta }}}_n(\alpha )\). This is done to highlight on the one hand their dependence on \(\alpha\), and on the other its rôle as a limit when \(p\rightarrow 1\). Next statement clarifies this last remark. Notations \(v_p(\cdot ,\alpha )\) and \(\theta _{n,p}(\alpha )\) (beginning of Section 4) are going to be employed.
Corollary 1
Fixed \(0< \alpha <1\), let \(v_p(t,\alpha )\) be the solution (5.8) while \(v(t,\alpha )\) designates the solution to (5.9) computed in Theorem 10. Then, the whole family \(v_p\), not merely a subsequence, satisfies
in \(L^1((0,b),t^{N-1}\ dt)\) for every \(b>0\). Moreover,
for all \(n\in {\mathbb {N}}\), where \({\theta _{n,p}}(\alpha )\) denotes the sequence of zeros of \(v_p\).
Proof
Convergence assertion (5.29) is a consequence of the uniqueness shown in Theorem 10.
To prove (5.30), we proceed by induction and firstly check that \({{\bar{\theta }}}_1^{\pm }={{\bar{\theta }}}_1\) (\(\alpha\) is omitted for simplicity). Thus, choose a subfamily \(v_{p'}\) so that \({{\bar{\theta }}}_{1,p'}\rightarrow {{\bar{\theta }}}_1^+\) while \(v_{p'}\rightarrow {{\tilde{v}}}\) a. e. in \((0,\infty )\) (Theorem 9). Thanks to Theorem 6 (finiteness of limits) and Theorem 7 (gaps between the limits), \({{\tilde{v}}}\ge 0\) in an interval \(({{\bar{\theta }}}_1^+-\delta ,{{\bar{\theta }}}_1^+)\) while \({{\tilde{v}}}\le 0\) in \(({{\bar{\theta }}}_1^+,{{\bar{\theta }}}_1^++\delta )\) for certain \(\delta >0\). But uniqueness entails that \({{\tilde{v}}} = v\) and so \({{\bar{\theta }}}_1^+\) must coincide with \({{\bar{\theta }}}_1\). Otherwise a discrepancy in signs should arise. By the same token, \({{\bar{\theta }}}^-_1\) must be \({{\bar{\theta }}}_1\).
Assume now that \({{\bar{\theta }}}_k^{\pm } = {{\bar{\theta }}}_k\) for \(1\le k \le n\). We are proving that \({{\bar{\theta }}}_{n+1}^+={{\bar{\theta }}}_{n+1}\). In fact, choose again a subfamily \(v_{p''}\) so that \(v_{p''}\rightarrow {{\hat{v}}}\) and satisfying \(\theta _{n+1,p''}\rightarrow {{\bar{\theta }}}_{n+1}^+\) (Theorem 6). Then, Theorem 7 provides some \(\eta >0\) such that \((-1)^{n+1}{{\hat{v}}}\ge 0\) in \(({{\bar{\theta }}}_{n+1}^+,{{\bar{\theta }}}_{n+1}^++\eta )\) and \((-1)^{n+1}{{\hat{v}}}\le 0\) in \(({{\bar{\theta }}}_{n+1}^+-\eta ,{{\bar{\theta }}}_{n+1}^+)\) (in fact this is just the sign in the whole interval \(({{\bar{\theta }}}_n,{{\bar{\theta }}}_{n+1}^+)\)). But again \({{\hat{v}}} = v\) and necessarily \({{\bar{\theta }}}_{n+1}^+ = {{\bar{\theta }}}_{n+1}\), to avoid inconsistency in the signs. For \({{\bar{\theta }}}_{n+1}^- = {{\bar{\theta }}}_{n+1},\) the argument is the same. \(\square\)
Figs. 2 and 3 show plottings of \(v_p(t,\alpha )\) and \(w_p(t,\alpha )\) with the parameters of Figs. 1 but p reduced to \(p=1.001\).
Drawing of \(v_p(t,\alpha )\) for \(N=2\), \(q = 2.5\), \(\alpha = 0.5\) and \(p=1.001\). Profile is dramatically steepened
Drawing of \(w_p(t,\alpha )\) for \(N=2\), \(q = 2.5\), \(\alpha = 0.5\) and \(p=1.001\). The Lipschitz nature of \(w_p\) is clearly reflected
5.4 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 \({{\tilde{w}}}(r) = w(\lambda r)\) and \(\beta _1(t)\) is just the function,
It is clear then that \(\beta u = |u|\).
On the other hand, distributions \(\hbox \mathrm{div\,}{\mathbf {z}}\), \(({\mathbf {z}}, Du)\) and |Du| are invariant under rotations in \({\mathbb {R}}^N\) (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 \(\varphi \in C_0^\infty (B_R)\), \(\varphi (x)=\psi (|x|)\). 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 \(\psi \in C^1[0,R]\) which vanishes near \(r=R\) and integrating by parts we obtain
which is the weak version of \(-\hbox \mathrm{div\,}{\mathbf {z}} = \lambda \beta -|{u}|^{{q}-2}{u}\) in polar coordinates.
Regarding the identity \(({\mathbf {z}}, Du)=|Du|,\) it suffices with checking it in \(D(\sigma ):= B_R\setminus \overline{B_\sigma }\) for \(0< \sigma < R\) small since it is clearly true near zero. Thus, define \(\varphi\) as in (5.17) where \(\psi \in C_0^\infty (\lambda \sigma ,{\bar{\theta }}_n(\alpha ))\). Then, \(\varphi \in C_0^\infty (D(\sigma )),\) while some computations show that
Taking the same test functions \(\psi\) and \(\varphi\), it can be seen that
Since (v, w) is the solution to (5.9), it follows that \((w, v_t)=|v_t|\) and so \(({\mathbf {z}}, Du)=|Du|\) as measures in \(D(\sigma )\). It yields \(({\mathbf {z}}, Du)=|Du|\) as measures in \(B_R\), so that the required coupling between \({\mathbf {z}}\) 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 \([{\mathbf {z}},\nu ] = {{\tilde{w}}}(R) = w({{\bar{\theta }}}_n(\alpha ))=(-1)^n\).
Parametrization (5.5) for \({{\bar{u}}}_{n,\lambda }\) together with its continuity in \(\alpha\) is provided by the expression for \(v(t,\alpha )\) and the smoothness of \({{\bar{\theta }}}_n\) and \(\alpha _n\) with respect to \(\alpha\) stated in Theorem 10. In addition, crucial relation (5.6) was the objective of Corollary 1.
The fact that \({{\bar{u}}}_{n,\lambda }\) bifurcates from zero at \(\lambda = {{\bar{\lambda }}}_n\) follows from (5.31) and the convergence \({{\bar{\lambda }}}_n(\alpha )\rightarrow 0\) as \(\alpha \rightarrow 0+\). Similarly, that \({{\bar{\lambda }}}_n(\alpha )\rightarrow \infty\) as \(\alpha \rightarrow 1-\) proves (5.4).
We next address the uniqueness issue in v). So, let \(u\in BV(B_R)\) be a radial solution in the sense of Definition 3, with associated function \(\beta (r)\) and field \({\mathbf {z}} = {{\tilde{w}}}(r)\frac{x}{r}\). It is also supposed that u satisfies the energy relation (5.7).
We start with the equation,
By testing with radial functions \(\psi (|x|)\in C_0^\infty (B_R),\) we obtain
By using the limit condition in (5.3), we arrive at
Since \(u\in L^\infty (B_R)\) (Theorem 2), it follows that \({{\tilde{w}}} \in W^{1,\infty }(0,R)\). 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, \(|\alpha |<1\) due to (2.12) (Theorem 2) and no generality is lost if we assume that \(\alpha \ge 0\). We now observe that (5.7) implies that the group \(\lambda |u|-\frac{|u|^q}{q}\) is nonincreasing. Therefore,
This in particular rules out the option \(\alpha = 0\).
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 \(\beta _1\) assumes the rôle of \(\beta\) in iii), being \((0,R\lambda )\) 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 \(b=R\lambda\). 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 \((0,R\lambda )\), the solution obtained in this theorem. Since the boundary condition (5.32) is only fulfilled at the points \({{\bar{\theta }}}_k(\alpha ),\) there must exist some n so that,
Thus, we have shown that solution \(u = {{\bar{u}}}_{n,\lambda }\) with \(\lambda = R^{-1}{{\bar{\theta }}}_n(\alpha )\). 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
Observe that a dead core \(\{{{\hat{u}}}_\lambda =0\}\) propagates toward \(x=0\) as \(\lambda \rightarrow \infty\). Many other families of solutions can be obtained. Of course, none of them satisfying the energy condition (5.7).
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Acknowledgements
Both authors have been supported by CECE (Generalitat Valenciana) under project AICO/2021/223. S. Segura de León has also been supported by MCIyU & FEDER, under project PGC2018–094775–B–I00. They would also like acknowledge to the anonymous referee for his/her careful reading of the manuscript.
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Sabina de Lis, J.C., Segura de León, S. p–Laplacian diffusion coupled to logistic reaction: asymptotic behavior as p goes to 1. Annali di Matematica 201, 2197–2240 (2022). https://doi.org/10.1007/s10231-022-01197-8
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DOI: https://doi.org/10.1007/s10231-022-01197-8