Regularity results for the best-Sobolev-constant function

Abstract

We derive some new regularity results for the best-Sobolev-constant function defined by

$$\begin{aligned} q\in [1,p^{\star }]\mapsto \lambda _{q}:=\inf \left\{ \left\| \nabla u\right\| _{p}^{p}\,/\,\left\| u\right\| _{q}^{p}:u\in W_{0}^{1,p} (\Omega )\setminus \{0\}\right\} , \end{aligned}$$

where $\Omega$ is a bounded and smooth domain of $\mathbb {R}^{N},\,1<p<N$ and $p^{\star }:=\frac{Np}{N-p}$. In a previous work, we proved that this function is absolutely continuous and thus its derivative $\lambda _{q}^{\prime }$ exists at almost all $q\in [1,p^{\star }]$. In this paper, we prove that $\lambda _{q}^{\prime }$ exists if, and only if, the functional

$$\begin{aligned} I_{q}(u):=\int \limits _{\Omega }\left| u\right| ^{q}\log \left| u\right| \hbox {d}x \end{aligned}$$

is constant on the set $E_{q}$ of the $L^{q}$-normalized extremal functions corresponding to $\lambda _{q}$. Moreover, we prove that the existence of $\lambda _{q}^{\prime }$ is also equivalent to the continuity at $q$ of the function $s\in [1,p^{\star })\mapsto I_{s}(u_{s})$, where $u_{s}$ is any function in $E_{s}$. It follows from these results that $\lambda _{q}^{\prime }$ exists and is continuous if $q\in [1,p]$ and $\Omega$ is a general bounded domain and also if $q\in (p,p^{\star })$ and $\Omega$ is a ball. After deriving some estimates for $I_{q}(u_{q})$, we also prove, under an expected asymptotic behavior of $u_{q}$, as $q\rightarrow p^{\star }$, that $\lambda _{q}$ is $\alpha$-Hölder continuous in $[1,p^{\star }]$ for any $0<\alpha <1$. As a consequence, this Hölder regularity holds for a general bounded domain $\Omega$ when $p=2$ and also for a ball when $p>1$.

1 Introduction

In this paper, we derive some regularity results for the best-Sobolev-constant function defined by

$$\begin{aligned} q\in [1,p^{\star }]\mapsto \lambda _{q}:=\inf \left\{ \mathcal {R} _{q}(u):u\in W_{0}^{1,p}(\Omega )\setminus \{0\}\right\} \end{aligned}$$

(1)

where $\Omega$ is a bounded and smooth domain of $\mathbb {R}^{N}$, $N\ge 2,\, 1<p<N,\, p^{\star }:=\dfrac{Np}{N-p}$ and

$$\begin{aligned} \mathcal {R}_{q}(u):=\frac{\left\| \nabla u\right\| _{p}^{p}}{\left\| u\right\| _{q}^{p}};\text { }u\in W_{0}^{1,p}(\Omega )\setminus \{0\} \end{aligned}$$

is the Rayleigh quotient associated with the Sobolev immersion $W_{0} ^{1,p}(\Omega )\hookrightarrow L^{q}(\Omega )$. (Here $\left\| \cdot \right\| _{s}:=\left( {\int _{\Omega }}\left| \cdot \right| ^{s}\hbox {d}x\right) ^{\frac{1}{s}}$ denotes the usual norm of $L^{s}(\Omega )$.)

It is well known that the immersion $W_{0}^{1,p}(\Omega )\hookrightarrow L^{q}(\Omega )$ is continuous if $1\le q\le p^{\star }$ and compact if $1\le q<p^{\star }$. Hence, there exists $u_{q}\in W_{0}^{1,p}(\Omega )\setminus \{0\}$ such that $\mathcal {R}_{q}(u_{q})=\lambda _{q}$, if $1\le q<p^{\star }$. Since $\mathcal {R}_{q}$ is homogeneous the extremal function $u_{q}$ associated with $\lambda _{q}$ can be chosen such that $\left\| u_{q}\right\| _{q}=1$. (From now on $u_{q}$ will denote any $L^{q}$-normalized extremal function corresponding to $\lambda _{q}$).

It is straightforward to verify that such a normalized extremal function $u_{q}$ is a weak solution of the Dirichlet problem

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta _{p}u = \lambda _{q}\left| u\right| ^{q-2}u &{} \quad \text {in }\Omega \\ u = 0 &{} \quad \text {on }\partial \Omega , \end{array} \right. \end{aligned}$$

(2)

for the $p$-Laplacian operator $\Delta _{p}u:=\mathrm{div }(\left| \nabla u\right| ^{p-2}\nabla u)$. Hence, classical results imply that $u_{q}\in C^{1,\alpha }(\overline{\Omega })$ for some $0<\alpha <1$. Thus, $u_{q}\in W_{0}^{1,p}(\Omega )\cap C^{1,\alpha }(\overline{\Omega })$ satisfies, for each $q\in [1,p^{\star })$:

$$\begin{aligned} \left\| u_{q}\right\| _{q}=1\quad \text { and }\quad \mathcal {R}_{q} (u_{q})=\left\| \nabla u_{q}\right\| _{p}^{p}=\lambda _{q}. \end{aligned}$$

(3)

Therefore, the infimum in (1) is actually a minimum if $1\le q<p^{\star }$. In the critical case, $q=p^{\star }$, one has $\lambda _{p^{\star }}=S^{p}$, where $S$ is the well-known Sobolev constant (see [4, 20]) and the minimum is not reached if $\Omega$ is a proper subset of $\mathbb {R}^{N}$. We recall that $S$ is explicitly given by

$$\begin{aligned} S:=\sqrt{\pi }N^{\frac{1}{p}}\left( \frac{N-p}{p-1}\right) ^{\frac{p-1}{p} }\left( \frac{\Gamma (N/p)\Gamma (1+N-N/p)}{\Gamma (1+N/2)\Gamma (N)}\right) ^{\frac{1}{N}} \end{aligned}$$

(4)

where $\Gamma (t)=\int \nolimits _{0}^{\infty }s^{t-1}e^{-s}\hbox {d}s$ is the Gamma Function.

We remark that $\lambda _{q}$ is simple if $1\le q\le p$ (see [14]), which means that extremal functions associated with $\lambda _{q}$ are scalar multiple one of the other. This property is still valid in the “super-linear” case $p<q<p^{\star }$ if $\Omega$ is a ball (see [1, 15]) or if $p=2$ and $2\le q\le 2+\epsilon$ (see [7]). However, for a general bounded $N$-dimensional domain, $\lambda _{q}$ is not known to be simple in the super-linear case. In fact, $\lambda _{q}$ is not simple if $\Omega$ is an annulus and $q$ is close enough to $p^{\star }$ (see [18]).

Thus, if

$$\begin{aligned} E_{q}:=\left\{ u\in W_{0}^{1,p}(\Omega ):\left\| u\right\| _{q}=1\text { and }\left\| \nabla u\right\| _{p}=\root p \of {\lambda _{q}}\right\} \end{aligned}$$

(5)

denotes the set of the $L^{q}$-normalized extremal functions, it follows that $E_{q}=\left\{ \pm u_{q}\right\}$ whenever $\lambda _{q}$ is simple. Otherwise, $E_{q}$ might be larger than $\left\{ \pm u_{q}\right\}$ and the question of determining the size of $E_{q}$ for a general domain in the “super-linear”case $p<q<p^{\star }$ is still an open problem for which our results might be useful.

Up to our knowledge, very little attention has been paid to the behavior of the function $q\in [1,p^{\star }]\mapsto \lambda _{q}$ in the literature.

In [5] Benci and Cerami considered the function $(q,\mu )\in (2,2^{\star })\times [0,+\infty )\mapsto m(q,\mu )$ where

$$\begin{aligned} m(q,\mu ):=\inf \left\{ \left\| \nabla u\right\| _{2}^{2}+\mu \left\| u\right\| _{2}^{2}:u\in W_{0}^{1,2}(\Omega )\text { and }\left\| u\right\| _{q}=1\right\} . \end{aligned}$$

(6)

This minimizing problem is closely associated with Dirichlet problem

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u+\mu u = \left| u\right| ^{q-2}u &{}\quad \text {in }\Omega \\ u = 0 &{}\quad \text {on }\partial \Omega . \end{array} \right. \end{aligned}$$

(7)

Benci and Cerami proved that for each fixed $\mu \ge 0$ there exists $q_{\mu }\in (2,2^{\star })$ such that (7) has at least $\mathrm{cat }\Omega$ positive solutions whenever $q_{\mu }\le q\le 2^{\star }$ (here $\mathrm{cat }\Omega$ denotes the Lyusternik–Shnirel’man category of $\overline{\Omega }$ in itself). In order to reach this remarkable multiplicity result they verified (see [5, Leema 4.1]) that for each $\mu \ge 0$ fixed the function $q\in (2,2^{\star })\mapsto m(q,\mu )$ is (left) continuous at $q=2^{\star }$. We observe that for $\mu =0$ (and $p=2$) this means that the function $q\in [1,2^{\star }]\mapsto \lambda _{q}$ is (left) continuous at $q=2^{\star }$. (We remark that Benci and Cerami [5] also proved in the same multiplicity result for all $\mu \ge \mu _{q}$ where $\mu _{q}\ge 0$ is obtained from each fixed $q\in (2,2^{\star })$.)

With respect to $p>1$, Huang proved in [13, Thm 2.1] that the function $q\in [1,p^{\star }]\mapsto \lambda _{q}$ is continuous in the open interval $(1,p)$ and lower semi-continuous in the open interval $(p,p^{\star })$.

We have recently started, in [10], a study on the behavior of the function $q\in [1,p^{\star }]\mapsto \lambda _{q}$ proving that it is decreasing, Lipschitz continuous in $[1,p^{\star }-\epsilon ]$ for each $\epsilon >0$ and left-continuous at $q=p^{\star }$, thus obtaining its absolute continuity in $[1,p^{\star }]$. Therefore, for almost all $q\in [1,p^{\star }]$, the derivative $\lambda _{q}^{\prime }$ of $\lambda _{q}$ exists. Using a different method, these results were extended in [2], to allow $q\in (0,1)$ and $p\ge N$ as well (so, $p^{\star }=\infty$ in this case).

In [9] we studied the asymptotic behavior, as $q\rightarrow p$, of the positive solutions of the “resonant Lane–Emden problem”

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta _{p}u = \lambda _{p}\left| u\right| ^{q-2}u &{}\quad \text {in }\Omega \\ u = 0 &{}\quad \text {on }\partial \Omega \end{array} \right. \end{aligned}$$

and obtained, as a byproduct of our main result, the existence of $\lambda _{p}^{\prime }$.

In the present paper, we give two characterizations of the existence of $\lambda _{q}^{\prime }$ and prove a Hölder regularity result for $\lambda _{q}$.

In Sect. 2, we first prove that the derivative $\lambda _{q}^{\prime }$ exists if, and only if the functional

$$\begin{aligned} I_{q}(u):=\int \limits _{\Omega }\left| u\right| ^{q}\log \left| u\right| dx \end{aligned}$$

(8)

is constant on the set $E_{q}$. It follows that $I_{q}$ is constant on $E_{q}$ for almost all $q\in (p,p^{\star })$ which is a new and nontrivial property that, at least in principle, could be used to prove results on simplicity of $\lambda _{q}$ for suitable domains or even to give estimates for the number of extremal functions associated with $\lambda _{q}$.

Thereafter, still in Sect. 2, we prove that the existence of $\lambda _{q}^{\prime }$ is also equivalent to the continuity of the function $s\in [1,p^{\star }]\mapsto I_{s}(u_{s})$ at $s=q$.

As consequence of these characterizations, we conclude that the function $s\in [1,p^{\star }]\mapsto \lambda _{s}$ is continuously differentiable at $s=q$ whenever $\lambda _{q}$ is simple. Thus, for a general bounded $\Omega$ and $p>1$ one always has $\lambda _{s}\in C^{1}([1,p])$. If $p=2$ one also has $\lambda _{s}\in C^{1}([1,2+\epsilon ])$, for some $\epsilon >0$ and if $p>1$ and $\Omega$ is a ball then $\lambda _{s}\in C^{1}([1,p^{\star }))$.

In order to obtain both characterizations of the differentiability, we first prove that

$$\begin{aligned} \lambda _{q}^{\prime }+\frac{p}{q}I_{q}(u_{q})\lambda _{q}=0 \end{aligned}$$

at each point $q\in [1,p^{\star }]$ where the derivative $\lambda _{q}^{\prime }$ exists. Thus, $\lambda _{q}$ satisfies a simple linear and homogeneous ordinary differential equation in the Carathéodory sense (that is, almost everywhere). Then, we combine this fact with the absolute continuity of the function $q\in [1,p^{\star }]\mapsto \lambda _{q}$ to obtain the following representation formula:

$$\begin{aligned} \lambda _{q}=\lambda _{1}\exp \left( -p\int \limits _{1}^{q}\frac{1}{s}\int \limits _{\Omega }\left| u_{s}\right| ^{s}\log \left| u_{s}\right| \hbox {d}x\hbox {d}s\right) ,\quad \text {for all }q\in [1,p^{\star }], \end{aligned}$$

(9)

where each extremal function $u_{s}$ can be arbitrarily chosen in $E_{s}$ for each $s\in [1,p^{\star })$.

In Sect. 3, we prove a Hölder regularity at $p^{\star }$ after deriving some estimates for $I_{q}(u_{q})$. More precisely, we prove that $\lambda _{q}$ is $\alpha$-Hölder continuous in the interval $[1,p^{\star }]$, for any $0<\alpha <1$, under the additional hypothesis

$$\begin{aligned} \limsup _{q\rightarrow p^{\star }} (p^{\star }-q)\left\| u_{q}\right\| _{\infty }^{\gamma }<\infty \end{aligned}$$

(10)

for some $\gamma >0$. This asymptotic behavior holds true, for instance, if $\Omega$ is a ball and $p>1$ and also if $p=2$ and $\Omega$ is an arbitrary bounded domain. Thus, at least in these two situations, our results guarantee that $\lambda _{q}$ is $\alpha$-Hölder continuous for any $0<\alpha <1$. However, we believe that (10) holds true for a general bounded and smooth domain, also if $1<p\not =2$.

2 Characterizations of the differentiability

From now on $E_{q}$ denotes the set of the $L^{q}$-normalized extremal functions, defined in (5), and $I_{q}$ denotes the functional defined by (8).

Theorem 1

For each $q\in [1,p^{\star })$, let $u_{q}$ be arbitrarily chosen in $E_{q}$. The following estimates hold for each $q\in (1,p^{\star })$:

$$\begin{aligned} \limsup _{s\rightarrow q^{+}}\frac{\lambda _{q}-\lambda _{s}}{q-s}\le -\frac{p}{q}I_{q}(u_{q})\lambda _{q}\le \liminf \limits _{s\rightarrow q^{-}} \frac{\lambda _{q}-\lambda _{s}}{q-s}. \end{aligned}$$

(11)

Therefore, the function $q\mapsto \lambda _{q}$ satisfies

$$\begin{aligned} \lambda _{q}^{\prime }+\frac{p}{q}I_{q}(u_{q})\lambda _{q}=0 \end{aligned}$$

(12)

at each point $q\in (1,p^{\star })$ where its derivative $\lambda _{q}^{\prime }$ exists.

Proof

Let $q\in (1,p^{\star })$. Since $\int _{\Omega }\left| u_{q}\right| ^{q}\hbox {d}x=1$ we have

$$\begin{aligned} \frac{\hbox {d}}{\hbox {d}s}\left[ \left( \int \limits _{\Omega }\left| u_{q}\right| ^{s}\hbox {d}x\right) ^{\frac{p}{s}}\right] _{s=q}&= \left( \int \limits _{\Omega }\left| u_{q}\right| ^{q}\hbox {d}x\right) ^{\frac{p}{q}}\left( \frac{\hbox {d}}{\hbox {d}s}\left[ \frac{p}{s}\log \left( \int \limits _{\Omega }\left| u_{q}\right| ^{s}\hbox {d}x\right) \right] _{s=q}\right) \\&= \frac{p}{q}I_{q}(u_{q}). \end{aligned}$$

It follows from (3) that

$$\begin{aligned} \lambda _{q}=\int \limits _{\Omega }\left| \nabla u_{q}\right| ^{p}\hbox {d}x\ge \lambda _{s}\left( \int \limits _{\Omega }\left| u_{q}\right| ^{s}\hbox {d}x\right) ^{\frac{p}{s}}. \end{aligned}$$

Therefore, the continuity of the function $q\mapsto \lambda _{q}$ and L’Hôpital’s rule yield

$$\begin{aligned} \liminf \limits _{s\rightarrow q^{-}}\frac{\lambda _{q}-\lambda _{s}}{q-s}&\ge \liminf \limits _{s\rightarrow q^{-}}\lambda _{s}\frac{\left( \int \nolimits _{\Omega }\left| u_{q}\right| ^{s}\hbox {d}x\right) ^{\frac{p}{s}}-1}{q-s}\\&=-\lambda _{q}\lim \limits _{s\rightarrow q^{-}}\frac{\hbox {d}}{\hbox {d}s}\left[ \left( \int \limits _{\Omega }\left| u_{q}\right| ^{s}\hbox {d}x\right) ^{\frac{p}{s}}\right] _{s=q}=-\lambda _{q}\frac{p}{q}I_{q}(u_{q}). \end{aligned}$$

Analogously,

$$\begin{aligned} \limsup _{s\rightarrow q^{+}}\frac{\lambda _{q}-\lambda _{s}}{q-s}\le \limsup _{s\rightarrow q^{+}}\lambda _{s}\frac{\left( \int \nolimits _{\Omega }\left| u_{q}\right| ^{s}\hbox {d}x\right) ^{\frac{p}{s}}-1}{q-s}=-\lambda _{q}\frac{p}{q}I_{q}(u_{q}). \end{aligned}$$

$\square$

We remark that Theorem 1 is valid for any choice of the extremal function $u_{q}$ in $E_{q}$. Therefore, the following consequence is immediate.

Corollary 2

Let $q\in (1,p^{\star })$ be such that $\lambda _{q}^{\prime }$ exists. Then the functional $I_{q}$ is constant on $E_{q}$.

In the sequel $f:[1,p^{\star })\rightarrow \mathbb {R}$ denotes the function defined by

$$\begin{aligned} f(q):=\frac{p}{q}I_{q}(u_{q}) \end{aligned}$$

where, for each $q\in [1,p^{\star })$, $u_{q}$ is arbitrarily chosen in $E_{q}$.

Now, we prove the representation formula (9) for the function $q\in [1,p^{\star }]\mapsto \lambda _{q}$.

Theorem 3

It holds

$$\begin{aligned} \lambda _{q}=\lambda _{1}e^{-\int \nolimits _{1}^{q}f(s)\mathrm{d}s} \quad \mathrm{for \; all }\,\quad q\in [1,p^{\star }]. \end{aligned}$$

(13)

Proof

Since the function $q\in [1,p^{\star }]\mapsto \lambda _{q}$ is absolutely continuous and its image is a closed interval $[a,b]\subset (0,\infty )$ we also have that the function $q\in [1,p^{\star }]\mapsto \log \lambda _{q}$ is absolutely continuous. Therefore,

$$\begin{aligned} \log (\lambda _{q}/\lambda _{1})=\int \limits _{1}^{q}\left( \log \lambda _{s}\right) ^{\prime }\hbox {d}s=-\int \limits _{1}^{q}f(s)\hbox {d}s \quad \hbox {for all}\quad q\in [1,p^{\star }] \end{aligned}$$

where the last equality follows from (12). Now, (13) follows after exponentiation. $\square$

Since $\lambda _{p^{\star }}=S^{p}$ we note from (13) that

$$\begin{aligned} \lambda _{q}=S^{p}\exp \left( p\int \limits _{q}^{p^{\star }}\frac{1}{s^{2}}\int \limits _{\Omega }\left| u_{s}\right| ^{s}\log \left| u_{s}\right| ^{s}\hbox {d}x\hbox {d}s\right) \text { for all }\quad q\in [1,p^{\star }]. \end{aligned}$$

(We recall that the Sobolev constant $S$ is explicitly given by (4).)

The following result is an immediate consequence of (13). We leave its proof to the reader.

Corollary 4

Let $q\in [1,p^{\star })$ be a point where $f$ is continuous. Then the derivative $\lambda _{q}^{\prime }$ exists and is given by the continuous expression $\lambda _{q}^{\prime }=-\lambda _{q}f(q)$.

Proposition 5

Suppose that $I_{q}$ is constant on $E_{q}$ for some $q\in [1,p^{\star })$. Then $f$ is continuous at $q$. In particular, $f$ is continuous at each point $q$ where $\lambda _{q}^{\prime }$ exists.

Proof

Let $q_{n}\rightarrow q$. Then $\lambda _{q_{n}}\rightarrow \lambda _{q}$ and, up to subsequences, $u_{q_{n}}$ converges in $C^{1}(\overline{\Omega })$ to a function $u$ satisfying (2) and such that $\left\| u\right\| _{q}=1$. This last claim follows by combining the classical Hölder regularity result (see [17]) with the fact that $\left\| u_{q_{n}}\right\| _{\infty }$ is bounded from above by a constant which is uniform with respect to $n$ (see Lemma 11 in the next section).

Therefore, $u\in E_{q}$ and thus

$$\begin{aligned} f(q_{n})\!=\!\frac{p}{q_{n}}\int \limits _{\Omega }\left| u_{q_{n}}\right| ^{q_{n} }\log \left| u_{q_{n}}\right| \hbox {d}x\rightarrow \frac{p}{q}\int \limits _{\Omega }\left| u_{q}\right| ^{q}\log \left| u_{q}\right| \hbox {d}x\!=\!\frac{p}{q}I_{q}(u)\!=\!\frac{p}{q}I_{q}(u_{q})=f(q), \end{aligned}$$

since $I_{q}$ is constant on $E_{q}$. $\square$

Corollary 6

The function $q\mapsto \lambda _{q}$ is continuously differentiable in

• 1. $[1,p]$ if $p>1$,

• 2. $[1,2+\epsilon ]$ (for some $\epsilon >0$) if $p=2$,

• 3. $(p,p^{\star })$ if $\Omega$ is ball.

Proof

Since $I_{q}(u_{q})=I_{q}(\left| u_{q}\right| )$, this corollary follows from the fact that $\lambda _{q}$ is simple in all of these cases. $\square$

Combining corollaries 2 and with Proposition 5 we obtain:

Theorem 7

The following assertions on a point $q\in [1,p^{\star })$ are equivalent:

• 1. $\lambda _{q}^{\prime }$ exists.

• 2. $I_{q}$ is constant on $E_{q}$.

• 3. The function $s\in [1,p^{\star }]\mapsto I_{s}(u_{s})$ is continuous at $s=q$.

We observe that the proof of Corollary also works for $q=p^{\star }$ if the limit $\lim \limits _{q\rightarrow p^{\star }}I_{q}(u_{q})$ exists. However, even the verification that $I_{q}(u_{q})$ is finite at $q=p^{\star }$ does not seem to be a simple task. One of the difficulties is that $\left\| u_{q}\right\| _{\infty }\rightarrow \infty$ as $q\rightarrow p^{\star }$. In fact, otherwise we reach a contradiction by applying a regularity result as in [21] and then making $q\rightarrow p^{\star }$ in the equation $-\Delta _{p}u_{q}=\lambda _{q}u_{q}^{q-1}$, thus obtaining a limit function $u\in W_{0}^{1,p}(\Omega )$ that minimizes the Rayleigh quotient $\mathcal {R}_{p^{\star }}(\Omega )$. Since $\Omega \not =\mathbb {R}^{N}$, this is absurd.

In the next section, we derive some estimates for $I_{q}(u_{q})$ and use them to prove that, under an additional (and natural) hypothesis on the behavior of $\left\| u_{q}\right\| _{\infty }$ (as $q$ tends to $p^{\star }$) the function $q\mapsto I_{q}(u_{q})$ belongs to $L^{r}([0,p^{\star }])$, for all $r>1$. Unfortunately, the $L^{r}$ bounds we obtain from these estimates are just linear with respect to $r$ and thus not enough to prove that $I_{q} (u_{q})$ stays bounded as $q\rightarrow p^{\star }$.

3 Hölder regularity

In order to obtain the absolute continuity of the function $\lambda _{q}$ in [10] we first prove that this function is Lipschitz continuous in each closed interval of the form $[1,p^{\star }-\epsilon ]\subset [1,p^{\star }]$. Of course, this fact guarantees that $\lambda _{q}$ is Hölder continuous (with any exponent $\alpha \in (0,1)$) in $[1,p^{\star }-\epsilon ]$. However, as pointed out in the end of the previous section, the precise behavior of $\lambda _{q}^{\prime }$ (or equivalently of $I_{q}(u_{q})$) as $q\rightarrow p^{\star }$ seems difficult to determine, even when $\Omega$ is a ball.

In this section we prove a Hölder regularity result for the function $\lambda _{q}$ in $[1,p^{\star }]$ by estimating $\big \Vert \lambda _{q}^{\prime }\big \Vert _{L^{r}([0,p^{\star }])}$ for any $r>1$. Taking into account (12), we first need to estimate $I_{q}(u_{q})$.

Lemma 8

The following estimates hold for each $0\not \equiv u\in W_{0}^{1,p}\left( \Omega \right)$:

$$\begin{aligned} \frac{1}{q}\left\| u\right\| _{q}^{q}\log (\left| \Omega \right| ^{-1}\left\| u\right\| _{q}^{q})<I_{q}(u)\le \left\| u\right\| _{q}^{q}\left( \log \left\| u\right\| _{q}+\frac{\log \left\| u\right\| _{p^{\star }}^{p^{\star }}-\log \left\| u\right\| _{q}^{p^{\star }}}{p^{\star }-q}\right) , \end{aligned}$$

(14)

where $\left| \Omega \right| :=\int \nolimits _{\Omega }\hbox {d}x$.

Proof

The continuous function $\varphi :[0,\infty )\rightarrow \mathbb {R}$ defined by $\varphi (\xi )=\xi \log \xi$, if $\xi >0$ and $\varphi (0)=0$ is strictly convex. Hence, for each $u\not \equiv 0$ Jensen’s inequality yields

$$\begin{aligned} \frac{1}{\left| \Omega \right| }I_{q}(u)=\frac{1}{q\left| \Omega \right| }\int \limits _{\Omega }\varphi (\left| u\right| ^{q} )\hbox {d}x>\frac{1}{q}\varphi (\left| \Omega \right| ^{-1}\left\| u\right\| _{q}^{q})=\frac{1}{q\left| \Omega \right| }\left\| u\right\| _{q}^{q}\log (\left| \Omega \right| ^{-1}\left\| u\right\| _{q}^{q}). \end{aligned}$$

Thus,

$$\begin{aligned} I_{q}(u)>\left\| u\right\| _{q}^{q}\log \left| \Omega \right| ^{-\frac{1}{q}}. \end{aligned}$$

(Note that the equality in the Jensen’s inequality occurs only if $u^{q} \equiv \int _{\Omega }\left| u\right| ^{q}\hbox {d}x$.)

Now, we prove the upper bound in (14) following [8]. Let $t\in [q,p^{\star }]$. It follows from Hölder inequality that

$$\begin{aligned} \left\| u\right\| _{t}\le \left\| u\right\| _{q}^{\alpha (t)}\left\| u\right\| _{p^{\star }}^{1-\alpha (t)},\quad q\le t\le p^{\star } \end{aligned}$$

with

$$\begin{aligned} \alpha (t):=\frac{q}{t}\frac{p^{\star }-t}{p^{\star }-q}=\left( \frac{p^{\star } }{t}-1\right) \frac{q}{p^{\star }-q}. \end{aligned}$$

Thus,

$$\begin{aligned} \log \left\| u\right\| _{t}\le \alpha (t)\log \left\| u\right\| _{q}+(1-\alpha (t))\log \left\| u\right\| _{p^{\star }} \end{aligned}$$

and

$$\begin{aligned} g(t):=\log \left\| u\right\| _{t}+\alpha (t)(\log \left\| u\right\| _{p^{\star }}-\log \left\| u\right\| _{q})-\log \left\| u\right\| _{p^{\star }}\le 0. \end{aligned}$$

Since $g(q)=0$, we obtain

$$\begin{aligned} g^{\prime }(q)=\lim _{t\rightarrow q^{+}}\frac{g(t)}{t-q}\le 0. \end{aligned}$$

But

$$\begin{aligned} g^{\prime }(t)&=\left( \frac{1}{t}\log \int \limits _{\Omega }\left| u\right| ^{t}\hbox {d}x\right) ^{\prime }+\alpha ^{\prime }(t)(\log \left\| u\right\| _{p^{\star }}-\log \left\| u\right\| _{q})\\&=\frac{1}{t}\left( \left\| u\right\| _{t}^{-t}\int \limits _{\Omega }\left| u\right| ^{t}\log \left| u\right| \hbox {d}x-\log \left\| u\right\| _{t}\right) -\frac{qp^{\star }}{t^{2}(p^{\star }-q)} (\log \left\| u\right\| _{p^{\star }}-\log \left\| u\right\| _{q}) \end{aligned}$$

and thus,

$$\begin{aligned} g^{\prime }(q)=\frac{1}{q}\left( \left\| u\right\| _{q}^{-q}\int \limits _{\Omega }\left| u\right| ^{q}\log \left| u\right| \hbox {d}x-\log \left\| u\right\| _{q}\right) -\frac{p^{\star }}{q}\frac{\log \left\| u\right\| _{p^{\star }}-\log \left\| u\right\| _{q} }{p^{\star }-q}\le 0 \end{aligned}$$

implying that

$$\begin{aligned} I_{q}(u)=\int \limits _{\Omega }\left| u\right| ^{q}\log \left| u\right| \hbox {d}x\le \left\| u\right\| _{q}^{q}\left( \log \left\| u\right\| _{q}+\frac{\log \left\| u\right\| _{p^{\star }}^{p^{\star }}-\log \left\| u\right\| _{q}^{p^{\star }}}{p^{\star }-q}\right) . \end{aligned}$$

$\square$

Remark 9

We emphasize that the estimates in (14) become simpler to handle if $\left| \Omega \right| \le 1=\left\| u\right\| _{q}$. In fact, under these conditions one has

$$\begin{aligned} 0<I_{q}(u)\le \frac{\log \left\| u\right\| _{p^{\star }}^{p^{\star }} }{p^{\star }-q}. \end{aligned}$$

(15)

We also note that a simple scaling argument gives

$$\begin{aligned} \lambda _{q}(\Omega _{1})=\lambda _{q}(\Omega )\left| \Omega \right| ^{\frac{p}{q}-\frac{p}{p^{\star }}} \end{aligned}$$

(16)

where $\Omega _{1}:=\left\{ x\in \mathbb {R}^{N}:x\left| \Omega \right| ^{\frac{1}{N}}\in \Omega \right\}$ is such that $\left| \Omega _{1}\right| =1$ and $\lambda _{q}(D)$ is defined as in (1) with $\Omega =D$.

It is also worth mentioning that $\frac{\log \left\| u_{q}\right\| _{p^{\star }}^{p^{\star }}}{p^{\star }-q}$ becomes an indeterminate form of the type $0/0$ as $q\rightarrow p^{\star }$, since

$$\begin{aligned} \lim _{q\rightarrow p^{\star }}\left\| u_{q}\right\| _{p^{\star }}=1. \end{aligned}$$

(17)

Indeed, since $1=\left\| u_{q}\right\| _{q}^{q}\le \left\| u_{q}\right\| _{p^{\star }}^{q}\left| \Omega \right| ^{1-\frac{q}{p\star }}$ one has

$$\begin{aligned} 1=\liminf _{q\rightarrow p^{\star }}\left| \Omega \right| ^{\frac{1}{p^{\star }}-\frac{1}{q}}\le \liminf _{q\rightarrow p^{\star }}\left\| u_{q}\right\| _{p^{\star }}. \end{aligned}$$

On the other hand, the inequality

$$\begin{aligned} \lambda _{p^{\star }}\left\| u_{q}\right\| _{p^{\star }}^{p}\le \left\| \nabla u_{q}\right\| _{p}^{p}=\lambda _{q} \end{aligned}$$

yields

$$\begin{aligned} \limsup _{q\rightarrow p^{\star }}\left\| u_{q}\right\| _{p^{\star }} \le \limsup _{q\rightarrow p^{\star }}\left( \frac{\lambda _{q}}{\lambda _{p^{\star }}}\right) ^{\frac{1}{p}}=1. \end{aligned}$$

Lemma 10

For each $\beta >0$ and all $1\le q<p^{\star }$ one has

$$\begin{aligned} I_{q}(u_{q})\le \log \left\| u_{q}\right\| _{\infty }\le \left| \Omega \right| ^{\frac{\beta }{q}}\beta ^{-1}\left\| u_{q}\right\| _{\infty }^{\beta }. \end{aligned}$$

(18)

Proof

The first inequality in (18) follows from (15) since

$$\begin{aligned} \log \left\| u_{q}\right\| _{p^{\star }}^{p^{\star }}\le \log \left( \left\| u_{q}\right\| _{\infty }^{p^{\star }-q}\int \limits _{\Omega }\left| u_{q}\right| ^{q}\hbox {d}x\right) =\log \left\| u_{q}\right\| _{\infty }^{p^{\star }-q}=(p^{\star }-q)\log \left\| u_{q}\right\| _{\infty }. \end{aligned}$$

Since $\log (x)\le \beta ^{-1}x^{\beta }$ for all $x\ge 1$ and $1=\left\| u_{q}\right\| _{q}\le \left\| u_{q}\right\| _{\infty }\left| \Omega \right| ^{\frac{1}{q}}$ the second inequality follows. $\square$

Lemma 11

The following estimate holds

$$\begin{aligned} \log \left\| u_{q}\right\| _{\infty }\le \frac{C}{p^{\star }-q},\quad \mathrm{for}\ \mathrm{all }\quad 1\le q<p^{\star } \end{aligned}$$

(19)

where $C$ is a positive constant which does not depend on $q$.

Proof

By taking $\sigma =q$ in Lemma 5 of Ercole [10] we obtain

$$\begin{aligned} \left\| u_{q}\right\| _{\infty }^{p^{\star }-q}\le K_{q}\left( \frac{\lambda _{q}}{\lambda _{p^{\star }}}\right) ^{\frac{N}{N-p}},\quad \mathrm{for}\ \mathrm{all }\quad 1\le q<p^{\star } \end{aligned}$$

where

$$\begin{aligned} K_{q}:=2^{\frac{N(p-1)+qp}{N-p}}\left( \frac{p+N(p-1)}{p}\right) ^{\frac{p^{\star }(N-1)}{N}}\le K_{p^{\star }},\quad 1\le q\le p^{\star }. \end{aligned}$$

Hence, (19) follows with

$$\begin{aligned} C:=\log \left( K_{p^{\star }}\max _{1\le q\le p^{\star }}\left( \frac{\lambda _{q}}{\lambda _{p^{\star }}}\right) ^{\frac{N}{N-p}}\right) . \end{aligned}$$

$\square$

Now, we are in position to prove a Hölder regularity result for the function $q\mapsto \lambda _{q}$ by assuming that

$$\begin{aligned} \limsup _{q\rightarrow p^{\star }} (p^{\star }-q)\left\| u_{q}\right\| _{\infty }^{\gamma }<\infty \end{aligned}$$

(20)

for some constant $\gamma >0$. Before proceeding, let us give some motivations for the assumption (20).

In [16] Knaap and Peletier proved the following asymptotic behavior for the case where $\Omega$ is the ball centered at the origin:

$$\begin{aligned} \lim _{q\rightarrow p^{\star }}(p^{\star }-q)\left\| u_{q}\right\| _{\infty }^{p/(p-1)}=A_{N,p,\Omega } \end{aligned}$$

(21)

being $A_{N,p,\Omega }$ a positive constant given explicitly in terms of $N$, $p$ and the volume of $\Omega$. Of course, (21) implies (20) with $\gamma =\frac{p}{p-1}$.

In the case where $p=2$, the asymptotic behavior (21) for a ball had already been proved by Atkinson and Peletier in [3]. In [6], still with $p=2$ and for a ball, Brezis and Peletier gave another proof of (21) and, among other important results, they conjectured that a similar asymptotic behavior should be true for a nonspherical bounded and smooth domain $\Omega$ (keeping $p=2$). This conjecture was then proved, independently, by Rey in [19] and by Han in [12]. The regular part of the Green function of $\Omega$ (associated with the Laplacian) played an essential role in the proofs presented in [12, 19]. Indeed, the proofs rely on the fact that, as $q$ goes to $2^{\star }$, the maximum points of $u_{q}$ concentrate at a point $x_{0}\in \Omega$ which is a critical point of the Robin function of $\Omega$ (the diagonal of the regular part of the Green function). In the case where $\Omega$ is a ball $x_{0}$ is its center.

However, to the best of our knowledge in the case $1<p\not =2$, the only result specifically related to the asymptotic behavior (21) is that by Knaap and Peletier in [16] for a ball. For a general bounded and smooth domain $\Omega$, Garcia Azorero and Peral Alonso showed in [11] that $u_{q}$ converges in the sense of measure to a multiple of the Dirac Delta Function concentrated at a point $x_{0}\in \Omega$. Thus, they reproduced for $1<p\not =2$ the concentration property known in the case $p=2$. After [11] some works have dealt with the convergence of the family $\left\{ u_{q}\right\}$ in the measures sense but, as far as we are aware, the generalization of (21) for a nonspherical bounded domains remains open if $1<p\not =2$.

In the sequel, we show that the asymptotic behavior (20) combined with (15) imply that $\lambda _{q}\in W^{1,r}([1,p^{\star }])$ for any $r\ge 1$ and hence that $\lambda _{q}\in C^{0,\alpha }([1,p^{\star }])$ for any $0<\alpha <1$ (by taking $\alpha :=1-\dfrac{1}{r}$). Thus, for a ball and $p>1$ and for a general bounded domain and $p=2$ we get more regularity at $p^{\star }$ than just absolute continuity.

We remark that $\lambda _{q}\in W^{1,1}([1,p^{\star }])$ for a general bounded domain $\Omega$ (because $\lambda _{q}$ is absolutely continuous in the closed interval $[1,p^{\star }]$).

Theorem 12

Assume that (20) happens for some $\gamma >0$. The function $q\mapsto \lambda _{q}$ belongs to the Sobolev space $W^{1,r} ([1,p^{\star }])$ for any $r>1$. Moreover,

$$\begin{aligned} \limsup _{r\rightarrow \infty }\left( r^{-1}\left\| \lambda _{q}^{\prime }\right\| _{L^{r}([1,p^{\star }])}\right) <\infty . \end{aligned}$$

(22)

Proof

By taking into account (16) we can assume, without loss of generality, that $\left| \Omega \right| \le 1$. Hence,

$$\begin{aligned} 0<I_{q}(u_{q})\le \beta ^{-1}\left\| u_{q}\right\| _{\infty }^{\beta } \end{aligned}$$

(23)

for any $\beta >0$, according to (18).

It follows from (12) that

$$\begin{aligned} \left\| \lambda _{q}^{\prime }\right\| _{L^{r}([1,p^{\star }])} \le \left\| \frac{p}{q}I_{q}(u_{q})\lambda _{q}\right\| _{L^{r} ([1,p^{\star }])}\le p\left( \max \limits _{1\le q\le p^{\star }}(\lambda _{q}/q)\right) \left\| I_{q}(u_{q})\right\| _{L^{r}([1,p^{\star }])}. \end{aligned}$$

Therefore, we just need to prove that $\left\| I_{q}(u_{q})\right\| _{L^{r}([1,p^{\star }])}$ is finite.

The hypothesis (20) implies that there exists $\overline{q} \in (1,p^{\star })$ such that

$$\begin{aligned} \left\| u_{q}\right\| _{\infty }\le c(p^{\star }-q)^{-\frac{1}{\gamma } }\quad \text { for all }\overline{q}\le q\le p^{\star } \end{aligned}$$

for some positive constant $c$ (which may depend on $\gamma$).

Combining this with (23) yields

$$\begin{aligned} \int \limits _{\overline{q}}^{p^{\star }}\left| I_{q}(u_{q})\right| ^{r} \hbox {d}q\le \beta ^{-r}c^{\beta r}\int \limits _{\overline{q}}^{p^{\star }}(p^{\star }-q)^{-\frac{\beta r}{\gamma }}\hbox {d}q. \end{aligned}$$

Thus, by taking $\beta :=\dfrac{\gamma }{2r}$ we obtain

$$\begin{aligned} \int \limits _{\overline{q}}^{p^{\star }}\left| I_{q}(u_{q})\right| ^{r} \hbox {d}q&\le \left( \frac{2r}{\gamma }\right) ^{r}c^{(\gamma /2)}\int \limits _{\overline{q} }^{p^{\star }}(p^{\star }-q)^{-\frac{1}{2}}\hbox {d}q\\&\le \left( \frac{2r}{\gamma }\right) ^{r}c^{(\gamma /2)}2(p^{\star }-\overline{q})^{\frac{1}{2}}\le k\left( \frac{2r}{\gamma }\right) ^{r}, \end{aligned}$$

where $k:=c^{(\gamma /2)}2(p^{\star }-\overline{q})^{\frac{1}{2}}$.

On the other hand, it follows from Lemma 11 that

$$\begin{aligned} \int \limits _{1}^{\overline{q}}\left| I_{q}(u_{q})\right| ^{r}\hbox {d}q&\le \int \limits _{1}^{\overline{q}}\left( \log \left\| u_{q}\right\| _{\infty }\right) ^{r}\hbox {d}q\\&\le \int \limits _{1}^{\overline{q}}\left( \frac{C}{p^{\star }-q}\right) ^{r} \hbox {d}q\le \left( \frac{C}{p^{\star }-\overline{q}}\right) ^{r}(\overline{q}-1). \end{aligned}$$

Thus, we conclude that $\lambda _{q}^{\prime }\in L^{r}([1,p^{\star }])$ and that

$$\begin{aligned} \int \limits _{1}^{p^{\star }}\left| I_{q}(u_{q})\right| ^{r}\hbox {d}q\le p\max \limits _{1\le q\le p^{\star }}(\lambda _{q}/q)\left[ k\left( \frac{2r}{\gamma }\right) ^{r}+\left( \frac{C}{p^{\star }-\overline{q}}\right) ^{r}(\overline{q}-1)\right] \le C_{1}C_{2}^{r}(1+r^{r}) \end{aligned}$$

for positive constants $C_{1}$ and $C_{2}$ which do not depend on $r$. Hence we obtain (22). $\square$

The following corollary follows immediately.

Corollary 13

Suppose that $p>1$ and $\Omega$ is a ball or that $p=2$ and $\Omega$ is a general bounded and smooth domain. Then $\lambda _{q}\in C^{0,\alpha }([1,p^{\star }])$ for any $0<\alpha <1$.