Asymptotic behavior for the radial eigenvalues of p-Laplacian in certain annular domains

TL;DR

This paper investigates the asymptotic behavior of radial eigenvalues for the p-Laplacian operator in annular domains, providing insights into their spectral properties as parameters vary.

Contribution

It establishes the asymptotic behavior of radial eigenvalues for the p-Laplacian in annular domains, a novel analysis in this geometric setting.

Findings

Derived asymptotic formulas for eigenvalues

Characterized eigenvalue behavior as parameters tend to limits

Enhanced understanding of spectral properties in annular domains

Abstract

In this paper we prove an asymptotic behavior for the radial eigenvalues to the Dirichlet $p$-Laplacian problem $-\Delta_p\,u = \lambda\,|u|^{p-2}u$ in $\Omega$, $u=0$ on $\partial\Omega$, where $\Omega$ is an annular domain $\Omega=\Omega_{R,\overline{R}}$ in $\mathbb{R}^N$.

Equations

$$\left\{\begin{array}[]{lll}\displaystyle-\Delta_{p}\,u=\lambda\,|u|^{p-2}u&\textup{in}&\Omega_{R,\overline{R}},\\ \displaystyle u=0&\textup{on}&\partial\Omega_{R,\overline{R}},\end{array}\right.$$

$$1<p\le N.$$

$$\left\{\begin{array}[]{rll}\displaystyle-\Delta\,u=\lambda\,u&\textup{in}&\Omega_{R,\overline{R}},\\ \displaystyle u=0&\textup{on}&\partial\Omega_{R,\overline{R}}.\end{array}\right.$$

$$\left\{\begin{array}[]{lll}\displaystyle(r^{N-1}|u^{\prime}(r)|^{p-2}u^{\prime}(r))^{\prime}+\lambda\,r^{N-1}|u(r)|^{p-2}u(r)=0&\textup{in}&(R,\bar{R}),\\ \displaystyle u(R)=u(\bar{R})=0.&&\end{array}\right.$$

$${\lambda }_k={\lambda }_k^{\text{mbox}rad}.$$

$$A=\frac{(R\overline{R}{)}^{\frac{N-p}{p-1}}}{{\overline{R}}^{\frac{N-p}{p-1}}-R^{\frac{N-p}{p-1}}}\text{mbox}andB=\frac{{\overline{R}}^{\frac{N-p}{p-1}}}{{\overline{R}}^{\frac{N-p}{p-1}}-R^{\frac{N-p}{p-1}}},$$

$$\left\{\begin{array}[]{lll}\displaystyle(|v^{\prime}(t)|^{p-2}v^{\prime}(t))^{\prime}+\lambda\,q(t)|v(t)|^{p-2}v(t)=0&\textup{in}&(0,1),\\ \displaystyle v(0)=v(1)=0.&&\end{array}\right.$$

$$q(t):=q_{R,\overline{R}}(t)={\left(\frac{p-1}{N-p}\right)}^p\frac{A^{\frac{(p-1)p}{N-p}}}{(B-t{)}^{\frac{p(N-1)}{N-p}}}.$$

$$q(t):=q_{R,\overline{R}}(t)={\left[R{\left(\frac{\overline{R}}{R}\right)}^t\ln \frac{\overline{R}}{R}\right]}^p.$$

$$q\in C^1([0,1]),q(t)>0\text{mbox}for0\le t\le 1.$$

$$0<{\lambda }_1<{\lambda }_2<\cdots <{\lambda }_k<{\lambda }_{k+1}<\cdots ,\underset{k\to \infty }{\lim }{\lambda }_k=\infty ,$$

$$\underset{R\to +\infty }{\lim }{\lambda }_k(R,R+1).$$

$${\pi }_p=\frac{2\pi (p-1{)}^{1/p}}{p\sin (\pi /p)},$$

$${\bar{q}}_{-}(R,\overline{R})={\int }_0^1\min \{1,q(t)\}dt$$

$${\bar{q}}_{+}(R,\overline{R})={\int }_0^1\max \{1,q(t)\}dt.$$

$${\left(\frac{k{\pi }_p}{{\bar{q}}_{+}(R,\overline{R})}\right)}^p\le {\lambda }_k(R,\overline{R})\le {\left(\frac{k{\pi }_p}{{\bar{q}}_{-}(R,\overline{R})}\right)}^p.$$

$${\varphi }^{\prime \prime }+\frac{N-1}{r}{\varphi }^{\prime }-\frac{{\alpha }_k}{r^2}\varphi =-{\lambda }_{k,j}(R,R+1)\varphi ,\text{mbox}in(R,R+1),$$

$$\varphi (R)=\varphi (R+1)=0,$$

$${\varphi }^{\prime \prime }=-{\lambda }_j\varphi ,\text{mbox}in(0,1),$$

$$\varphi (0)=\varphi (1)=0,$$

$$\underset{R\to +\infty }{\lim }{\lambda }_{k,j}(R,R+1)={\lambda }_j.$$

$$\underset{R\to +\infty }{\lim }{\bar{q}}_{+}(R,R+1)=\underset{R\to +\infty }{\lim }{\bar{q}}_{-}(R,R+1)=1.$$

$$\underset{R\to +\infty }{\lim }{\lambda }_k(R,R+1)={\left(k{\pi }_p\right)}^p,$$

$$\left\{\begin{array}[]{lll}\displaystyle(|v^{\prime}(t)|^{p-2}v^{\prime}(t))^{\prime}+\lambda\,|v(t)|^{p-2}v(t)=0&\textup{in}&(0,1),\\ \displaystyle v(0)=v(1)=0.&&\end{array}\right.$$

$$\underset{R\to +\infty }{\lim }{\bar{q}}_{+}(R,R+1)=\underset{R\to +\infty }{\lim }{\bar{q}}_{-}(R,R+1)=1.$$

$$\underset{R\to +\infty }{\lim }{\lambda }_k(R,R+1)={\left(k\pi \right)}^2,$$

$$\left\{\begin{array}[]{lll}\displaystyle v^{\prime\prime}(t)+\lambda\,v(t)=0&\textup{in}&(0,1),\\ \displaystyle v(0)=v(1)=0.&&\end{array}\right.$$

$$q(t)={\left[R{\left(\frac{\overline{R}}{R}\right)}^t\ln \frac{\overline{R}}{R}\right]}^p.$$

$$q^{\prime }(t)=p{\left[R{\left(\frac{\overline{R}}{R}\right)}^t\ln \frac{\overline{R}}{R}\right]}^{p-1}R{\left(\frac{\overline{R}}{R}\right)}^t{\left(\ln \frac{\overline{R}}{R}\right)}^2>0,$$

$$0<q(0)\le q(t)\le q(1),\forall t\in [0,1],$$

Full text

1. Introduction

This paper investigates an asymptotic behavior for the radial eigenvalues $\lambda_{k}=\lambda_{k}(R,\overline{R})$ (when $0<\overline{R}=R+1$ and $R\to+\infty$) to the following eigenvalue problem

[TABLE]

where $\Omega_{R,\overline{R}}=\{x\in\mathbb{R}^{N}:R<|x|<\overline{R}\}$, with $0<R<\overline{R}$ constants in $\mathbb{R}$, is the annular domain, and we suppose that

[TABLE]

In particular, when $p=2$, we obtain the Dirichlet Laplacian problem

[TABLE]

Since we are interested only in the radial eigenvalues of (1.1), we can rewrite (1.1) as the following $1$-dimensional eigenvalue problem

[TABLE]

We remark that for every $1\leq k\in\mathbb{N}$, if we denote by $\lambda_{k}$ the $k$-th eigenvalue of (1.3) and by $\lambda_{k}^{\mbox{rad}}$ the $k$-th radial eigenvalue of (1.1),

[TABLE]

In order to study the solution of (1.1), one can make a standard change of variables, see for example [1, 5].

If $N>p$, let $t=-\frac{A}{r^{(N-p)/(p-1))}}+B$ and $v(t)=u(r)$, where

[TABLE]

then the problem (1.3) (hence (1.1)) transforms into the boundary value problem for the nonautonomous ODE

[TABLE]

where

[TABLE]

In the case $p=N$, one sets $r=R(\frac{\overline{R}}{R})^{t}$ and $v(t)=u(r)$, obtaining again the problem (1.4), now with

[TABLE]

Let $\lambda_{k}$ be the $k$-th eigenvalue of (1.4) and let $\varphi_{k}$ be an eigenfunction corresponding to $\lambda_{k}$. Since $q$ satisfies

[TABLE]

It is known that

[TABLE]

and that $\varphi_{k}$ has exactly $k-1$ zeros in $(0,1)$, see [2, 3].

Motivated by the work of Zhang [7], whose purpose was to compute an estimate for eigenvalues of the Dirichlet $p$-Laplacian (1.4), $p>1$, we propose to prove an asymptotic behavior for the $\lambda_{k}(R,\overline{R})$ in the form:

[TABLE]

The following estimate is known, see Zhang [7, Remark 2.1]. We suppose that

[TABLE]

[TABLE]

and

[TABLE]

In [7] (with $T=1$), the author proved the inequality

[TABLE]

In S.S. Lin [4, Lemma A.1], the author proves an asymptotic behavior for all eigenvalues (that is, radial and non-radial eigenvalues), of Dirichlet problem (1.2), which is the following result.

Lemma 1.1** ([4]).**

Let $\lambda_{k,j}(R,R+1)$ be the $j$-th eigenvalue of

[TABLE]

[TABLE]

where $\alpha_{k}=k(k+N-2)$, and let $\lambda_{j}=j^{2}\pi^{2}$ be the $j$-th eigenvalue of

[TABLE]

[TABLE]

where $k=0,1,2\cdots$ and $j=1,2,3,\cdots$. Then

[TABLE]

In this paper, we will prove a generalization of the results of [4], for the radial eigenvalues of $p$-Laplacian, in the cases when $p=r+1$ and $N=2r+1$, with $r\in\mathbb{N}$; $p=N$ and $p=2<3\leq N$. The last case is another proof of Lin’s result for the radial eigenvalues. It is noteworthy that it is not yet known a characterization for all eigenvalues of (1.1). The results of [4] are consequence of some results of Bessel functions, while in our paper we use a totally different approach following the work of Zhang [7].

We state the main result.

Theorem 1.2**.**

(i)* Suppose that $N=p$. Then*

[TABLE]

In particular, by (1.8)

[TABLE]

that is, the eigenvalues $\lambda_{k}(R,R+1)$ converge asymptotically to the eigenvalues of the problem

[TABLE]

(ii)* Suppose that $p=2$ and $N\geq 3$. Then,*

[TABLE]

In particular, by (1.8)

[TABLE]

that is, the eigenvalues $\lambda_{k}(R,R+1)$ converge asymptotically to the eigenvalues of the problem

[TABLE]

2. Proofs

Proof of Theorem 1.2.

(i): By (1.6), we have

[TABLE]

Therefore,

[TABLE]

hence, the function $q$ is increasing at $t$ and

[TABLE]

that is,

[TABLE]

If $R>0$ and $\overline{R}=R+1$, we have

[TABLE]

and

[TABLE]

Therefore,

[TABLE]

and

[TABLE]

where we used the Euler Number

[TABLE]

By (2.2) and (2.3), we obtain that

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As

[TABLE]

we conclude that

[TABLE]

As

[TABLE]

we conclude that

[TABLE]

It follows from (1.8) that

[TABLE]

By (2.4), (2.5) and by limits in (2.6), we obtain

[TABLE]

The proof of $(\pi_{p}k)^{p}$ is a solution of (1.11), according to Zhang [7] (see also del Pino and Manasevich [6]).

Proof of (ii): By (1.5), if $N>p$, we have

[TABLE]

Therefore,

[TABLE]

and the function $q$ is increasing at $t$. Hence,

[TABLE]

Since $p=2$ and $N\geq 3$, by(2.7), we have

[TABLE]

In particular,

[TABLE]

and

[TABLE]

If $R>0$ and $\overline{R}=R+1$, we have

[TABLE]

and we conclude that

[TABLE]

where we used that

[TABLE]

as $R\to+\infty$.

Similarly,

[TABLE]

and we conclude that

[TABLE]

Therefore, by (2.8) and (2.9), we obtain that

[TABLE]

As

[TABLE]

we conclude that

[TABLE]

As

[TABLE]

we conclude that

[TABLE]

Since by (1.7) we have $\pi_{2}=\pi$, it follows from (1.8) that

[TABLE]

By (2.10), (2.11) and by limits in (2.12), we obtain

[TABLE]

This proves the item ($ii$). ∎

3. Additional results

Similar to Theorem 1.2, we can get the following result. Let $r\in\mathbb{N}$. Suppose that $p=r+1$ and $N=2r+1$. Then,

[TABLE]

In particular, by (1.8)

[TABLE]

Indeed, since $p=r+1$ and $N=2r+1$, then

[TABLE]

and by (2.7), we have

[TABLE]

If $R>0$ and $\overline{R}=R+1$, we have

[TABLE]

In particular,

[TABLE]

and

[TABLE]

Therefore,

[TABLE]

and

[TABLE]

By (3.3) and (3.4), we obtain that

[TABLE]

As

[TABLE]

we conclude that

[TABLE]

As

[TABLE]

we conclude that

[TABLE]

It follows from (1.8) that

[TABLE]

By (3.5), (3.6) and by limits in (3.7) we obtain

[TABLE]