The Gauss sums involving 24-order character and their recursive properties

1.   Introduction

An expedient feature of the $p$-Laplacian eigenvalue problem is that the eigenfunctions may be multiplied by constant factors (in other words, the fact that if $u$ is an eigenfunction, so is $ku$). Unfortunately, the $p(x)$-Laplacian eigenvalue problem does not possess this expedient property. It is important to stress that the loss of the property under consideration is not only a consequence of the dependence on $x$, but it can also occur in presence of unbalanced growth. For example, the double phase operator (that does not depend on $x$)

loses this property. In this paper we are interested in considering that the operator has both peculiarities: It depends on $x$ and it is unbalanced.

Let $\Omega \subset \mathbb{R}^{N} (N\geq2)$ be a bounded domain with Lipschitz boundary $\partial\Omega$. This article studies an eigenvalue problem coming from the minimization of the Rayleigh quotient:

among all $u\in W_{0}^{1, \mathcal{H}_{n}}(\Omega), u\not \equiv0$. These functions belong to an appropriate Musielak-Orlicz Sobolev space with variable exponents; see its definition in section two. The function $a:\bar{\Omega}\rightarrow [0, +\infty)$ is a $C^{1}$ differentiable function.

Put

and define the first eigenvalue as

By a similar proof of Proposition 3.1 in ^[1], we can show that the following equation

is the Euler-Lagrange equation corresponding to the minimization of the Rayleigh quotient (1.2), where $\lambda_{(p_{n}(\cdot), \, q_{n}(\cdot))} = \lambda^{1}_{(p_{n}(\cdot), \, q_{n}(\cdot))}$.

Here, we impose the following hypotheses on the variable exponents $p_{n}(x)$ and $q_{n}(x)$.

(H1): Assume that $p_{n}(x)$ and $q_{n}(x)$ are two sequences of $C^{1}$ functions in $\overline{\Omega}$, $q_{n}(\cdot) > p_{n}(\cdot)$ for every $n\geq1$ and

(H2): The following two quotients are bounded, namely,

where for a function $g$ we denote

(H3): We also assume that

then we can find a positive and continuous function $\theta\, (0 < \theta < +\infty)$, such that

uniformly for all $x\in \Omega$.

The differential operator in (1.5) is the double-phase operator with variable exponents, which can be given by

This operator is the classical double phase operator (1.1) when $p_{n}(x)$ and $q_{n}(x)$ are constants. Moreover, special cases of (1.12), studied extensively in the literature, occur when $\inf_{\overline{\Omega}}\mu > 0$ (the weighted $(q(x), p(x))$-Laplacian) or when $\mu\equiv0$ (the $p(x)$-Laplacian).

The energy functional related to the double-phase operator (1.12) is given by

whose integrand switches two different elliptic behaviors. The integral functional (1.13) was first introduced by Zhikov ^[2,3,4,5], who obtained that the energy density changed its ellipticity and growth properties according to the point in order to provide models for strongly anisotropic materials. Moreover, double phase differential operators (1.12) and corresponding energy functionals (1.13) have several physical applications. We refer to the works of ^[6] on transonic flows, ^[7] on quantum physics and ^[8] on reaction diffusion systems. Finally, we mention a recent paper that is very close to our topic. For related works dealing with the double phase eigenvalue problems, we refer to the works of Colasuonno-Squassina ^[9], who proved the existence and properties of related variational eigenvalues. By using the Rayleigh quotient of two norms of Musielak-Orlicz space, the author of this paper has defined the eigenvalue, which has the same properties as the $p$-Laplace operator. Recently, Liu-Papageorgiou has considered an eigenvalue problem for the Dirichlet $(p, q(\cdot))-$Laplacian by using the Nehari method (see ^[10]), a nonlinear eigenvalue problem for the Dirichlet $(p, q)-$Laplacian with a sign-changing Carathéodory reaction (see ^[11]) and a nonlinear eigenvalue problem driven by the anisotropic $(p(\cdot), q(\cdot))-$Laplacian (see ^[12]). Motivated by ^[9], Yu^[13] discuss the asymptotic behavior of an eigenvalue for the double phase operator. However, to the author's knowledge, the eigenvalue problem for variable exponents double phase operator has remained open. Our article fits into this general field of investigation.

Assume that $\delta:\Omega\rightarrow [0, \infty)$ is the distance function, which is given by

This function is a Lipschitz continuous function. For all $x\in\Omega$, we get $|\nabla \delta| = 1$. Define

It is known from the paper ^[1] that

Define

and

The following are the main results of this paper.

Theorem 1.1. Let $u\in C(\Omega)$ be a weak solution of problem (1.5), then it is also a viscosity solution of the problem (3.2).

Theorem 1.2. Let hypotheses (H1)–(H3) be satisfied, $\lambda^{1}_{(p_{n}(\cdot), \, q_{n}(\cdot))}$ and $\Lambda_{\infty}$ be defined by (1.4) and (1.14), respectively. In addition, assume that $u_{n}$ normalized by $\|u_{n}\|_{\mathcal{H}_{n}} = 1$ is the positive first eigenfunction, then,

(1)

(2) there exists a nonnegative function $u_{\infty}$ such that $u_{\infty}\in C^{\alpha}(\Omega)\setminus\{0\}$ and $||u_{\infty}||_{L^{\infty}(\Omega)} = 1;$

(3) we can extract a subsequence, which is still denoted by $u_{n}$, such that

in the space $C^{\alpha}(\Omega)$, where $\alpha \, (0 < \alpha < 1)$ is a constant;

(4) we can obtain that the function $u_{\infty}(x)$ is a nontrivial viscosity solution of the problem

To the best of our knowledge, this is the first work dealing with the double phase eigenvalue problem (1.5). The rest of this paper is organized as follows. In section two, we collect some notations and facts about the Musielak-Orlicz space $L^{\mathcal{H}}(\Omega)$ and $W_{0}^{1, \mathcal{H}}(\Omega)$, which will be used in this paper. Section three and section four are devoted to prove Theorems 1.1 and 1.2, respectively.

2.   Preliminaries

In this section, we recall some known results about the Musielak-Orlicz spaces $L^{\mathcal{H}}(\Omega)$ and $W_{0}^{1, \mathcal{H}}(\Omega)$. For more detail, please see references ^{[9,14,15,16,17]}.

We follow the notation of ^[9]. Let $N(\Omega)$ denote the set of all generalized $N$-functions. Let us introduce the nonlinear function $\mathcal{H}:\Omega\times[0, +\infty)\rightarrow [0, +\infty)$ defined as

with $1 < p(x) < q(x)$ and $0\leq a(\cdot)\in L^{1}(\Omega)$. It is clear that $\mathcal{H}\in N(\Omega)$ is a locally integrable and generalized $N$-function. In addition, it fulfills the $\Delta_{2}$ condition, namely,

Therefore, in correspondence to $\mathcal{H}$, we define the Musielak-Orlicz space $L^{\mathcal{H}}(\Omega)$ as

which can be equipped with the norm

where

which is called $\mathcal{H}$-modular.

Similarly, we can define the Musielak-Orlicz Sobolev spaces. The space $W^{1, \mathcal{H}}(\Omega)$ is given by

with the norm

We denote by $W_{0}^{1, \mathcal{H}}(\Omega)$ the completion of $C_{0}^{\infty}(\Omega)$ in $W^{1, \mathcal{H}}(\Omega)$. With these norms, the spaces $L^{\mathcal{H}}(\Omega)$, $W^{1, \mathcal{H}}(\Omega)$ and $W_{0}^{1, \mathcal{H}}(\Omega)$ are separable, reflexive and uniformly convex Banach spaces.

From Proposition 2.16 (ⅱ) in ^[18], if

then the following Poincar$\acute{e}$-type inequality

holds for all $u\in W_{0}^{1, \mathcal{H}}(\Omega)$, where $C$ is a positive constant independent of $u$. Therefore, in this paper, we equip $W_{0}^{1, \mathcal{H}}(\Omega)$ with the equivalent norm $\|\nabla u\|_{\mathcal{H}}$ for all $u\in W_{0}^{1, \mathcal{H}}(\Omega)$.

Proposition 2.1. ^[18] If $u\in L^{\mathcal{H}}(\Omega)$ and $\rho_{\mathcal{H}}(u)$ is the $\mathcal{H}$-modular, then the following properties hold.

(1) $If\: u\neq0$, then $\|u\|_{\mathcal{H}} = \lambda$ if, and only if, $\varrho_{\mathcal{H}}(\frac{u}{\lambda}) = 1;$

(2) $\|u\|_{\mathcal{H}} < 1 \ ( = 1; > 1)$ if, and only if, $\varrho_{\mathcal{H}}(u) < 1 \ ( = 1; > 1);$

(3) If $\|u\|_{\mathcal{H}} \leq 1$, then $\|u\|_{\mathcal{H}}^{q^{+}} \leq \rho_{\mathcal{H}}(u)\leq \|u\|_{\mathcal{H}}^{p^{-}};$

(4) If $\|u\|_{\mathcal{H}} \geq 1$, then $\|u\|_{\mathcal{H}}^{p^{-}} \leq \rho_{\mathcal{H}}(u)\leq \|u\|_{\mathcal{H}}^{q^{+}};$

(5) $\|u\|_{\mathcal{H}}\rightarrow 0$ if, and only if, $\rho_{\mathcal{H}}(u)\rightarrow 0;$

(6) $\|u\|_{\mathcal{H}}\rightarrow 0$ if, and only if, $\rho_{\mathcal{H}}(u)\rightarrow 0.$

3.   The proof of Theorem 1.1

Given $u\in C(\Omega)\bigcap W^{1, \mathcal{H}_{n}}_{0}(\Omega)$ and $\phi\in C^{2}(\Omega)$. Define

and

where $\triangle_{\infty}\phi$ is the $\infty$-Laplacian.

Here, we are now in a position to give the following definition of weak solutions to problem (1.5).

Definition 3.1. We call $u\in W^{1, \mathcal{H}_{n}}_{0}(\Omega)\backslash\{0\}$ a weak solution of problem (1.5) if

is satisfied for all test functions $v \in W^{1, \mathcal{H}_{n}}_{0}(\Omega)$. If $u\neq 0$, we say that $\lambda_{(p_{n}(\cdot), q_{n}(\cdot))}$ is an eigenvalue of (1.5) and that $u$ is an eigenfunction corresponding to $\lambda_{(p_{n}(\cdot), q_{n}(\cdot))}$.

In (1.5), we replace $u$ by $\phi$ and keep $S_{n}$, $K_{n}$ and $k_{n}$ unchanged, then

We first recall the definition of viscosity solutions. Assume we are given a continuous function

where $\mathcal{S}(N)$ denotes the set of $N \times N$ symmetric matrices.

Consider the problem

where

Definition 3.2. Assume that $x_{0}\in \Omega$, $u\in C(\Omega)$, $\psi\in C^{2}(\Omega)$ and $\varphi\in C^{2}(\Omega)$.

(1) Let $u(x_{0}) = \psi(x_{0})$ and suppose that $u-\psi$ attains its strict maximum value at $x_{0}$. If

for all of such $x_{0}$, then the function $u$ is said to be a viscosity subsolution of Eq (3.2).

(2) Let $u(x_{0}) = \varphi(x_{0})$ and suppose that $u-\varphi$ attains its strict minimum value at $x_{0}$. If

for all of such $x_{0}$, then the function $u$ is said to be a viscosity supersolution of Eq (3.2).

(3) If $u$ is both a subsolution and a supersolution of the problem (3.2), then $u$ is a viscosity solution of the problem (3.2).

Proof of Theorem 1.1. Claim: $u$ is a viscosity supersolution of (3.2).

Let $x_{0}\in \Omega$ and $\varphi\in C^{2}(\Omega)$. Assume that $u(x_{0}) = \varphi(x_{0})$ and the function $u-\varphi$ obtains its strict minimum value at the point $x_{0}$. Our goal is to show that

If

then by continuity there exists a positive constant $r$ such that $B(x_{0}, 2r)\subset\Omega$, $u > \varphi$ in this ball, except for the point $x_{0}$ and

for all $x\in B(x_{0}, 2r)$. Thus, if $x\in B(x_{0}, r)$, we have

If $x\in \partial B(x_{0}, r)$, the minimum value of the function $u-\varphi$ is defined as $m$. Let $\Phi(x): = \varphi(x)+\frac{m}{2}$. Note that $m > 0$ and the above inequality still holds if the function $\varphi(x)$ is replaced by $\Phi(x)$, namely,

Define $\eta(x): = (\Phi-u)^{+}\geq0$, then if $x\in \partial B(x_{0}, r)$, we have $\eta(x)\equiv0$.

Let

We multiply (3.5) by the function $\eta(x)$ and integrate over $B(x_{0}, r)$, then the inequality

is true.

If we define

and use $\eta_{1}(x)$ as a test function in (3.1), then we get

Subtracting (3.7) from (3.6), we arrive at

The first integral is nonnegative due to the elementary inequality

which holds for all $p > 1$. Here, we take $p = p_{n}(x)$. We get a contradiction. Hence, (3.4) holds. Similarly, we conclude that $u$ is a viscosity subsolution of (3.2) and we omit the details. □

4.   The proof of Theorem 1.2

Let $n\in N$ be large enough such that $p_{n}\geq r > N$, which results in $W_{0}^{1, \mathcal{H}_{n}}(\Omega)\hookrightarrow W_{0}^{1, r}(\Omega)$ (see Proposition 2.16 (1) of Blanco, Gasiński, Harjulehto and Winkert ^[18]). It follows that $u_{n}$ are continuous functions. The reason is that the space $W_{0}^{1, r}(\Omega)\hookrightarrow\hookrightarrow C^{\alpha}(\Omega)$, $0 < \alpha < 1$. Moreover, it is known (see ^[9]) that for each $n\in \mathbb{N}$ fixed, we have $u_{n} > 0$.

In order to prove Theorem 1.2, we only need to prove the following conclusions.

Lemma 4.1. Let $h:\overline{\Omega}\rightarrow (1, \infty)$ be a given continuous function, then

for all $v \in W_{0}^{1, \mathcal{H}}(\Omega)$ and $s\in (1, p^{-})$.

Proof. Since $\left\|\frac{\nabla v}{||\nabla v||_{\mathcal{H}}}\right\|_{\mathcal{H}} = 1$, it follows from Proposition 2.1 that

Thus,

Invoking Proposition 2.1 again, we conclude that

which implies (4.1). □

Lemma 4.2. If $u\in L^{\infty}(\Omega)$, then we have

Proof. Step1: To show that the following inequality holds,

If $k_{n}(u)\leq k_{\infty}(u)$, the above inequality is true. Thus, we can assume that $k_{n}(u) > k_{\infty}(u)$, and since $q_{n}(x) > p_{n}(x) > 1$, we have

which implies (4.5) holds.

Step2: To show that the following inequality holds,

Case1: $k_{\infty}(u) = 0$. It is easy to find that (4.6) holds.

Case2: $k_{\infty}(u) > 0$. Given $\varepsilon > 0$, there exists a nonempty set $\Omega_{\varepsilon}\subset\Omega$ such that, for all $x\in \Omega_{\varepsilon}$, $|u| > k_{\infty}(u)-\varepsilon$. Ignoring those indices $n$ that $k_{n}(u)\geq k_{\infty}(u)-\varepsilon$, we have

which gives

The arbitrariness of $\varepsilon$ implies that (4.6) is true. Consequently, (4.4) holds. □

Remark 4.1. If $|\nabla u|\in L^{\infty}(\Omega)$, we can argue as Lemma 4.2 to obtain that

Lemma 4.3. If the assumptions of Theorem 1.2 hold, then

(1) (1.18) holds;

(2) there exists a nonnegative function $u_{\infty}$ such that $u_{\infty}\in C^{\alpha}(\Omega)\setminus\{0\}$ and $||u_{\infty}||_{L^{\infty}(\Omega)} = 1;$

(3) we can extract a subsequence, which is still denoted by $u_{n}$, such that

in the space $C^{\alpha}(\Omega)$, where $\alpha \, (0 < \alpha < 1)$ is a constant.

Proof. Assume for simplicity that the following inequality holds

Step 1: To show that,

Inserting $u(x) = \delta(x)$ into (1.4) gives

Note that by Lemma 4.2 and Remark 4.1, we have

Step 2: We now claim that $u_{\infty}\in W_{0}^{1, \infty}(\Omega)$.

Since (4.8) holds, for all $n\in N$ sufficiently large, we can assume that $\lambda^{1}_{(p_{n}(\cdot), \, q_{n}(\cdot))}\leq\Lambda_{\infty}+1$. Thus, we have

Note that the sequence $\{||\nabla u_{n}||_{\mathcal{H}_{n}}\}$ is bounded.

Let $r\in[1, \infty)$ be arbitrary. We can find an integer $n_{r}$, for all $n\geq n_{r}$, such that $p_{n}(\cdot)\geq r$ and

Hence, the sequence $\{u_{n}\}$ is bounded in the reflexive Banach space $W_{0}^{1, r}(\Omega)$. We can find a subsequence, still defined by $\{u_{n}\}$, and a function $u_{\infty}\in W_{0}^{1, r}(\Omega)$, such that $\nabla u_{n}\rightharpoonup\nabla u_{\infty}$ in $W_{0}^{1, r}(\Omega)$ and $u_{n}\rightarrow u_{\infty}$ in $L^{r}(\Omega)$.

Define

and it follows that

and

Using Hölder's inequality and the above inequality, we have

Thus, (4.1) and (4.10) ensure that

We choose an arbitrary positive real number $r_{1}$ such that $B(x, r_{1})\subset\Omega$, where the point $x\in\Omega$ is a Lebesgue point such that $|\nabla u_{\infty}|\in L^{1}(\Omega)$, then we find that

Passing to the limit as $r\rightarrow +\infty$ in the above inequality, gives

Letting $r_{1}\rightarrow0^{+}$ in the above inequality, gives

for a.e. $x\in\Omega$, which implies that $\nabla u_{\infty}\in L^{\infty}(\Omega)$, as claimed.

Step 3: We want to prove that $u_{n}\rightarrow u_{\infty}$ in $C^{\alpha}(\Omega)$ ($0 < \alpha < 1$) and $\|u_{\infty}\|_{L^{\infty}(\Omega)} = 1$.

Keeping in mind that $r\in[1, \infty)$ is an arbitrary constant, we can assume that $r > N$. Therefore, this combined with the fact that $W^{1, r}_{0}(\Omega)\hookrightarrow\hookrightarrow C^{\alpha}(\Omega)\, (0 < \alpha < 1)$ implies that there exists a nonnegative function $u_{\infty}\in C^{\alpha}(\Omega)\setminus\{0\}$, such that $u_{n}\rightarrow u_{\infty}$ in $C^{\alpha}(\Omega)$ and $u_{n}$ converges uniformly to $u_{\infty}$ in $\Omega$. Given $\varepsilon\in(0, 1)$, we can find a constant $N_{\varepsilon}\in \mathbb{N}$ such that

for all $x\in\Omega, n\geq N_{\varepsilon}$. It follows that

and

for all $n\geq N_{\varepsilon}$. Letting $n\rightarrow \infty$ in (4.14) and (4.15) yields

Thus, the inequality

holds. In view of Lemma 4.2 and (4.16), we can get

Step 4: To show that $\liminf_{n\rightarrow \infty}\lambda^{1}_{(p_{n}(\cdot), \, q_{n}(\cdot))}\geq\Lambda_{\infty}$.

Since $\nabla u_{n}\rightharpoonup\nabla u_{\infty}$ in $W_{0}^{1, r}(\Omega)$, $||u_{n}||_{\mathcal{H}_{n}} = 1$ and the inequality (4.11) holds, we have

Letting $r\rightarrow \infty$ and using Proposition 7 in ^[19] and equality (4.17), we get

Thus, (4.8) and (4.18) imply that (1.18) holds. The proof is complete. □

Remark 4.2. We can again argue with Step 3 to obtain

The function $u_{\infty}(x)$ also has the following property.

Lemma 4.4. If the assumptions of Theorem 1.2 hold, we can deduce that $u_{\infty}(x)$ is a nontrivial viscosity solution of the problem (1.20).

Proof. For the first part we only need to show that $u_{\infty}$ is a viscosity subsolution of (1.20). Let $x_{0}\in \Omega$ and $\psi\in C^{2}(\Omega)$. Assume that $u_{\infty}-\psi$ attains its strict maximum value of zero at $x_{0}$, namely, $u_{\infty}(x_{0})-\psi(x_{0}) = 0$.

Claim: We want to show that

By Lemma 4.3, we know that the convergence of $u_{n}$ to $u_{\infty}$ in $\Omega$ is uniform. Therefore, there exists a sequence $\{x_{n}\}\subset \Omega$ such that $x_{n}\rightarrow x_{0}$ (as $n\rightarrow \infty$), $u_{n}(x_{n}) = \psi(x_{n})$ and $u_{n}-\psi$ attains its strict maximum value at $x_{n}$.

Employing Theorem 1.1, it turns out that for any $n\in \mathbb{N}$ large enough, $u_{n}$ are continuous viscosity solutions of (1.5) with $\lambda_{(p_{n}(\cdot), \, q_{n}(\cdot))} = \lambda^{1}_{(p_{n}(\cdot), \, q_{n}(\cdot))}$. Thus, we have

Case 1: $\psi(x_{0}) = u_{\infty}(x_{0}) > 0$.

Continuing (4.21), for $n\in \mathbb{N}$ sufficiently large, we have $|\nabla \psi(x_{n})| > 0$. Let us assume the assertion is not true, then by (4.21) and continuity, we have $\psi(x_{0})\leq0$. This leads to a contradiction.

Dividing both sides of (4.21) by

we see that the following inequality holds

Now, letting $n\rightarrow \infty$, we deduce that

Taking the lower limit in inequality (4.22) and employing the limits above, we have

Note that by (4.17), (4.19) and $u_{\infty}(x_{0}) = \psi(x_{0}) > 0$, we have

and

Claim:

Assume that $\Lambda_{\infty}\psi(x_{0}) > |\nabla \psi(x_{0})|$, then (4.24) and (1.11) imply

and

Thus, choosing $\varepsilon > 0$ small enough, we have

for all $n\in \mathbb{N}$ sufficiently large. By (4.29), we get

From (4.23), (4.27) and (4.30), we see that

which is a contradiction. Hence, (4.26) holds.

Claim:

Suppose that the above inequality is not true, then we have

Thus, choosing $\varepsilon_{1} > 0$ small enough, we have

for all $n\in \mathbb{N}$ sufficiently large. We are led to

In view of $(\psi(x_{0}))^{\theta(x_{0})}K_{\infty}(u_{\infty})-|\nabla \psi(x_{0})| > 0$ and (4.25),

Therefore, this fact along with (4.23) shows that (4.31) holds. This is a contradiction. Thus we deduce that (4.32) holds.

Claim:

Taking (4.24) and (4.26) into account, we have

At the same time, by (4.25) and (4.32), we also deduce that (4.36) holds. If we assume that inequality (4.35) does not hold, then by (4.23) and (4.36), there is a contradiction. Thus, we deduce that (4.35) holds.

Case 2: $\psi(x_{0}) = u_{\infty}(x_{0}) = 0$.

Note that if $|\nabla \psi(x_{0})| = 0$ (in this case, we have $\triangle_{\infty}\psi(x_{0})$ = 0), the inequality (4.20) trivially holds. Hence, let us assume that $|\nabla \psi(x_{0})| > 0$, then $|\nabla \psi(x_{n})| > 0$ for $n\in \mathbb{N}$ large enough. We can use very similar arguments as Case 1 to conclude that (4.20) holds. The same argument can be used in order to show that $u_{\infty}$ is a viscosity supersolution of (1.20). □

By Lemmas 4.3 and 4.4, it follows that Theorem 1.2 holds.

Remark 4.3. In the particular case where $p_{n}(x) = np(x)$ and $q_{n}(x) = nq(x)$, Theorems 1.1 and 1.2 are also true.

5.   Conclusions

In this paper, we studied a double-phase eigenvalue problem with large variable exponents. As we know, for $p$-Laplace operator eigenvalue problems, there is an important feature that if $u$ is an eigenfunction, so is $ku$, where $k$ is an arbitrary constant. However, the double-phase operator with variable exponents looses this property. To overcome the above mentioned shortcoming, we defined the eigenvalue by using the Rayleigh quotient of two norms of Musielak-Orlicz space. Moreover, in the particular case where $p_{n}(\cdot) = p_{n}$ and $q_{n}(\cdot) = q_{n}$, Theorems 1.1 and 1.2 are also true (see ^[13]).

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (No.12001196) and the Natural Science Foundation of Henan (No. 232300421143).

Conflict of interest

The authors declare that they have no competing interests.