Faber polynomial coefficients estimates for certain subclasses of $q$-Mittag-Leffler-Type analytic and bi-univalent functions

1.   Introduction and main results

In this paper, we investigate the existence of a least energy sign-changing solution for the following fractional $p$-Laplacian problem:

where $s\in (0, 1)$, $2 < \frac{N}{s}: = p$, $(-\Delta)^{s}_{p}$ is the fractional $p$-Laplacian operator which, up to a normalizing constant, may be defined by setting

along functions $u \in C_0^{\infty} (\mathbb{R}^N)$, where $B_{\varepsilon}(x)$ denotes the ball of $\mathbb{R}^N$ centered at $x \in \mathbb{R}^N$ and radius $\varepsilon > 0.$ In addition, the potential $V\in C(\mathbb{R}^N, \mathbb{R})$, the nonlinear $f$ has exponential critical growth, and such nonlinear behavior is motivated by the Trudinger-Moser inequality (Lemma 2.2).

Recently, the study of nonlocal problems driven by fractional operators has piqued the mathematical community's interest, both because of their intriguing theoretical structure and due to concrete applications such as obstacle problems, optimization, finance, phase transition, and so on. We refer to ^[18] for more details. In fact, when $p = 2$, problem (1.1) appears in the study of standing wave solutions, i.e., solutions of the form $\psi(x, t) = u(x) e^{-i \omega t}$, to the following fractional Schrödinger equation:

where $\hbar$ is the Planck constant, $W: \mathbb{R}^{N} \rightarrow \mathbb{R}$ is an external potential, and $f$ is a suitable nonlinearity. Laskin ^[25,26] first introduced the fractional Schrödinger equation due to its fundamental importance in the study of particles on stochastic fields modeled by Lévy processes.

After that, remarkable attention has been devoted to the study of fractional Schrödinger equations, and a lot of interesting results were obtained. For the existence, multiplicity and behavior of standing wave solutions to problem (1.2), we refer to ^{[1,9,10,14,19,33]} and the references therein.

For general $p$ with $2 < p < \frac{N}{s}$, problem (1.1) becomes the following fractional Laplacian problem:

We emphasize that the fractional $p$-Laplacian is appealing because it contains two phenomena: the operator's nonlinearity and its nonlocal character. In fact, for the fractional $p$-Laplacian operator $(-\Delta)_{p}^{s}$ with $p\neq2$, one cannot obtain a similar equivalent definition of $(-\Delta)_{p}^{s}$ by the harmonic extension method in ^[10]. For those reasons, the study of (1.3) becomes attractive. In ^[13], the authors obtained infinitely many sign-changing solutions of (1.3) by using the invariant sets of descent flow. Moreover, they also proved (1.3) possesses a least energy sign-changing solution via deformation Lemma and Brouwer degree. We stress that, by using a similar method, Wang and Zhou ^[35], Ambrosio and Isernia ^[3] obtained the least energy sign-changing solutions of (1.3) with $p = 2$. For more results involving the fractional $p$-Laplacian, we refer to ^{[2,17,20,21,32]} and the references therein.

Another motivation to investigate problem (1.1) comes from the fractional Schrödinger equations involving exponential critical growth. Indeed, we shall study the case where the nonlinearity $f(t)$ has the maximum growth that allows us to treat problem (1.1) variationally in $W^{s, \frac{N}{s}}(\mathbb{R}^N)$ (see the definition in (1.5)). If $p < \frac{N}{s}$, Sobolev embedding (Theorem 6.9 in ^[18]) states $W^{s, p}(\mathbb{R}^N)\hookrightarrow L^{p_{s}^{*}}(\mathbb{R}^N)$, where $p_{s}^{*} = \frac{Np}{N-sp}$, and $p_{s}^{*}$ is called the critical Sobolev exponent. Moreover, the same result ensures that $W^{s, \frac{N}{s}}(\mathbb{R}^N)\hookrightarrow L^{\lambda}(\mathbb{R}^N)$ for any $\lambda\in[\frac{N}{s}, +\infty)$. However, $W^{s, \frac{N}{s}}(\mathbb{R}^N)$ is not continuously embedded in $L^{\infty}(\mathbb{R}^N)$ (for more details, we refer to ^[18]). On the other hand, in the case $p = \frac{N}{s}$, the maximum growth that allows us to treat problem (1.1) variationally in Sobolev space $W^{s, \frac{N}{s}}(\mathbb{R}^N)$, which is motivated by the fractional Trudinger-Moser inequality proved by Ozawa ^[30] and improved by Kozono et al. ^[23] (see Lemma 2.2). More precisely, inspired by ^[23], we say that $f(t)$ has exponential critical growth if there exists $\alpha_0 > 0$ such that

On the basis of this notation of critical, many authors pay their attention to investigating elliptic problems involving the fractional Laplacian operator and nonlinearities with exponential growth. When $N = 1, s = \frac{1}{2}$, $p = 2$ and replacing $\mathbb{R}$ by $(a, b)$, problem (1.1) becomes the following fractional Laplacian equation:

When $f$ is $o(|t|)$ at the origin and behaves like $\exp(\alpha t^2)$ as $|t|\rightarrow \infty$, by virtue of the Mountain Pass theorem, Iannizzotto and Squassina ^[22] proved the existence and multiplicity of solutions for (1.4). Utilizing the constrained variational methods and quantitative deformation lemma, Souza et al. ^[16] considered the least energy sign-changing solution of problem (1.4) involving exponential critical growth.

For problem (1.1) with exponential critical growth nonlinearity $f$, we would like to mention references ^[8,27,34]. In ^[27], by applying variational methods together with a suitable Trudinger-Moser inequality for fractional space, the authors obtained at least two positive solutions of (1.1). In ^[8], the authors considered problem (1.1) with a Choquard logarithmic term and exponential critical growth nonlinearity $f$. They proved the existence of infinite many solutions via genus theory. In ^[34], by using Ljusternik-Schnirelmann theory, Thin obtained the existence, multiplicity and concentration of nontrivial nonnegative solutions for problem (1.1). For more recent results on fractional equations involving exponential critical growth, see ^{[8,15,16,22,27,31,34]} and the references therein. We also refer the interested readers to ^[11,29] for general problems with Trudinger-Moser-type behavior.

Inspired by the works mentioned above, it is natural to ask whether problem (1.1) has sign-changing solutions when the nonlinearity $f$ has exponential critical growth. To our knowledge, there are few works on it except ^[16]. Souza et al. ^[16] considered the case when $N = 1, p = 2$ and $s = \frac{1}{2}$. However, compared with ^[16], the situation when $p > 2$ is quite different. In particular, the decomposition of functional $I$ (see the definition in (1.8)) is more difficult than that in ^[16]. Therefore, some difficulties arise in studying the existence of a least energy sign-changing solution for problem (1.1), and this makes the study interesting.

In order to study problem (1.1), from now on, we fix $p = \frac{N}{s}$, $p' = \frac{p}{p-1} = \frac{N}{N-s}$ and consider the following assumptions on $V$ and $f$ :

$(V)$ $V(x) \in C\left(\mathbb{R}^{N}\right)$ and there exists $V_{0} > 0$ such that $V(x) \geq V_{0}$ in $\mathbb{R}^{N}$. Moreover, $\lim _{|x| \rightarrow \infty} V(x) = +\infty$;

$(f_{1})$ The function $f\in C^{1}\left(\mathbb{R}\right)$ with exponential critical growth at infinity, that is, there exists a constant $\alpha_0 > 0$ such that

$(f_{2})$ $\lim \limits_{t \rightarrow 0} \frac{|f(t)|}{|t|^{p-1}} = 0;$

$(f_{3})$ There exists $\theta > p$ such that

where $F(t) = \int_{0}^{t} f(s) d s$; $(f_{4})$

There are two constants $q > p$ and $\gamma > 1$ such that

$(f_{5})$ $\frac{f(t)}{|t|^{p-1}}$ is an increasing function of $t\in \mathbb{R}\backslash \{0\}$.

Before stating our results, we recall some useful notations. The fractional Sobolev space $W^{s, p}(\mathbb{R}^N)$, where $p = \frac{N}{s}$, is defined by

where $[u]_{s, p}$ denotes Gagliardo seminorm, that is,

$W^{s, p}(\mathbb{R}^N)$ is a uniformly convex Banach space (see ^[18]) with norm

Let us consider the work space

with the norm

$X$ is a uniformly convex Banach space, and thus $X$ is a reflexive space. By the condition $(V)$, we have that the embedding from $X$ into $W^{s, p}(\mathbb{R}^N)$ is continuous.

Definition 1.1. We say that $u\in X$ is a weak solution of problem (1.1) if

for all $\varphi \in X$.

Define the energy functional $I:X\rightarrow \mathbb{R}$ associated with problem (1.1) by

By the Trudinger-Moser type inequality in ^[23] (see Lemma 2.2), we prove that $I(u)\in C^{1}(X, \mathbb{R})$, and the critical point of functional $I$ is a weak solution of problem (1.1) (see Remark 2.2 in Section 2).

For convenience, we consider the operator $A:X\rightarrow X^{*}$ given by

where $X^{*}$ is the dual space of $X$. In the sequel, for simplicity, we denote $\langle \cdot, \cdot \rangle_{X^{*}, X}$ by $\langle \cdot, \cdot \rangle$. Moreover, we denote the Nehari set $\mathcal{N}$ associated with $I$ by

Clearly, $\mathcal{N}$ contains all the nontrivial solutions of (1.1). Define $u^{+}(x): = \max \left\{ u(x), 0 \right\}$ and $u^{-}(x): = \min \left\{ u(x), 0 \right\}$, and then the sign-changing solution of (1.1) stays on the following set:

Set

and

Now, we can state our main results.

Theorem 1.1. Assume that $(V)$ and $\left(f_{1}\right)$ to $\left(f_{5}\right)$ hold, and then problem (1.1) possesses a least energy sign-changing solution $u^{*}\in X$, provided that

where $\alpha^{*}$ is a constant defined in Lemma 2.2,

and

The proof of Theorem 1.1 is based on the arguments presented in ^[7]. We first check that the minimum of functional $I$ restricted on set $\mathcal{M}$ can be achieved. Then, by using a suitable variant of the quantitative deformation Lemma, we show that it is a critical point of $I$. However, due to the nonlocal term

the functional $I$ no longer satisfies the decompositions

which are very useful to get sign-changing solutions of problem (1.1) (see for instance ^[4,5,6,7,12]). In fact, due to the fractional $p$-Laplacian operator $(-\Delta)^{s}_p$ with $s\in (0, 1)$ and $p\neq 2$, one cannot obtain a similar equivalent definition of $(-\Delta)^{s}_p$ by the harmonic extension method (see ^[10]). In addition, compared with ^[16], problem (1.1) contains nonlocal operator $(-\Delta)_{p}^{s}$ with $p > 2$, which brings some difficulties while studying problem (1.1). In particular, for problem (1.4), one has the following decomposition:

where $J$ is the energy functional of (1.4). However, for the energy functional $I$, it is impossible to obtain a similar decomposition like (1.15) due to the nonlinearity of the operator $(-\Delta)_{p}^{s}$. In order to overcome this difficulty, we divide $\mathbb{R}^{2N}$ into several regions (see Lemma 2.6) and decompose functional $I$ on each region carefully. Furthermore, another difficulty arises in verifying the compactness of the minimizing sequence in $X$ since problem (1.1) involves the exponential critical nonlinearity term. Fortunately, thanks to the Trudinger-Moser inequality in ^[23], we overcome this difficulty by choosing $\gamma$ in assumption $(f_4)$ appropriately large to ensure the compactness of the minimizing sequence. To achieve this, a more meticulous calculation is needed in estimating $m$.

On the other hand, by a similar argument in ^[13], we also consider the energy behavior of $I(u^{*})$ in the following theorem, where $u^{*}$ is the least energy sign-changing solution obtained in Theorem 1.1.

Theorem 1.2. Assume that $(V)$ and $\left(f_{1}\right)$ to $\left(f_{4}\right)$ are satisfied. Then, $c > 0$ is achieved, and

where $u^{*}$ is the least energy sign-changing solution obtained in Theorem 1.1.

The paper is organized as follows: Section 2 contains the variational setting of problem (1.1), and it establishes a version of the Trudinger-Moser inequality for (1.1). In Section 3, we give some technical lemmas which will be crucial in proving the main result. Finally, in Section 4, we combine the minimizing arguments with a variant of the Deformation Lemma and Brouwer degree theory to prove the main result.

Throughout this paper, we will use the following notations: $L^{\lambda}(\mathbb{R}^N)$ denotes the usual Lebesgue space with norm $|\cdot|_{\lambda}$; $C, C_1, C_2, \cdots$ will denote different positive constants whose exact values are not essential to the exposition of arguments.

2.   Preliminaries

In this section, we outline the variational framework for problem $(1.1)$ and give some preliminary Lemmas. For convenience, we assume that $V_0 = 1$ throughout this paper. Recalling the definition of fractional Sobolev space $X$ in (1.6), we have the following compactness results.

Lemma 2.1. Suppose that $(V)$ holds. Then, for all $\lambda \in [p, \infty)$, the embedding $X\hookrightarrow L^{\lambda}(\mathbb{R}^N)$ is compact.

Proof. Define $Y = L^{\lambda}\left(\mathbb{R}^{N}\right)$ and $B_{R} = \left\{x \in \mathbb{R}^{N}:|x| < R\right\}, B_{R}^{c} = \mathbb{R}^{N} \backslash \overline{B_{R}}$. Let $X(\Omega)$ and $Y(\Omega)$ be the spaces of functions $u \in X, u \in Y$ restricted onto $\Omega \subset \mathbb{R}^{N}$ respectively. Then, it follows from Theorems 6.9, 6.10 and 7.1 in ^[18] that $X\left(B_{R}\right) \hookrightarrow Y\left(B_{R}\right)$ is compact for any $R > 0$. Define $V_{R} = \inf\limits_{x\in B_{R}^{c}} V(x)$. By $(V)$, we deduce that $V_R\rightarrow \infty$ as $R \rightarrow \infty$. Therefore, when $\lambda = \frac{N}{s} = p$, we have

which implies

By virtue of Theorem 7.9 in ^[24], we can see that $X\hookrightarrow L^{p}(\mathbb{R}^N)$ is compact.

When $\lambda > p$, by the interpolation inequality, one can also obtain that $X\hookrightarrow L^{\lambda}(\mathbb{R}^N)$ is compact. This completes the proof.

To study problems involving exponential critical growth in the fractional Sobolev space, the main tool is the following fractional Trudinger-Moser inequality, and its proof can be found in Zhang ^[37]. First, to make the notation concise, we set, for $\alpha > 0$ and $t\in \mathbb{R}$,

where $S_{k_{p}-2}(\alpha, t) = \sum_{k = 0}^{k_{p}-2} \frac{\alpha^{k}}{k!} |t|^{p'k}$ with $k_p = \min\{k\in \mathbb{N}; k\geq p\}.$

Lemma 2.2. Let $s\in (0, 1)$ and sp = N. Then, there exists a positive constant $\alpha^{*} > 0$ such that

for all $u \in W^{s, p}(\mathbb{R}^N)$ with $\|u\|_{W^{s, p}(\mathbb{R}^N)} \leq 1$.

For $\Phi(\alpha, u)$, we also have the following properties, which have been proved in ^[8,27]. For the reader's convenience, we sketch the proof here.

Lemma 2.3. ^[27] Let $\alpha > 0$ and $r > 1$. Then, for every $\beta > r$, there exists a constant $C =$ $C(\beta) > 0$ such that

Proof. Noticing that

we deduce

On the other hand, it holds that

and the lemma follows.

Lemma 2.4. ^[8] Let $\alpha > 0.$ Then, $\Phi(\alpha, u) \in L^{1}\left(\mathbb{R}^{N}\right)$ for all $u \in X$.

Proof. Let $u \in X \backslash\{0\}$ and $\varepsilon > 0$. Since $C_{0}^{\infty}\left(\mathbb{R}^{N}\right)$ is dense in $X$, there exists $\phi \in C_{0}^{\infty}\left(\mathbb{R}^{N}\right)$ such that $0 < \|u-\phi\|_{X} < \varepsilon$. Observe that for each $k \geq k_{p}-1$,

Consequently,

From Lemma 2.2, choosing $\varepsilon > 0$ sufficiently small such that $\alpha 2^{p'} \varepsilon^{p'} < \alpha^{*}$, we have

On the other side, since $\exp \left(\alpha 2^{p'}|\phi|^{p'}\right) = \sum_{k = 0}^{+\infty} \frac{\alpha^{k}}{k!} 2^{p' k}|\phi|^{p' k}$, there exists $k_{0} \in \mathbb{N}$ such that $\sum_{k = k_{0}}^{+\infty} \frac{\alpha^{k}}{k!} 2^{p' k}|\phi|^{p' k} < \varepsilon$. This fact, combined with the fact that $p' k > 0$ for all $k_{p}-1 \leq k \leq k_{0}$, gives us

Therefore, $\Phi(\alpha, u) \in L^{1}\left(\mathbb{R}^{N}\right)$ for all $u \in X$.

Remark 2.1. From Lemmas 2.2–2.4, we deduce that $\Phi(\alpha, u)^l \in L^{1}(\mathbb{R}^N)$ for all $u \in X$, $\alpha > 0$ and $l\geq 1$.

The next lemma shows the growth behavior of the nonlinearity $f$.

Lemma 2.5. Given $\varepsilon > 0$, $\alpha > \alpha_0$ and $\zeta > p$, for all $u\in X$, it holds that

and

Proof. We will only prove the first result. Since the second inequality is a direct consequence of the first one due to assumption $(f_3)$, we omit it here.

In fact, by $(f_2)$, for given $\varepsilon > 0$, there exists $\delta > 0$ such that $|f(u)|\leq \varepsilon |u|^{p-1}$ for all $|u| < \delta$. Now, as $\zeta > p$, there exists $r > 0$ such that $\zeta = p+r.$ Hence, once $z^{\zeta-1}, z^{q'k}$ for all $k\geq k_p-1$ and $z^r$ are increasing functions, if $|u| > \delta$, it follows from $(f_1)$ that

This completes the proof.

Remark 2.2. It follows from Lemmas 2.1–2.3 that $I$ is well-defined on $X$. Moreover, $I\in C^{1}(X, \mathbb{R}^N)$, and

for all $v\in X$. Consequently, the critical point of $I$ is the weak solution of problem (1.1).

We seek the sign-changing solution of problem (1.1). As we saw in Section 1, one of the difficulties is the fact that the functional $I$ does not possess a decomposition like (1.14). Inspired by ^[13,35], we have the following:

Lemma 2.6. Let $u \in X$ with $u^{\pm}\neq 0$. Then,

(i) $I(u) > I(u^{+})+I(u^{-})$,

(ii) $\langle I^{\prime}(u), u^{\pm} \rangle > \langle I^{\prime}(u^{\pm}), u^{\pm} \rangle$.

Proof. Observe that

By density (see Di Nezza et al. Theorem 2.4 ^[18]), we can assume that $u$ is continuous. Define

Then, for $u\in X$ with $u^{\pm}\neq 0$, by a straightforward computation, one can see that

Similarly, we also have

Taking into account (2.4)–(2.6), we deduce that $I(u) > I(u^{+})+I(u^-)$. Analogously, one can prove $(ii)$.

In the last part of this section, we prove the following inequality, which will play an important role in estimating the upper bound for $m: = \inf\limits_{u\in \mathcal{M}}I(u)$.

Lemma 2.7. For all $u\in X$ with $u^{\pm}\neq 0$ and constants $\sigma, \tau > 0$, it holds that

Furthermore, the inequality in (2.7) is an equality if and only if $\sigma = \tau$.

Proof. First, we claim the following elementary inequality holds true:

where $a, b\geq 0$, $\sigma, \tau > 0$ and $p = \frac{N}{s} > 2$.

Indeed, if $a = 0$ or $b = 0$, one can easily check (2.8). Thus, we can assume that $a, b > 0$. Setting $\kappa = \frac{a}{a+b}$ and $t = \frac{\sigma}{\tau}$, (2.8) becomes

Let us define $g(t): = \kappa t^{p}+(1-\kappa)- (\kappa t +(1-\kappa))^{p}$, and then $g'(t) = \kappa p \left[ t^{p-1}-\left(\kappa t + (1-\kappa)\right)^{p-1} \right].$ Noting that $0 < \kappa < 1$, we can observe that $g'(1) = 0$, $g'(t) < 0$ for $0 < t < 1$, and $g'(t) > 0$ for $t > 1$. Therefore, $g(t) > g(1) = 0$ for all $t > 0$ and $t\neq 1$, which implies (2.9). Consequently, (2.8) holds true.

Now, let us consider the inequality (2.7). By a straightforward computation, one can see from (2.8) that

which implies (2.7), and Lemma 2.7 follows.

3.   Some technical lemmas

The aim of this section is to prove some technical lemmas related to the existence of a least energy nodal solution. Firstly, we collect some preliminary lemmas which will be fundamental to prove our main result.

Lemma 3.1. Under the assumptions of Theorem 1.1, we have

(i) For all $u\in \mathcal{N}$ such that $\left\| u \right\|_{X} \rightarrow \infty$, $I(u)\rightarrow \infty;$

(ii) There exist $\rho, \mu > 0$ such that $\left\|u^{\pm}\right\|_{X} \geq \rho$ for all $u \in \mathcal{M}$ and $\left\| u \right\|_{X} > \mu$ for all $u\in \mathcal{N}$.

Proof. $(i)$ Since $u\in\mathcal{N}$ and $(f_3)$ holds, we see that

Hence, the above inequality ensures that $I(u)\rightarrow \infty$ as $\left\| u \right\|_{X}\rightarrow \infty$.

$(ii)$ We claim that there exists $\mu > 0$, such that $\|u\|_{X} > \mu$ for all $u\in \mathcal{N}.$ By contradiction, we suppose that there exists a sequence $\{u_n\}\subset \mathcal{N}$ such that $\left\| u_n \right\|_{X} \rightarrow 0$ in $X$.

Then, it follows from Lemmas 2.1, 2.3, 2.5 and Hölder's inequality that

where $\alpha > \alpha_0$, $\zeta > p$ and $r > 1$ with $\frac{1}{r}+\frac{1}{r'} = 1$.

On the other hand, since $\left\| u_n \right\|_{X} \rightarrow 0$, there exists $N_0\in \mathbb{N}$ and $\vartheta > 0$ such that $\left\| u_n \right\|_{X}^{p'} < \vartheta < \frac{\alpha^{*}}{\alpha_0}$ for all $n > N_0$. Choosing $\alpha > \alpha_0$, $r > 1$ and $\beta > r$ satisfying $\alpha \vartheta < \alpha^{*}$ and $\beta \alpha \vartheta < \alpha^{*}$, for all $n > N_0$, we deduce from Lemmas 2.2 and 2.3 that

Combining (3.3) with (3.2), and choosing $\varepsilon = \frac{1}{2}$ in (3.2), we can see that

where $C'$ is a constant independent of $n$. Obviously, (3.4) contradicts with $\left\| u_n \right\|_{X}\rightarrow 0$, and we have proved the claim.

For $u_n\in \mathcal{M}$, we have $\langle I'(u_n), u^{\pm}_n \rangle = 0$. Hence, it follows from Lemma 2.6 that

which implies

By arguments as with (3.2) and (3.3), we deduce that there exist $\rho_1, \rho_2 > 0$ such that $\left\| u^{+}_n \right\|_{X} > \rho_1$ and $\left\| u^{-}_n \right\|_{X} > \rho_2.$ Select $\rho = \min (\rho_1, \rho_2)$, and then $\left\| u^{\pm}_n \right\|_{X} > \rho$, and this completes the proof.

Lemma 3.2. Under the assumptions of Theorem 1.1, for any $u\in X\backslash \{0\}$, there exists a unique $\nu_u \in \mathbb{R}^{+}$ such that $\nu_u u \in \mathcal{N}$. Moreover, for any $u\in\mathcal{N}$, we have

Proof. Given $u\in X\backslash \{0\}$, we define $g(\nu) = I(\nu u)$ for all $\nu\geq 0$, i.e.,

It follows from $(f_2)$ and $(f_4)$ that the function $g(\nu)$ possesses a global maximum point $\nu_u$, and $g^{\prime}(\nu_u) = 0$, i.e.,

which implies $\nu_u u \in\mathcal{N}$. Now, we claim the uniqueness of $\nu_u$. Suppose, on the contrary, there exists $\tilde{\nu}_u\neq \nu_{u}$ such that $\tilde{\nu}_{u} u \in \mathcal{N}$. Then, it holds that

Without loss of generality, we may assume $\tilde{\nu}_{u} > \nu_{u}$. Combining (3.6), (3.7) and $(f_5)$, we deduce that

which leads to a contradiction. Thus, $\nu_{u}$ is unique. Obviously, for any $u\in \mathcal{N}$, $\nu_{u} = 1$, and (3.5) follows. This completes our proof.

Since we are considering the constrained minimization problem on $\mathcal{M},$ in the following, we will show that the set $\mathcal{M}$ is nonempty.

Lemma 3.3. If $u \in X$ with $u^{\pm} \neq 0$, then there exists a unique pair $\left(\sigma_{u}, \tau_{u}\right)$ of positive numbers such that

Consequently, $\sigma_{u} u^{+} + \tau_{u} u^{-} \in \mathcal{M}$.

Proof. Let $G:(0, +\infty)\times (0, +\infty) \rightarrow \mathbb{R}^2$ be a continuous vector field given by

for every $(\sigma, \tau)\in (0, +\infty)\times (0, +\infty).$ By virtue of Lemmas 2.5 and 2.6, we deduce that

where $\alpha > \alpha_0$ and $\zeta > q$. Choose $\varepsilon = \frac{1}{2}$, $\sigma$ small enough such that $\left\| \sigma u^{+} \right\|^{p'}_{X} < \frac{\alpha^{*}}{\alpha_0}$. Taking into account (3.8) and arguing as with (3.3) in Lemma 3.1, one can see that

and similarly

Hence, it follows from (3.9) and (3.10) that there exists $R_{1} > 0$ small enough such that

and

On the other hand, by $(f_3)$, there exist constants $D_1, D_2 > 0$ such that

Then, we have

where $A^{+} \subset \operatorname{supp}\left(u^{+}\right)$ is a measurable set with finite and positive measure $|A^{+}|.$ Due to the fact that $\theta > p$, for $R_2$ sufficiently large, we get

Similarly, we get

Hence, taking into account (3.6), (3.7), (3.15), (3.16) and thanks to the Miranda theorem ^[28], there exists $(\sigma_u, \tau_u)\in [R_1, R_2]\times[R_1, R_2]$ such that $G(\sigma_u, \tau_u) = 0$, which implies $\sigma_u u^{+}+\tau_u u^{-} \in \mathcal{M}$.

Now, we are in the position to prove the uniqueness of the pair $(\sigma_u, \tau_u)$. First, we assume that $u = u^{+}+u^{-}\in \mathcal{M}$ and $(\sigma_u, \tau_u)\in (0, +\infty)\times (0, \infty)$ is another pair such that $\sigma_u u^{+} + \tau_u u^{-}\in \mathcal{M}$. In this case, we just need to prove that $(\sigma_u, \tau_u) = (1, 1)$. Notice that

and

Without loss of generality, we may assume $\sigma_u \leq \tau_u$. Then, by a direct computation, one has

which together with (3.18) implies

Combining (3.20) with (3.17), we deduce that

Hence, by $(f_5)$ and since $u^{+}\neq 0$, we obtain $\sigma_u\geq 1$. Moreover, using similar arguments as in (3.19), we can deduce that

Therefore, it follows from (3.17), (3.18) and (3.21) that

which together with $(f_5)$ implies $\tau_u\leq 1.$ Thus, we conclude the proof of the uniqueness of the pair $(1, 1)$.

For the general case, we suppose that $u$ does not necessarily belong to $\mathcal{M}$. Let $(\sigma_u, \tau_u)$, $(\sigma'_u, \tau'_u) \in (0, +\infty)\times(0, \infty).$ We define $v = v^{+}+v^{-}$ with $v^{+} = \sigma_u u^{+}$ and $v^{-} = \tau_u u^{-}$. Therefore, we have $v\in \mathcal{M}$ and $\frac{\sigma'_u}{\sigma_u} v^{+}+\frac{\tau'_u}{\tau_u} v^{-}\in \mathcal{M},$ which implies $(\sigma_u, \tau_u) = (\sigma'_u, \tau'_u)$, and this completes the proof.

The following two lemmas will be useful in proving Theorem 1.1.

Lemma 3.4. Under the assumptions of Theorem 1.1, and let $u\in X$ with $u^{\pm}\neq 0$ such that $\langle I^{\prime}(u), u^{\pm}\rangle\leq 0.$ Then, the unique pair of positive numbers obtained in Lemma 3.3 satisfies $0 < \sigma_u, \tau_u \leq 1$.

Proof. Here we will only prove $0 < \sigma_u\leq 1$. The proof of $0 < \tau_u\leq 1$ is analogous, and we omit it here. Since $\langle I^{\prime}(u), u^{+}\rangle\leq 0,$ it holds that

Without loss of generality, we can assume that $\sigma_u\geq \tau_u > 0$, and $\sigma_u u^{+}+\tau_u u^{-} \in \mathcal{M}$. Therefore, utilizing a similar argument as in (3.19), we deduce that

Taking into account (3.22) and (3.23), we obtain

which together with assumption $(f_5)$ and the fact $u^{+}\neq 0$ shows that $\sigma_u \leq 1.$ Hence, we finish the proof.

Lemma 3.5. Under the assumptions of Theorem 1.1, let $u\in X$ with $u^{\pm}\neq 0$ and $(\sigma_u, \tau_u)$ be the unique pair of positive numbers obtained in Lemma 3.3. Then, $(\sigma_u, \tau_u)$ is the unique maximum point of the function $h^{u}:[0, +\infty) \times[0, +\infty) \rightarrow \mathbb{R}$ given by

Proof. In the demonstration of Lemma 3.3, we saw that $(\sigma_u, \tau_u)$ is the unique critical point of $h^{u}$ in $(0, +\infty)\times(0, +\infty)$. In addition, by the definition of $h^{u}$ and (3.13), we have

where $A^{+} \subset \operatorname{supp}\left(u^{+}\right)$ and $A^{-} \subset \operatorname{supp}\left(u^{-}\right)$ are measurable sets with finite and positive measures $|A^{+}|$ and $|A^{-}|$. Since $\theta > p$, we conclude that $h^{u}(\sigma, \tau)\rightarrow -\infty$ as $|(\sigma, \tau)|\rightarrow \infty$. In particular, one can easily check that there exists $R > 0$ such that $h^{u}(a, b) < h^{u}(\sigma_u, \tau_u)$ for all $(a, b) \in$ $(0, \infty) \times(0, \infty) \backslash \overline{B_{R}(0)}$, where $\overline{B_{R}(0)}$ is a closure of the ball of radius $R$ in $\mathbb{R}^{2}$.

To end the proof, we just need to verify that the maximum of $h^{u}$ does not occur in the boundary of $[0, +\infty)\times [0, +\infty)$. Suppose, by contradiction, that $(0, b)$ is a maximum point of $h^{u}$. Then, for $a > 0$ small enough, one can see from (3.9) that $h^{u}(a, 0) = I(a u^{+}) > 0$. Hence, it follows from Lemma 2.6 that

for $a > 0$ small enough. However, this contradicts with the assumption that $(0, b)$ is a maximum point of $h^{u}$. The case $(a, 0)$ is similar, and we complete the proof.

Since Lemma 3.3 shows $\mathcal{M}$ is nonempty, and Lemma 3.1 implies that $I(u) > 0$ for all $u\in \mathcal{M}$, $I$ is bounded below in $\mathcal{M}$, which means that $m: = \inf\limits_{u\in \mathcal{M}}I(u)$ is well-defined. Now, we shall prove an upper bound for $m$ to recover the compactness, which urges us to prove that $m$ can be achieved.

Lemma 3.6. Under the assumptions of Theorem 1.1 and that $\theta$ is the constant given by $(f_3)$,

Proof. Due to $\mathcal{M}\subset \mathcal{N}$, we have $m\ge c: = \inf\limits_{u\in \mathcal{N}}I(u)$. Moreover, for all $u\in \mathcal{N}$, by Lemma 3.1 it holds that

On the other hand, by the similar procedure used in ^[13], there exists $w \in \mathcal{M}_q$ with $w^{\pm}\neq 0$, such that $I_{q}(w) = m_q$ and $\langle I'_{q}(w), w^{\pm} \rangle = 0$. Therefore, it holds that

In addition, by virtue of Lemma 3.3, there exist $\sigma, \tau > 0$ such that $\sigma w^{+}+\tau w^{-} \in \mathcal{M}$. Therefore, it holds that

which together with $(f_4)$ implies that

Now, from (3.26) and thanks to Lemma 2.7, we conclude that

Therefore, by the definition of $\gamma$ in Theorem 1.1, we obtain (3.24).

Lemma 3.7. Under the assumptions of Theorem 1.1, let $\{u_n\}\subset \mathcal{M}$ be a minimizing sequence for $m$, and then

and

hold for some $u\in X$.

Proof. We will only prove the first result, since the second limit is a direct consequence of the first one. Since $\{u_n\}\subset \mathcal{M}$ is a minimizing sequence for $m$, $I(u_n)\rightarrow m$, and it follows from (3.25) that

which implies $\{u_n\}$ is bounded in $X$. Then, by virtue of Lemma 2.1, up to a subsequence, there exists $u\in X$ such that

Hence,

Moreover, utilizing (3.29) again, we deduce that there exist $n_0\in \mathbb{N}$ and $\vartheta > 0$ such that $\left\| u_n \right\|^{p'}_{X} < \vartheta < \frac{\alpha^{*}}{\alpha_0}$ for $n > n_0$. Choose $\alpha > \alpha_0$, $r > 1$ and close to 1, and $\beta > r$ satisfying $\alpha \vartheta < \alpha^{*}$ and $\beta \alpha \vartheta < \alpha^{*}$, and then for all $n > n_0$, it follows from Lemmas 2.2 and (3) that

where $C$ is a constant independent of $n$. Thus, by virtue of Lemma 2.5 and Hölder's inequality, for every Lebesgue measurable set $A\subset \mathbb{R}^N$ and $n > n_0$, it holds that

Due to (3.33) and the fact that $u^{\pm}_n \rightarrow u^{\pm}$ in $L^{p}(\mathbb{R}^N)$ and $L^{\zeta r'}(\mathbb{R}^N)$, we conclude that for any $\varepsilon > 0$ and $n > n_0$, there exists $\delta > 0$ such that for every Lebesgue measurable set $A\subset \mathbb{R}^N$ with ${\rm{meas}}(A)\leq \delta$, it holds that

Similarly, for any $\varepsilon > 0$ and $n > n_0$, there exists $R > 0$ such that

Therefore, by (3.31), (3.34), (3.35) and thanks to Vitali's convergence theorem, one can prove

Thus, we finish the proof.

4.   Proof of main results

In this section, we will prove Theorems 1.1 and 1.2. To this end, we consider the minimization problem

Firstly, let us start with the existence of a minimizer $u^{*}\in\mathcal{M}$ of $I.$

Lemma 4.1. Under the assumptions of Theorem 1.1, the infimum $m$ is achieved.

Proof. By using Lemma 3.1, we know that there exists a minimizing sequence $\{u_n \}_n \subset \mathcal{M}$ bounded in $X$, such that

Without loss of generality, we may assume up to a subsequence that there exists $u^{*}$ such that

Then, by Lemmas 2.6, 3.1 and 3.7, we conclude that

which implies $(u^{*})^{\pm}\neq 0$, and consequently $u^{*} = (u^{*})^{+}+(u^{*})^{-}$ is sign-changing. Hence, by Lemma 3.3, there exist $\sigma, \tau > 0$ such that

Now, we claim that $\sigma = \tau = 1$. Indeed, since $u_n\in\mathcal{M}$, $\langle I^{\prime}(u_n), u^{\pm}_{n}\rangle = 0$, i.e.,

and

where $(\mathbb{R}^{N})_{n}^{+}: = \{x\in \mathbb{R}^N : u_{n}(x)\ge 0 \}$ and $(\mathbb{R}^{N})_{n}^{-}: = \{x\in \mathbb{R}^N : u_{n}(x)\leq 0 \}$. Notice the functional $\left\| u \right\|^{p}_{X}$ is weakly lower semicontinuous on $X$, and we see that

Moreover, it follows from Lemma 3.7 that

Taking into account (4.3)–(4.6) and thanks to Fatou's lemma, we deduce that

which together with Lemma 3.4 implies that $0 < \sigma, \tau\leq 1$. In the following, we will show that $\sigma = \tau = 1$, and $I(u^{*}) = m$. In fact, by (4.2), $(f_5)$, Fatou's lemma and the definition of $\mathcal{M}$, one has

Let us observe that by the above calculation we can infer that $\sigma = \tau = 1$. Thus, $u^{*}\in\mathcal{M}$, and $I(u^{*}) = m.$

Now we are ready to prove Theorem 1.1. The proof is based on the quantitative deformation lemma and Brouwer degree theory. For more details, we refer to the arguments used in ^[7].

Proof of Theorem 1.1. We assume by contradiction that $I^{\prime}(u^{*}) \neq 0$. Then, there exist $\delta, \kappa > 0$ such that

Define $D: = \left[1-\delta_1, 1+\delta_1\right] \times\left[1-\delta_1, 1+\delta_1\right]$ and a map $\xi: D \rightarrow X$ by

where $\delta_1\in (0, \frac12)$ small enough such that $\left\|\xi (\sigma, \tau)-u^{*} \right\|_{X}\leq 3\delta$ for all $(\sigma, \tau)\in \bar{D}.$ Thus, by virtue of Lemma 4.1, we can see that

Therefore,

By using [36,Theorem 2.3] with

and $c: = m$. By choosing $\varepsilon: = \min \left\{\frac{m-\beta}{4}, \frac{\kappa \delta}{8}\right\}$, we deduce that there exists a deformation $\eta \in C([0, 1] \times X, X)$ such that:

$(i)$ $\eta(t, v) = v$ if $v \notin I^{-1}\left(\left[m-2 \varepsilon, m+2 \varepsilon\right]\right)$; $(ii)$ $I(\eta(1, v)) \leq m-\varepsilon$ for each $v \in X$ with $\|v-u^{*}\|_{X} \leq \delta$ and $I(v) \leq m+\varepsilon$; $(iii)$ $I(\eta(1, v)) \leq I(v)$ for all $u \in X$.

By $(ii)$ and $(iii)$ we conclude that

Therefore, to complete the proof of this Lemma, it suffices to prove that

Indeed, if (4.10) holds true, then by the definition of $m$ and (4.9), we get a contradiction.

In the following, we will prove (4.10). To this end, let us define ${\bf{\Psi}}^{u^{*}}:[0, +\infty) \times[0, +\infty) \rightarrow \mathbb{R}^{2}$ by

Furthermore, for $(\sigma, \tau)\in \overline{D}$, we define

Since $\eta(1, \xi (\sigma, \tau)) = \xi (\sigma, \tau)$ on $\partial D$, by the Brouwer degree theory (see Theorem D.9 ^[36]), we have

Now, we assert that $\deg({\bf{\Psi}}^{u^{*}}, D, 0) = 1.$ If this assertion holds true, then $\widetilde{{\bf{\Psi}}}(\sigma_0, \tau_0) = 0$ for some $(\sigma_0, \tau_0)\in D$. Thus, there exists $u_0: = \eta(1, \xi (\sigma_0, \tau_0))\in \mathcal{M}$ and (4.10) follows.

In fact, let us first define

Clearly, $C_p = D_p > 0$, $A_p, B_p > 0$. Notice that $u\in\mathcal{M}$, and we can see that

Moreover, $(f_5)$ guarantees

Then, by a direct computation, we have

and

In addition,

Taking advantage of (4.12)–(4.16), we deduce that

Notice that $u^{*}\in \mathcal{M}$, and thanks to Lemmas 3.3 and 3.5, it holds that

which together with (4.11) implies $\deg({\bf{\Psi}}^{u^{*}}, D, 0) = 1.$ This completes our proof.

Lemma 4.2. For any $v \in \mathcal{M}$, there exist $\tilde{\sigma}_{v}, \tilde{\tau}_{v} \in(0, 1)$ such that $\tilde{\sigma}_{v} v^{+}, \tilde{\tau}_{v} v^{-} \in \mathcal{N}$.

Proof. We just prove $\tilde{\sigma}_{v} \in(0, 1)$. The other case can be obtained by similar arguments. Since $v \in \mathcal{M}$, i.e., $\langle I^{\prime}(v), v^{+} \rangle = 0$, by Lemma 2.6, we obtain

On the other hand, by Lemma 3.2, there exists $\tilde{\sigma}_{v} > 0$ such that $\tilde{\sigma}_{v} v^{+} \in \mathcal{N}$, which implies that

Taking into account (4.9) and (4.10), we deduce that

Thus, it follows from $(f_4)$ that $\tilde{\sigma}_{v} < 1$.

Proof of Theorem 1.2. Using a similar idea from the proof of Lemma 4.1, we find $\bar{u}\in \mathcal{N}$ such that $I(\bar{u}) = c > 0$, where $c: = \inf\limits_{u\in \mathcal{N}}I(u)$. Furthermore, utilizing the same steps of the proof of Theorem 1.1, we show that $I^{\prime}(\bar{u}) = 0$. Thus, $\bar{u}$ is a ground state solution of problem (1.1). Hence, to complete the proof of Theorem 1.2, we need to study the energy behavior of $I(u^{*})$, where $u^{*}$ is the sign-changing solution of (1.1) obtained in Theorem 1.1.

In fact, by Lemma 4.2, there exist $0 < \tilde{\sigma}_{u^{*}}, \tilde{\tau}_{u^{*}} < 1$ such that $\tilde{\sigma}_{u^{*}}(u^{*})^{+}, \tilde{\tau}_{u^{*}}(u^{*})^{-}\in\mathcal{N}$. Therefore, we deduce from $(f_5)$ and Lemma 3.2 that

which completes the proof of Theorem 1.2.

5.   Conclusions

This manuscript has employed the variational method to study the fractional $p$-Laplacian equation involving Trudinger-Moser nonlinearity. By using the constrained variational methods, quantitative Deformation Lemma and Brouwer degree theory, we prove the existence of least energy sign-changing solutions for the problem.

Acknowledgments

K. Cheng was supported by the Jiangxi Provincial Natural Science Foundation (20202BABL211005), W. Huang was supported by the Jiangxi Provincial Natural Science Foundation (20202BABL211004), and L. Wang was supported by the National Natural Science Foundation of China (No. 12161038) and the Science and Technology project of Jiangxi provincial Department of Education (Grant No. GJJ212204).

Conflict of interest

The authors declare that there is no conflict of interest.