Group invariant solutions for the planar Schrödinger-Poisson equations

1.   Introduction

The present paper is concerned with the existence of solution to the planar Schrödinger-Poisson equations

where $p\geq2$, $V, f$ are continuous, mirror symmetric or rotationally periodic functions, and $f(x, t)$ has exponential critical growth in the Trudinger-Moser sense (^[1]).

In last decades, considerable attention has been paid to the following Schrödinger-Poisson equations:

with various conditions on the parameters $p, N$ and functions $V, K, f$. These kinds of equations arise in many contexts of physics, such as, in quantum mechanics ^[2,3,4] and semiconductor theory^[5,6,7,8]. In ^[5], Eq (1.2) was introduced as a model describing solitary waves for nonlinear stationary equations of Schrödinger type interacting with an electrostatic field where the unknown functions $u$ and $\phi$ wave function for particles and potential, respectively. Let $p\in(1, 6]$ and $K\in L^\infty(\mathbb{R}^3)$. For each $u\in H^1(\mathbb{R}^3)$, the second equation in (1.2) determines the Newton potential $\phi_{u}$ in $D^{1, 2}(\mathbb{R}^3)$, i.e.,

Many minimization techniques, such as minimizing on a constraint set ^[9,10] and the Mountain Pass Theorem ^{[11,12,13,14,15]}, were used in the Eq (1.2).

When $K(x)\equiv0$, Eq (1.2) becomes the Schrödinger equation. In this case, there are many results to Eq (1.2) with the dual method if $V$ and $f$ satisfy some certain conditions, such as a positive lower bound on $V$ or a monotonicity condition on $f$ (see ^{[16,17,18,19,20]} and references therein).

In the following, let us focus on the two-dimensional case. Stubbe ^[21] considered the equations

where $\lambda\in\mathbb{R}$ is a constant. They set up a variational framework for Eq (1.3) with a subspace $Z$ of $H^1(\mathbb{R}^2)$:

They proved that there exists a unique radial ground state solution for any $\lambda\geq0$. In addition, they proved that there exists a negative number $\lambda^{\ast}$, such that for any $\lambda\in(\lambda^{\ast}, 0)$ there are two radial ground states with different $L^2$ norms. Cigolani and Weth ^[22] considered Eq (1.1) with $p = 2$ and $f(x, u) = b|u|^{\sigma-2}u$. Specifically, $V\in C(\mathbb{R}^2, (0, \infty))$ is $\mathbb{Z}^2$ periodic. Using the concentration-compactness theory, they proved that Eq (1.1) has a ground state $u\in X_2$ and a solution sequence $\{u_n\}_n\subset X_2$, such that $\lim_{n\rightarrow \infty}J(u_n) = \infty$. Here,

and $J$ are the energy functionals associated with Eq (1.1).

Chen and Tang ^[23] considered Eq (1.1) with $p = 2$, i.e.,

where $V\in C(\mathbb{R}^2, [0, \infty))$ is axially symmetrical and $f\in C(\mathbb{R}^2\times\mathbb{R})$ is of subcritical or critical exponential growth in the sense of Trudinger-Moser. More precisely, we say that $f(x, t)$ has subcritical exponential growth at $t = \pm\infty$ if it verifies

(F1') For every $A > 0$,

and

for any $\alpha > 0$;

and the function $f(x, t)$ is said to have the critical exponential growth at $t = \pm\infty$ if it verifies

(F1) The nonlinearity $f$ satisfies (1.5) and there exists $\alpha_0 > 0$ such that, for any $\alpha > \alpha_0$, (1.6) holds but

This notion of criticality can be referred to ^[24].

For the critical growth case, Chen and Tang ^[23] established the existence of a ground state solution for Eq (1.4) by assuming following conditions on $V$ and $f$:

(V0) $V\in C(\mathbb{R}, [0, \infty))$ and $\liminf_{|x|\rightarrow \infty} V(x) > 0$;

(CF1) $V(x): = V(x_1, x_2) = V(|x_1|, |x_2|)$ for all $x\in\mathbb{R}^2$, $f(x, t): = f(x_1, x_2, t) = f(|x_1|, |x_2|, t)$ for all $(x, t)\in\mathbb{R}^2\times\mathbb{R}$;

(CF2) $f(x, t)t > 0$ for all $(x, t)\in\mathbb{R}^2\times\mathbb{R}\backslash\{0\}$, and there exists $M_0 > 0$ and $t_0 > 0$ such that

where $F(x, t): = \int_0^tf(x, s)ds;$

(CF3) $\liminf_{|t|\rightarrow \infty}\frac{t^2F(x, t)}{e^{\alpha_0 t^2}} > \frac{2}{\alpha_0^2\rho^2}$ uniformly on $x\in\mathbb{R}^2$, where $\rho\in(0, 1/2)$ satisfying $\rho^2\max_{|x|\leq\rho}V(x)\leq1$;

(CF4) $\frac{f(x, t)-V(x)t}{|t|^{3}}$ is non-decreasing on $t\in\mathbb{R}\backslash\{0\}$.

Recently, Cao et al. ^[25] considered the equations

where $\sigma\geq2p, \; b\geq0$ and $V\in C(\mathbb{R}^2, (0, \infty))$ are $\mathbb{Z}^2$ periodic. With a similar method in ^[22], they obtained the existence of a positive ground state solution of Eq (1.7) in $X_p$, where

Here, we will prove the existence of a nontrivial solution to Eq (1.1), not only for all $p \geq 2$, but also for general nonlinearities $f$ and potentials $V$.

To describe our main results, we introduce the following notations: Let us view $\mathbb{R}^2$ as $\mathbb{C}$, let $k \in \mathbb{N}, k\geq 2$ and we say that $v \in {\mathcal P}_k$ if $v(ze^{2\pi i/k}) = v(z)$ over $\mathbb{C}$. We define

We say that $v$ is mirror symmetric denoted by $v \in \mathcal{M}$ if $v(\overline z) = v(z)$ in ${\mathbb C}$. Let

Finally, we write the associated functional of Eq (1.1) in the following form

and the associated Nehari manifold of the functional (1.8) is

Our main result is stated as follows.

Theorem 1.1. Let $p\geq2$, $V$ and $f$ satisfy $\rm{(V0), \rm(F1)}$ and the following conditions

$\text{(VF)} \; V\in\mathcal{V}_{k, 1}\; \mathit{\mbox{and}}\; f\in\mathcal{F}_{k, 1}$ with $k\geq4$ or $V\in\mathcal{V}_{k, 2}\; \mathit{\mbox{and}}\; f\in\mathcal{F}_{k, 2}$ with $k\geq2$;

$\text{(F2)} \; f(x, t)t \ge 0$ for all $(x, t)\in\mathbb{R}^2\times\mathbb{R}\backslash\{0\}$, and there exists $M_0 > 0$ and $t_0 > 0$ such that $F(x, t)\leq M_0|f(x, t)|$ for $x\in\mathbb{R}^2$, $|t|\geq t_0;$

$\text{(F3)}$ There exists $q\in\mathbb{R}$ such that $\liminf_{|t|\rightarrow \infty}\frac{|t|^qF(x, t)}{e^{\alpha_0 t^2}} = +\infty;$

$\text{(F4)} \; g_p(x, t)$ is non-decreasing on $t\in(-\infty, 0)$ and $t\in(0, \infty)$, where

for some $\mu < 1$.

$\rm{(F5)}$ If $p = 2$, $f(x, t) = o(t)$ as $t\rightarrow0$ uniformly on $\mathbb{R}^2;$ and if $p > 2$, $f(x, t) = O(t^{s_0})$ with $s_0 > 1$ as $t\rightarrow0$ uniformly on $\mathbb{R}^2.$

Then, Eq (1.1) has a nontrivial solution $\bar{u}$. Moreover, if $V\in\mathcal{V}_{k, 1}$ and $f\in\mathcal{F}_{k, 1}$, then $\bar{u}\in E_{k, p}$ satisfies

if $V\in\mathcal{V}_{k, 2}$ and $f\in\mathcal{F}_{k, 2}$, then $\bar{u}\in T_{k, p}$ satisfies

Remark 1.2. Comparing to [23, Theorem 4], we have weakened the assumptions ${\rm(CF1)–(CF3)}$ to ${\rm(VF)}$ and ${\rm(F2)–(F3)}$, respectively. More precisely,

● ${\rm(CF1)}$ means $V\in\mathcal{V}_{2, 2}$ and $f\in\mathcal{F}_{2, 2}$, hence, it is a special case of ${\rm(VF)}$;

● The condition ${\rm(F3)}$ is less restrictive than ${\rm(CF3)}$ for the behavior of $f$ at infinity;

● ${\rm(F2)}$ improves slightly ${\rm(CF2)}$ where $f(x, t)t > 0$ is replaced by $f(x, t)t\geq0$;

Here is an example of $f$, which satisfies ${\rm(VF)}$ and ${\rm(F1)–(F5)}$, but not ${\rm(CF3)}$. Let $\theta > 0, p_0\geq p, q_0 > 2$ and

with odd extension to $t < 0$. Finally, it seems that ^[23] used implicitly $f(t) = o(t)$ as $t\rightarrow0$ with $p = 2$ in ${\rm(F5)}$ (see the proof of Lemma 2.6 there).

Our approach works also for the subcritical case.

Theorem 1.3. Let $p\geq2$, $V$ and $f$ satisfy (V0), (VF), (F1'), (F5) and the following condition:

(F4') $f(x, t)t > 0$ for all $(x, t)\in\mathbb{R}^2\times(\mathbb{R}\backslash\{0\})$ and there exists $\nu\in(2, \infty), t_1\in(0, \infty)$ such that

Furthermore, if $p > 2$, we assume that

where

Then, Eq (1.1) has a nontrivial solution $\bar{u}$. Moreover, $\bar{u}\in E_{k, p}$ if $V\in\mathcal{V}_{k, 1}$ and $f\in\mathcal{F}_{k, 1}$, and $\bar{u}\in T_{k, p}$ if $V\in\mathcal{V}_{k, 2}$ and $f\in\mathcal{F}_{k, 2}$.

This paper is organized as follows: In Section 2, we present some basic results; in particular we show that the energy functional corresponding to the nonlocal term is non positive, which is our key observation and different from the available results, see Lemma 2.3. In Section 3, we prove a mountain pass type theorem using a new test function, see Lemma 3.2 below. In Sections 4 and 5, we give the proof of Theorems 1.1 and 1.3, respectively.

2.   Preliminaries

In this section, we will give some preliminary definitions and basic facts about inequalities, such as the Moser-Trudinger inequality, the energy estimate of the nonlocal term. In the following, the letter $C$ denotes generic positive constants and $\|\cdot\|_q$ denotes the standard norm in $L^q({\mathbb{R}^2})$.

The function space $X_p$ is a Banach space equipped with the norm

where

while

is induced by the scalar product

We will use the following bilinear functionals (see ^[21]):

By the Hardy-Littlewood-Sobolev inequality (see ^[26]), there exists $C > 0$ such that for any $u, v\in L^{4/3}(\mathbb{R}^2)$,

Corresponding to (2.5)–(2.7), we define

The following bound for $I_2(u)$ is a direct consequence of (2.8):

We can rewrite the associated functional of Eq (1.1) in the following form

Next, we state several lemmas.

Lemma 2.1. ${\rm(i)}$ Let $u\in H^{1}(\mathbb{R}^2)$, then for any $\alpha > 0$,

$\rm(ii)$ Given $M > 0$, $\alpha\in(0, 4\pi)$, there exists a constant $C(M, \alpha)$ such that for all $u\in H^{1}(\mathbb{R}^2)$ satisfying $\|\nabla u\|_{2}\leq 1$, $\|u\|_2\leq M$, there holds

The statements (i) and (ii) of the above lemma were first established by [27, Lemma 1] and [1, Lemma 2.1], respectively (see also ^[28,29]).

Lemma 2.2. Assume that $V$ and $f$ satisfy ${\rm(V0), (F1)}{\rm(}$or ${\rm(F1')}{\rm)}, {\rm(F5)}$. Then, $I_i, \Phi\in C^1(X_p, \mathbb{R})$ and

For the sake of completeness, we present a proof of Lemma 2.2 in the appendix. The following lemma is our first key observation.

Lemma 2.3. For any $u\in X_p$, we have $I_{0}(u)\leq0.$

Proof. First, let $u\in C_0^\infty(\mathbb{R}^2)$ with ${\rm supp}(u) \subset B_{\frac{1}{2}}(0)$. Then

Consider now $u\in{C}_{0}^{\infty}(\mathbb{R}^2, \mathbb{R})$. Take $R > 0$ such that $\mathrm{supp}(u)\subset B_R(0)$ and let $w(x) = u(2Rx)$, so ${\rm supp}(w) \subset B_{\frac{1}{2}}(0)$ and $\phi_w(x): = \frac{1}{4R^2}\phi_u(2Rx).$ Hence,

We conclude by the density argument. For any $R > 0$, let $\varphi_R(r)$ be a $C_0^\infty$ cut-off function such that $0\leq\varphi\leq1$, $\varphi_R\equiv1$ on $[0, R]$ and $\varphi_R\equiv0$ on $[R+1, \infty)$. Let $\eta$ be the standard mollifier and $\eta_{\delta}(x): = \frac{1}{\delta^2}\eta(\frac{x}{\delta})$, where $\delta > 0$. Given $\epsilon > 0$, since $\sqrt{V}u\in L^2(\mathbb{R}^2), \; [\ln(1+|\cdot|)]^{1/p}u\in L^p(\mathbb{R}^2)$, [30, Pages 264 and 714], we can choose $\delta$ small enough such that

Therefore, for any $u \in X_p$, there exists $\{u_n\}_n\subset C_0^\infty(\mathbb{R}^2)$ such that $\lim_{n\rightarrow \infty}\|u_n-u\|_{X_p} = 0.$ By the fact $I_0 = I_1-I_2$ and Lemma 2.2, we conclude that $I_{0}(u) \leq0.$

Corollary 2.4. Assume that $V$ and $f$ satisfy ${\rm(V0)}$ and ${\rm(F1)}{\rm(}$or ${\rm(F4')}{\rm)}$. Then

Proof. For any $\omega\in X_p\backslash\{0\}$, there exists $\delta > 0$ such that $m\{|\omega(x)|\geq\delta\} > 0$. For a critical case, by Lemma 2.3 and (F1), one has

For a subcritical case, we choose large enough $R > 0$ such that $m(G) > 0$, where

By (F4') and choosing $M: = \frac{\|\omega\|^2}{m(G)} > 0$, there exists $t_M > 0$ such that

which together with Lemma 2.3 implies

as $t\rightarrow \infty$, and we complete the proof.

The following lemma is inspired by [31, Lemma 2.2].

Lemma 2.5. Assume that $V$ and $f$ satisfy ${\rm(V0)}$ and ${\rm(VF)}$. Then there exists $C_k > 0$ such that

In particular, since $T_{k, p} \subset E_{k, p}$, (2.13) holds for $u, v\in T_{k, p}$.

Proof. Let $\Omega_1 : = \{(x_1, x_2)\in\mathbb{R}^2:x_i\geq0\}$, $\Omega_2 = -\Omega_1$. For any $x\in \Omega_1$ and $y\in\Omega_2$, one has

Then, it follows from the definition of $E_{k, p}$ and $k\geq4$ that

so we obtain (2.13).

3.   Variational framework

In this section, we will quote a version of Mountain Pass Theorem and prepare the proof of Theorems 1.1 and 1.3.

Lemma 3.1. Let $Y$ be a real Banach space and $I\in C^1(Y, \mathbb{R})$. Let $S$ be a closed subset of $Y$, which disconnects $Y$ into distinct connected $Y_1$ and $Y_2$. Suppose further that $I(0) = 0$ and

(i) $0\in Y_1$, and there exists $\alpha > 0$ such that $I|_{S}\geq\alpha,$

(ii) There is $e\in Y_2$ such that $I(e)\leq0.$

Then, $I$ possesses a Cerami sequence with $c\geq\alpha > 0$ given by

where

and a Cerami sequence means a sequence $\{u_n\}\subset X$ such that

The proof of the above lemma can be found in [32, Theorem 3]. We state another result that serves as a bridge between the mountain pass structure (see Lemma 3.3) and Theorem 1.1.

Lemma 3.2. Assume that $V$ and $f$ satisfy ${\rm(V0)}$, ${\rm(F1)}$ and ${\rm(F3)–(F5)}$. Then there exists $n_0\in\mathbb{N}$ such that

where

Proof. Without loss of generality, we can fix $q\geq2$. Direct computation yields

where $\delta_n = O(\frac{1}{\ln{n}})$ as $n \to \infty$. By (F3), there exists $t_0 > 0$ such that

There are three cases for the value of $t$.

Case (i): $0 \leq t \le \sqrt{\frac{3\pi}{\alpha_0}}$. For large $n$, then it follows from (3.2) and Lemma 2.3 that

Case (ii): $\sqrt{\frac{3\pi}{\alpha_0}} \le t\le \sqrt{\frac{8\pi}{\alpha_0}}$. For large $n$, we have $t\omega_n(x)\geq t_0$ for $x\in B_{(\ln{n})^{q/2}/n}$. Then it follows from (3.2), (3.3) and Lemma 2.3 that

where

Let $t_n > 0$ be the unique maximum of $\varphi_n$ in $\mathbb{R}_+$, then (as $n\to\infty$)

and

Combining (3.5)–(3.7), one has

Case (iii): $t\ge \sqrt{\frac{8\pi}{\alpha_0}}$. As in the above case (ii), we have

for large $n$. To get the third inequality, we used the fact that the function

is decreasing on $t\ge \sqrt{\frac{8\pi}{\alpha_0}}$ when $n$ is large enough. Combining the conclusions for cases (i)–(iii), the proof is completed.

Now we show the existence of the Cerami sequence.

Lemma 3.3. Assume that $V$ and $f$ satisfy ${\rm(V0)}$, ${\rm(VF)}$, ${\rm(F1)}{\rm(}$or ${\rm(F1')}$ and ${\rm(F4')}{\rm)}$ and ${\rm(F5)}$. Then there exists a constant $\tilde{c}\in(0, \sup_{t\geq0}\Phi(t\omega_{n_0})]$ and a Cerami sequence $\{u_n\}\subset E_{k, p}$ such that

Proof. Applying the Sobolev embedding theorem for given $s\in[2, \infty)$, there exists $\gamma_s > 0$ such that

By (F1) (or (F1')) and (F5) for any $\epsilon > 0$, there exists some constant $C_\epsilon > 0$ such that

On the other hand, in view of Lemma 2.1, one has

Let $\epsilon = \frac{1}{4\gamma_2^{2}}$ from (3.11)–(3.13), and there holds

Hence, it follows from (2.11) and (3.14) that if $\|u\|\leq\sqrt{\frac{\pi}{\alpha_0}}$,

Therefore, there exists $\kappa_0 > 0$ and $0 < \rho < \sqrt{\frac{\pi}{\alpha_0}}$ such that

By (V0), (F1) (or (F4')) and Corollary 2.4, we have $\lim_{t\rightarrow \infty}\Phi(t\omega_{n_0}) = -\infty$, and then we can choose $t^* > 0$ such that $e = t^*\omega_{n_0}\in Y_2: = \{u\in E_{k, p}:\|u\| > \rho\}$ and $\Phi(e) < 0$. Let $Y_1: = \{u\in E_{k, p}:\|u\|\leq\rho\}$, then in view of Lemma 3.1, one deduces that there exists $\tilde{c}\in[\kappa_0, \sup_{t\geq0}\Phi(t\omega_{n_0})]$ and a Cerami sequence $\{u_n\}\subset E_{k, p}$ satisfying (3.10).

4.   Proof of Theorem 1.1

The proof of Theorem 1.1 is based on the following lemmas. As in Lemma 2.5, we only consider the $E_{k, p}$ case.

Lemma 4.1. Assume that $V$ and $f$ satisfy ${\rm(V0), (VF), (F1), (F4)}$ and ${\rm(F5)}$, then we have

${\rm(i)}$ Let $m_1: = \inf_{\mathcal{N}_1}\Phi(u)$, then there exists a constant $c_\ast\in(0, m_1]$ and a sequence $\{u_n\}\subset E_{k, p}$ satisfying

${\rm(ii)}$ For any $u\in E_{k, p}\backslash\{0\}$, there exists a unique $t_u > 0$ such that $t_uu\in\mathcal{N}_1$. Moreover, we have

Proof. We will prove that

and then we can get the statement (i). Indeed, if (4.2) holds the same as [33, Lemma 3.2], we can choose $v_k\in\mathcal{N}_1$ such that

For any $v_k$, similarly to Lemma 3.3, we can obtain a Cerami sequence $\{u_{k, n}\}_n\subset E_{k, p}$ such that

with $c_k\in(0, \sup_{t\geq0}\Phi(tv_k)]$. By (4.2) and the diagonal rule, we can verify (4.1), and now we prove (4.2). By (2.11) and (2.12), one has

According to the fact $u\in \mathcal{N}_1$ and $\min_{t\geq0}(t^{2p}-pt^2+p-1)$ attained at $t = 1$, then (4.2) holds.

Next, we consider statement (ii). Let $u\in E_{k, p}\backslash\{0\}$ be fixed and $\zeta(t): = \Phi(tu)$ on $[0, \infty)$. By the definition (2.11),

Using (3.15), (F1) and Lemma 2.3, one has $\zeta(0) = 0$, $\zeta(t) > 0$ for $t > 0$ small and $\zeta(t) < 0$ for $t$ large. Therefore $\max_{t\in(0, \infty)}\zeta(t)$ is achieved at some $t_u > 0$ so that $\zeta^{\prime}(t_u) = 0$ and $t_uu\in\mathcal{N}_1$. Now, we claim that $t_u$ is unique. In fact, for any given $u\in E_{k, p}\backslash\{0\}$, let $t_1, t_2 > 0$ such that $\zeta^{\prime}(t_1) = \zeta^{\prime}(t_2) = 0$. By (4.3), taking $t = \frac{t_2}{t_1}$ and $t = \frac{t_1}{t_2}$ respectively, it implies

where $g(t): = t^{2p}-pt^2+p-1$. Therefore, we must have $t_1 = t_2$, since $g(s) > 0$ for any $s > 0$, $s\ne 1$.

Lemma 4.2. Assume that $V$ and $f$ satisfy ${\rm(V0), (VF), (F1), (F4)}$ and ${\rm(F5)}$. Then any sequence satisfying (4.1) is bounded w.r.t. $\|\cdot\|$.

Proof. We only consider the case $p > 2$. The case $p = 2$ is obtained by [23, Lemma 2.11]. First, we prove that

Indeed, by (F4), there holds

By (4.4), one has

Here, we also used (2.11), (2.12) and (4.1). Therefore, we complete the proof.

Proof of Theorem 1.1 completed. Applying Lemmas 4.1 and 4.2, we deduce that there exists a sequence $\{u_n\}\subset E_{k, p}$ satisfying (4.1) and $\|u_n\|\leq C < \infty$. Now, we prove

Indeed, let $p \ge 2$, and by (2.11), (2.12) and (4.1) there holds

hence (4.6) holds true. Next, we complete the proof of Theorem 1.1 in three steps.

Step 1: $\{u_n\}$ is bounded in $E_{k, p}$.

We first prove that $\delta_0: = \limsup_{n\rightarrow \infty}\|u_n\|_p > 0$. Suppose the contrary $\delta_0 = 0$, then from the Gagliardo-Nirenberg inequality (see [34, Page 125]):

where $2\leq p < t < \infty, \theta = \frac{p}{t}$. Hence, $u_n\rightarrow0$ in $L^{\eta}(\mathbb{R}^2)$ for $\eta\in(2, +\infty)$. Given any $\varepsilon\in(0, M_0C_{10}/t_2)$, we choose $M_\varepsilon > M_0C_{10}/\varepsilon$, then it follows from (F2) and (4.6) that

Applying (F5), one has

and

By the arbitrariness of $\varepsilon > 0$, we deduce from (F2), (4.8) and (4.9) that

Hence, by (2.10) we have

By Lemmas 3.2 and 3.3, we know that $\bar{\varepsilon}: = \frac{1}{3}(1-\frac{\alpha_0\tilde{c}}{2\pi}) > 0$, which together with (2.11), (3.10), (4.11), (4.12) and the fact $I_1(u_n)\geq0$ implies

Now, let $d\in(1, \frac{p}{p-1})$ satisfy

By (F1), there exists $C > 0$ such that

It follows from (4.13)–(4.15) and Lemma 2.1 that

As $d^{\prime} = \frac{d}{d-1} > p$, using (4.16) there holds

Combining (2.10)–(2.12), (3.10), (4.10) and (4.17), we arrive at

This contradiction shows that $\delta_0 > 0$. Now, from (2.10), (4.5) and Lemma 2.3, one has

which, together with Lemma 2.5, implies that $\|u_n\|_{\ast}$ is bounded and $\{u_n\}$ is bounded in $E_{k, p}$.

Step 2: $\Phi'(\bar{u}) = 0$ in $E_{k, p}^{\prime}$ and $\Phi(\bar{u}) = m_1$.

We can assume by [25, Lemma 2.3] and passing to a subsequence again if necessary, that $u_n\rightharpoonup\bar{u}$ in $E_{k, p}$, $u_n\rightarrow \bar{u}$ a.e. on $\mathbb{R}^2$ and

where $s\in[2, \infty)$ if $p = 2$ and $s\in(2, \infty)$ if $p > 2$. First, we need prove that

Since (4.6) and the condition (F1), (F2) and (F5) hold the same as [23, Assertions 2 and 3], (4.19) still holds. Next, we prove that

Indeed, noting that $u_n\rightarrow \bar{u}$ in $L^{\frac{4p}{3}}(\mathbb{R}^2)$ by [35, Lemma A.1], there exists $w_0\in L^{\frac{4p}{3}}(\mathbb{R}^2)$ such that

a.e., for a subsequence if necessary, which together with the Lebesgue dominated convergence theorem and Hardy-Littlewood-Sobolev inequality implies

as $n\rightarrow \infty$ and (4.20) is proved. Now, we claim that

Indeed, similar as (4.20), we also have

By [25, Lemma 3.3], one has

which together with (3.10), (4.19) and Fatou's Lemma implies

Hence, we can obtain

Since $\bar{u}\neq0$, by Lemma 4.1 there exists $\bar{t}\in(0, 1]$ such that $\bar{t}\bar{u}\in\mathcal{N}_1$. By (3.10), (4.4) and (4.25), the weak lower semi-continuity of norm, Lemma 4.1, the condition (F4) and Fatou's Lemma, we have

which implies (4.22) and

By (4.19), (4.20), (4.22) and (4.27) and the weak lower semi-continuity of norm and Fatou's Lemma, one has

which implies

Hereafter, we claim that

where $|v_n|^p: = |u_n|^{p-2}|u_n-\bar{u}|^2$. Indeed, we have

Then, we estimate

For any $\epsilon > 0$, there exists $R_\epsilon > 0$ such that

Now, we split

where

for large enough $n$ and

which together with (4.31) implies

Hence, by Fatou's Lemma, we have

Since $I_1(u_n)\rightarrow I_1(\bar{u})$ and $A_1(|u_n|^p, |v_n|^{p})\geq0$, we conclude with (4.30). Finally, by [25, Lemma 3.2] we obtain $\|u_n-\bar{u}\|_\ast\rightarrow0$, which together with (4.27) and (4.29) implies

Step 3: $\Phi'(\bar{u}) = 0$ in $X_{p}^{\prime}$.

By using the group action on the space $X_p$, we will conclude $\Phi^{\prime}(\bar{u}) = 0$. Let $G\subset\mathcal{O}(2)$ be a finite group of transforms acting on $X_p$, where $\mathcal{O}(2)$ denotes the group of orthogonal transformations in $\mathbb{R}^2$. The action of $G$ on the space $X_p$ is a continuous map (see [35, Definition 1.27]):

Assume that $\varphi\in C^1(X_p, \mathbb{R})$ is invariant by $G$; that is, $\varphi(w\circ\tau) = \varphi(w)$ for any $\tau \in G$, $w \in X_p$. Let $u$ be a critical point of $\varphi$ in $X_{p, G}$, where

Then $\varphi^{\prime}(u) = 0$ in $X_p^\prime$. In fact, given any $v\in X_p$, we define

where $\#(G)$ denotes the cardinal of $G$. For any $\tau_0\in G$, since $G\subset\mathcal{O}(2)$, we have

which implies $\bar{v}\in X_{p, G}$. Therefore, one has

For the second line, we used the fact $u \in X_{p, G}$, so we have $\varphi^{\prime}(u) = 0$ in $X_p^\prime$.

The two cases in Theorem 1.1 are direct consequences of the above discussion. Indeed, let $G_1$ be the subgroup of $\mathcal{O}(2)$ generated by $z\mapsto ze^{2\pi i/k}$, then $X_{p, G_1} = E_{k, p}$. If $G_2$ is generated by $z\mapsto ze^{2\pi i/k}$ and $z\mapsto \bar z$, $X_{p, G_2} = T_{k, p}$, so $\Phi'(\bar{u}) = 0$ in $X_p'$.

5.   Proof of Theorem 1.3

We prove Theorem 1.3 in three steps for subcritical case and the Ambrosetti-Rabinowitz condition (F4'). As in the proof of Theorem 1.1, we only consider the function space $E_{k, p}$.

Step 1: $\|u_n\|$ is bounded.

We only consider the case $p > 2$. Same as critical case, the case $p = 2$ can be obtained by [23, Lemma 2.11]. Applying Lemma 3.3 and (5.1), there exists a sequence $\{u_n\}\subset E_{k, p}$ satisfying (3.10). By (3.10) and (F4'), we can choose a constant $\lambda_0\in\left(\frac{1}{\nu}, \frac{1}{2}-\frac{M_{t_1}}{\gamma^2}\right)$ and then we have

and then $\|u_n\|$ is bounded.

Step 2: $\{u_n\}$ is bounded in $E_{k, p}$.

As in the proof of the critical case, we first prove $\delta_0: = \limsup_{n\rightarrow \infty}\|u_n\|_p > 0$. Suppose the contrary $\delta_0 = 0$. Denoting $M_\ast: = \sup_{n}\|u_n\|$ and $M_{\ast\ast}: = \sup_{n}\|u_n\|_2$. By (F1') and (F5), choosing $\alpha\in(0, \frac{p-1}{pM_\ast^2})$, one has

By (5.2) and Lemma 2.1, we have

Hence, by (5.3) and Lemma 3.3, we know

This contradiction shows that $\delta_0 > 0$. Now, from (2.10), (5.1) and Lemma 2.3, one has

which, together with Lemma 2.5, implies that $\|u_n\|_{\ast}$ is bounded and $\{u_n\}$ is bounded in $E_{k, p}$.

Step 3: $\Phi'(\bar{u}) = 0$ in $X_p^{\prime}$.

We may assume by [25, Lemma 2.3] and passing to a subsequence again if necessary, that $u_n\rightharpoonup\bar{u}$ in $E_{k, p}$, $u_n\rightarrow \bar{u}$, a.e., on $\mathbb{R}^2$ and

where $s\in[2, \infty)$ if $p = 2$ and $s\in(2, \infty)$ if $p > 2$. Let $M: = \sup_{n}\|\nabla u_n\|_2$. By (F1'), we can choose $\alpha > 0$ small enough such that $M^2 < \frac{4\pi}{\alpha}$. Therefore, there is $\beta > p$ big enough such that $\frac{M^2\beta}{\beta-1} < \frac{4\pi}{\alpha}$. Without loss of generality, we may assume that $s_0\in(1, 2)$ in (F5). Then, it follows (F5) and Lemma 2.1 that

Similarly, one has

Furthermore, it follows from (2.9), (2.10) and the Hölder inequality that

By [25, Lemma 3.3], we have

Combining (2.11), (2.12), (3.10), and (5.5)–(5.8), there holds

By (2.10), (5.9) and Lemma 2.1, we have

where $|v_n|^p: = |u_n|^{p-2}|u_n-\bar{u}|^2$ for every $n\in\mathbb{N}$. By (3.10) and (5.10), one has

which, together with Lemma 2.5, implies

From (5.11) and [25, Lemma 3.3], one has

where we used the following inequality

and $C$ is independent of $a, b\in\mathbb{R}$. Combining with (5.9), we have $u_n\rightarrow \bar{u}$ in $E_{k, p}$. Hence, $0 < \tilde{c} = \lim_{n\rightarrow \infty}\Phi(u_n) = \Phi(\bar{u})$ and $\Phi^{\prime}(\bar{u}) = 0$ in $E_{k, p}^\prime$. We conclude that $\Phi'(\bar{u}) = 0$ in $X_{p}^{\prime}$, as in the proof of step 3 of Theorem 1.1.

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 is partially supported by National Natural Science Foundation of China, No. 12071189. Thanks to the reviewers for their valuable suggestions and comments. The author would like to express the sincere gratitude to his tutors, professors Dong Ye and Feng Zhou, for their valuable guidance and insights that helped to improve the paper.

Conflict of interest

The authors declare there are no conflicts of interest.

Appendix

In this section, we give the proof of Lemma 2.2. For any $u\in X_p$, we denote $\Psi(u): = \int_{\mathbb{R}^2}F(x, u)dx$. In fact, we just need to prove $\Psi\in C^{1}(X_p, \mathbb{R})$, and the readers can refer to [25, Lemma 2.3] for the rest. First, given any $u, v\in X_p$, for almost every $x\in\mathbb{R}^2$

On the other hand, we can choose a large enough number $t_1 > 0$ such that

By (F1), (F5) and Lemma 2.1, one has, for any $u\in X_p$,

Then, for any $u\in X_p$, $f(x, u)\in L^2(\mathbb{R}^2)$, it implies that the Gateaux derivative $\Psi_g^{\prime}(u)$ exists and $\Psi_g^{\prime}(u)\in X_p^{\prime}$.

Now let $\{u_n\}\subset X_p$, $\|u_n-\bar{u}\|_{X_p}\rightarrow0$. Hence, $u_n\rightarrow \bar{u}$ in $H^1(\mathbb{R}^2)$. Let us prove $\Psi'(\bar u) = \lim_{n\to \infty}\Psi'(u_n)$. It suffices to prove

Define that $M: = \sup_{n}\|\nabla u_n\|_2$, then we prove this lemma in two cases.

Special case: $M\leq\sqrt{\frac{\pi}{2\alpha_0}}$.

For any given $\varepsilon\in(0, 1)$, we choose large enough $R_\varepsilon > 0$ such that

and

By Lemma 2.1 and $\alpha_0\le \frac{\pi}{2M^2}$, it is easy to verify that there is a $C_0 > 0$ such that $\|u\|_4\leq C_0\|u\|_{X_p}$ for all $u\in X_p$ and

Now, we claim that

The proof of (A1) is in spirit of [36, Lemma 2.1]. As $L^2(B_{R_\varepsilon})$ is a Hilbert space, we need only to prove

for the first part of (A1). Let $M^{\prime}$ be large enough such that

By the dominated convergence theorem and Fatou's Lemma, one has

where $h_n(x): = \left||f(x, u_n(x))|^2\chi_{\{|u_n| < M^{\prime}\}\cap B_{R_\varepsilon}}-|f(x, \bar{u}(x))|^2\chi_{\{|u_n| < M^{\prime}\}\cap B_{R_\varepsilon}}\right|$, and we use the fact

Therefore, we get (A1). By (A1), for large $n$, one has

where $\frac{1}{p}+\frac{1}{p'} = 1$ and $\gamma: = \inf_{u\in X}{\frac{\|u\|}{\|u\|_{H^1(\mathbb{R}^2)}}} > 0$ (see [31, Lemma 2.1]).

General case: $M > 0$.

For any $R > 0$, let $\varphi_R(r)$ be a $C_0^\infty$ cut-off function such that $0\leq\varphi\leq1$, $\varphi_R\equiv1$ on $[0, R]$ and $\varphi_R\equiv0$ on $[R+1, \infty)$. Let $\delta > 0$ (to be determined later), we can choose large enough bounded domain $B_{R}(0)$ and its bounded open coverage $\{\Omega_\ell\}_{\ell \le N_c}$ which has a partition of unity $w_\ell \; (1 \le \ell \le N_c)$ such that\\

where

Choosing $\delta > 0$ small enough and repeating now the proof of the special case, we can prove

and

Therefore, one has

So we obtain Lemma 2.2.