Existence and regularity results for critical $(p, 2)$-Laplacian equation

1.   Introduction and main results

Let us consider the following critical $(p, 2)$-Laplacian equation

where $N\geqslant3$, $1 < p < N$, $0 < \alpha < N$, and $\Delta_{p}$ is the $p$-Laplacian with $\Delta_{p} = \nabla(|\nabla u|^{p-2}\nabla u)$. $I_{\alpha}$ is the Riesz potential defined by

Equation (1.1) is closely related to the following nonlocal quasilinear equation:

where $1 < p, q < N$ and $\mu:\mathbb{R}^N\mapsto[0, \infty)$ is supposed to be Lipschitz continuous. The operator involved in (1.2) is the so-called double phase operator whose behavior switches between two different elliptic situations. The pioneering work to treat such operators comes from Zhikov ^[36,37], who introduced such classes to provide models of strongly anisotropic materials. For more details and recent works about double phase problems, we refer to ^[15,22].

When $N = 3$, $p = q = 2$, $\mu = 1$, and $F(u) = u^{2}$, then Eq (1.1) is the well-known Choquard equation:

Equation (1.3) appears in several physical models like the quantum theory of polarons ^[29], Hartree-Fock theory ^[18], and self-gravitating matter ^[30]. After the pioneer work of Lieb ^[18] and Lions ^[20], the existence of weak solutions for Choquard equations have been a fascinating topic in past decades. For more related work, we refer to ^[2,31] for the subcritical case, ^[8,13] for the upper critical case, ^[9,26] for the lower critical case, and ^[21,32] for the double critical case.

When $q = 2\not = p$ and $\mu = 1$, Eq (1.2) reduces to (1.1). It appears in many different disciplines of physics and has a wide range of applications, such as chemical reaction design ^[5], quantum field theory ^[4], biophysics ^[10], and plasma physics ^[33]. From a mathematical point of view, the main difficulty in (1.1) is the non-homogeneity of the operator $-\Delta_p-\Delta$. For this reason, equations involving such operator or its variant have been received increasing attention from various authors. In particular, Gasiński-Papageorgiou ^[14] considered Eq (1.1) when the nonlinearity takes the following form:

where $p > 2$ and $\Omega\in\mathbb{R}^N$ is a bounded $C^2$ domain. Under the assumption that $f(x, u)$ exhibits asymmetric behaviour as $u\to\pm\infty$, more precisely $f(x, u)$ is superlinear in the positive direction (without satisfying the Ambrosetti-Rabinowitz condition) and sublinear resonant in the negative direction, the authors obtained the existence and multiplicity results of (1.4) via variational tools and Morse theory methods. Later, Papageorgiou-Rădulescu-Repovš ^[28] imposed certain assumptions on $f(x, u)$ to make it double resonant at both $\pm\infty$ and $0$. By virtue of variational tools and critical groups, the authors obtained the existence and multiplicity results of (1.4).

In ^[27], the authors considered the following Dirichlet problem:

where $p > 2$, $\lambda > 0$, $\Omega\subset\mathbb{R}^N$ with a $C^2$ boundary, and $f(x, u)$ is a Carathéodory function. Based on critical point theory, together with suitable truncation and comparison techniques, Papageorgiou-Rădulescu-Repovš ^[27] obtained the existence and multiplicity results of (1.5) when $\lambda$ is near the principal eigenvalue $\lambda_1(p) > 0$ of $(-\Delta_{p}, W_0^{1, p}(\Omega))$. Subsequently, their work was extended by Bhattacharya-Emamizadeh-Farjudian ^[6] to the case of $1 < p < 2$. By applying the fibering method and spectrum analysis, a priori bounds and regularity results of (1.5) were investigated. Moameni-Wong ^[24] studied the case of $f(x, u)$ in (1.5) satisfying supercritical growth. By using a variational principle on convex subsets of a Banach space, the authors proved the existence of at least one nontrivial solution of (1.5). Equation (1.5) with Neumann boundary condition ($\frac{\partial u}{\partial\nu} = 0$) has been considered recently in Mihăilescu ^[23]. The authors showed that the eigenvalue set of this problem consists of $0$ and an unbounded open interval from the first eigenvalue of $-\Delta_{p}-\Delta$ $(p > 2)$ to infinity. After that, Fărcăşeanu et al. ^[12] extended the results in ^[23] to $0 < p < 2$ by means of the determination of a critical point on the Nehari manifold ^[3]. For more results related to the $(p, 2)$-Laplacian equation, one can refer to ^[1,16]

Recently, Moroz-Van Schaftingen ^[25] established the $W_{loc}^{2, q}(\mathbb{R}^{N})$ regularity $(q > 1)$ and Pohožaev identity of weak solutions for the following generalized Choquard equation:

where $N\geqslant3$, $\alpha\in(0, N)$, and $F$ satisfies the subcritical Berestycki-Lions type condition, namely:

$(H_{1})$ There exists $t_{0}\in\mathbb{R}\backslash\{0\}$ such that $F(t_{0})\not = 0$, where $F:t\in\mathbb{R}\to\int_{0}^{t}f(\zeta)\mathrm{d}\zeta$.

$(H_{2})$ There exists $C > 0$ such that for every $t\in\mathbb{R}$, $|tf(t)|\leqslant C(|t|^{\frac{N+\alpha}{N}}+|t|^{\frac{N+\alpha}{N-2}})$.

$(H_{3})$

Li-Ma ^[17] studied Eq (1.6) with a perturbation. By virtue of the subcritical approximation and the Pohožaev constraint method, they obtained the regularity and Pohožaev identity of weak solutions. Cassani-Du-Liu ^[7] studied Eq (1.6) with $N = 2$ and $I_\alpha = \ln\frac{1}{|x|}$. By using an asymptotic approximation approach, the existence of positive solutions of (1.6) is obtained.

Up to our knowledge, no results have been reported regarding the existence and regularity of weak solutions for the $(p, 2)$-Laplacian equation with critical Hartree-type nonlinearity. Inspired by the above cited results, the main objective of this paper is to fill this gap. The novelty of this paper lies in two aspects. On one hand, due to the existence of the $(p, 2)$-Laplacian operator, problem (1.1) becomes non-homogeneous. Therefore, the method used in ^[25] is invalid. To overcome this difficulty, we introduce some new ideas and establish new estimates to improve the integrability of weak solutions of Eq (1.1). On the other hand, we are the first to consider a class of $(p, 2)$-Laplacian equation with critical Hartree-type nonlinearity.

Before we present our results, we suppose that $f$ satisfies the following conditions:

$(F_{1})$ There exists $C > 0$ such that for every $t\in\mathbb{R}$, $|tf(t)|\leqslant C(|t|^{2_{\alpha}^{\sharp}}+|t|^{2_{\alpha}^{*}})$, where $2_{\alpha}^{\sharp} = \frac{N+\alpha}{N}$ and $2_{\alpha}^{*} = \frac{N+\alpha}{N-2}$.

$(F_{2})$ $F(u) = \frac{1}{2_{\alpha}^{\sharp}} |u|^{2_{\alpha}^{\sharp}} + \frac{\lambda}{2_{\alpha}^{*}} |u|^{2_{\alpha}^{*}}$.

Now we can formulate our main results in this paper.

Theorem 1.1. Let $N\geqslant 3$, $1 < p < N$, $0 < \alpha < N$, and condition $(F_{1})$ holds. If $u$ is a nontrivial solution of Eq (1.1), then

(i) $u\in L^{q}(\mathbb{R}^{N})$ for any $q\in[2, \infty]$;

(ii) the following Pohožaev identity holds:

Theorem 1.2. Let $N\geqslant 3$, $1 < p < N$, $0 < \alpha < N$, and condition $(F_{2})$ hold. Then, there exists $\Lambda > 0$ such that for any $\lambda\in(0, \Lambda)$, Eq (1.1) possesses a nonnegative radially symmetric ground state solution, where

At the end of this section, we outline our method. We introduce this into two parts.

Regularity: First, by applying the Minty-Browder theorem ^[11] and the decomposition of Riesz potential, we improve the integrability of weak solutions to Eq (1.1). Then, under a different range of $2_{\alpha}^{*}$, we use two different iteration approaches to establish an $L^{\infty}(\mathbb{R}^{N})$ estimate for weak solutions of Eq (1.1). As a result, the Pohožaev identity of Eq (1.1) is established.

Existence: With delicate analysis and optimal range of $\lambda$, we give an exact estimate of the minimum on the Pohožaev manifold. Using this fact, one can show that the minimizing sequences in Pohožaev manifold are non-vanishing in $L^{2}(\mathbb{R}^{N})$ and $L^{2^{*}}(\mathbb{R}^{N})$. This, together with a compactness lemma (see Proposition 2.2), the existence of ground state solutions of Eq (1.1) is obtained. Finally, we prove these ground solutions are radially symmetric.

This paper is organized as follows. In Section 2, we introduce some basic notations and technical lemmas. In Section 3, we study the regularity of weak solutions and Pohožaev identity of Eq (1.1). In Section 4, we study the existence and symmetry of ground state solutions of Eq (1.1).

2.   Preliminaries

In this section, we give some definitions and results which will be used later. $C$, $C_{i}\; (i = 1, 2, \cdots)$ denote positive constants which can be changed line by line. Let $X$ be a Banach space, and use $X_{rad}$ to denote the radial subspace of $X$.

In this work, our working space can be defined by

equipped with the norm

Proposition 2.1. (^[19]) Let $s, t > 1$, and $\alpha\in(0, N)$ with $\frac{1}{s}+\frac{1}{t} = 1+\frac{\alpha}{N}$. Then, there exists $C(N, \alpha, s, t) > 0$ such that for any $u\in L^{s}(\mathbb{R}^{N})$ and $v\in L^{t}(\mathbb{R}^{N})$,

If $s = t = \frac{2N}{N+\alpha}$, then $C(N, \alpha, s, t) = C_{N, \alpha} = \pi^{\frac{N-\alpha}{2}} \frac{\Gamma(\frac{\alpha}{2})}{\Gamma(\frac{N+\alpha}{2})} \left[\frac{\Gamma(\frac{N}{2})}{\Gamma(N)}\right]^{-\frac{\alpha}{N}}$.

Proposition 2.2. (^[34]) Let $N\geqslant 3$, and $\{u_{n}\}\subset E$ be any bounded sequence satisfying

Then, the sequence $\{u_{n}\}$ converges weakly and a.e. to $u\not\equiv0$ in $L^{2}_{loc}(\mathbb{R}^{N})$.

The following inequalities can be viewed as a consequence of Proposition 2.1, which is useful in the following estimation:

and

where $\mathcal{S}_{1}$ and $\mathcal{S}_{2}$ are the embedding constants.

Lemma 2.1. For any $x, y\in\mathbb{R}^{N}$, the following assertions are valid:

(i) If $1 < p < 2$, then

(ii) If $2\leqslant p < \infty$, then

3.   Regularity of weak solutions and the Pohožaev identity

In this section, we study the regularity of weak solutions of Eq (1.1).

Lemma 3.1. (^[25]) Let $q, r, w, t\in [1, \infty)$ and $\zeta\in[0, 2]$ such that

If $\mu\in(0, 2)$ satisfies

then for every $H\in L^w(\mathbb{R}^{N})$, $K\in L^t(\mathbb{R}^{N})$, and $u\in L^q(\mathbb{R}^{N}) \cap L^r(\mathbb{R}^{N})$,

Similar to the proof of [25, Lemma 3.2], we get the following lemma without proof.

Lemma 3.2. Let $N\geqslant3$, $0 < \alpha < N$, and $0 < \theta < 2$. If $H, G\in L^\frac{2N}{\alpha} (\mathbb{R}^{N})+L^\frac{2 N}{\alpha+2}(\mathbb{R}^{N})$, and $\frac{\alpha}{N} < \theta < 2$, then for every $\epsilon > 0$ there exists $C_{\epsilon, \theta}\in\mathbb{R}$ such that, for every $u\in E$,

Proposition 3.1. (^[11]) Let $X$ be a reflexive Banach space. Let $\Phi$ be a (nonlinear) continuous mapping from $X$ into its dual space $X^{-1}$ such that

$(\mathrm{i})$ $\langle \Phi u-\Phi v, u-v \rangle > 0, \ \ \forall u, v\in X, \; u\neq v${;

$(\mathrm{ii})$ $\lim\limits_{\|u\|_{X}\rightarrow \infty}\frac{\langle \Phi u, u \rangle}{\|u\|_{X}} = +\infty$.

Then, for every $g\in X^{-1}$, there exists a unique $u\in X$ such that $\Phi u = g$.

Lemma 3.3. Let

Then, $\Phi$ satisfies the following conditions:

$(\mathrm{i})$ $\langle \Phi u-\Phi v, u-v \rangle > 0, \ \ \forall u, v\in E, \; u\neq v$;

$(\mathrm{ii})$ $\lim\limits_{\|u\|_{E}\rightarrow \infty}\frac{\langle \Phi u, u \rangle}{\|u\|_{E}} = +\infty$.

Proof. Under direct calculation, we can compute

Now, we give the verifications of $\mathrm{(i)}$–$\mathrm{(ii)}$. By Lemma 2.1, we have that for $1 < p < 2$,

and for $2\leqslant p < \infty$,

Combining (3.2) and (3.3), for $p\in(1, \infty)$ there exists $C > 0$ such that

In view of Lemma 3.2 with $\theta = 1$, there exists $\tau > 0$ such that for each $v\in E$, we have

Taking this together with (3.4) and (3.5), we obtain

So, condition $(\mathrm{i})$ follows. By Lemma 3.2, it is easy to verify condition $(\mathrm{ii})$. The proof is complete. □

Lemma 3.4. Suppose that $H, G\in L^{\frac{2N}{\alpha}}(\mathbb{R}^{N})+L^{\frac{2N}{\alpha+2}}(\mathbb{R}^{N})$ and $u\in E$ solve

Then, $u\in L^{q}(\mathbb{R}^{N})$ for each $q\in\left[2, \frac{2^{*}N}{\alpha}\right]$.

Proof. Using Lemma 3.2 with $\theta = 1$, there exists $\tau > 0$ such that, for every $\varphi\in E$,

Let sequences $\{H_{n}\}, \{G_{n}\}\in L^{\frac{2N}{\alpha}} (\mathbb{R}^{N})$ such that $|H_{n}|\leqslant|H|$ and $|G_{n}|\leqslant|G|$, and $H_{n}\to H$ and $G_{n}\to G$ almost everywhere in $\mathbb{R}^{N}$. In what follows, we claim that there exists a unique solution $u_{n}\in E$ satisfying

where $u\in E$ is the given solution of (3.6). The duality is given in this case by

In view of Lemma 3.3, it is easy to verify that $\Psi$ satisfies all the conditions described in Proposition 3.1. Applying Proposition 3.1 with $g(u) = (\tau-1)u$, we get the desired results.

Moreover, we also claim that the sequence $\{u_{n}\}$ converges weakly to $u$ in $E$ as $n\to\infty$. Multiplying both sides of (3.8) by $u_{n}$ and integrating it over $\mathbb{R}^{N}$, then

Combining this with (3.7), the Hölder inequality, and the Young inequality, one has

By this, we obtain

which implies that $\{u_{n}\}$ is bounded in $E$. Then, there exists $\tilde{u}\in E$ such that $u_{n}\rightharpoonup\tilde{u}$ in $E$ and $u_{n}\to\tilde{u}$ almost everywhere in $\mathbb{R}^{N}$. By $H_{n}\in L^{\frac{2N}{\alpha+2}}(\mathbb{R}^{N})$, it is easy to verify $H_{n}u_{n}$ is bounded in $L^{\frac{2N}{N+\alpha}}(\mathbb{R}^{N})$. Hence, we get $H_{n}u_{n}\rightharpoonup H\tilde{u}$ in $L^{\frac{2N}{N+\alpha}}(\mathbb{R}^{N})$. Moreover, for any $\varphi\in C_{0}^{\infty}(\mathbb{R}^{N})$, by $|G_{n}|\leqslant|G|$ and the Lebesgue dominated convergence theorem, we can deduce $G_{n}\varphi\to G\varphi$ in $L^{\frac{2N}{N+\alpha}}(\mathbb{R}^{N})$. Then, we have

Thus, $\tilde{u}$ is a weak solution of

By Proposition 3.1, we know that Eq (3.11) admits a unique solution. Then, $u = \tilde{u}$.

For $\theta > 0$, we define the truncation $u_{n, \theta}: \mathbb{R}^{N}\to\mathbb{R}$ by

For any $q > 2$, it is easy to check $|u_{n, \theta}|^{q-2} u_{n, \theta}\in E$. Taking $|u_{n, \theta}|^{q-2} u_{n, \theta}\in E$ as a test function in Eq (3.8), we can see that

Applying Lemma 3.2 with $\theta = \frac{2}{q}$, where $q\in[2, \frac{2N}{\alpha})$, there then exists $C > 0$ such that

Taking this together with the above two chain of inequalities and making use of the Hölder inequality and the Young inequality, we can infer

By $q\in[2, \frac{2N}{\alpha})$ and Proposition 2.1, then

with $\frac{1}{s} = \frac{N+\alpha}{2N} - \frac{1}{2} + \frac{1}{q}$ and $\frac{1}{t} = \frac{N+\alpha}{2N} + \frac{1}{2} - \frac{1}{q}$.

Using the fact that $u_{n}\in L^{q}(\mathbb{R}^{N})$ and $H_{n}, G_{n}\in L^{\frac{2N}{\alpha}}(\mathbb{R}^{N})$, we get $|H_{n}u_{n}|\in L^{s}(\mathbb{R}^{N})$ and $|G_{n}||u_{n}|^{q-1}\in L^{t}(\mathbb{R}^{N})$. By applying the Lebesgue dominated convergence theorem, we have

Inserting this into (3.12) and taking $\theta\to\infty$, by the Sobolev embedding theorem we can deduce

Taking into account (3.10), (3.13), and the Fatou lemma, we get that

which means that $u\in L^{q}(\mathbb{R}^{N})$ for any $q\in\left[2, \frac{2^{*}N}{\alpha}\right]$. The proof is complete. □

Lemma 3.5. (^[19]) Let $1\leqslant s\leqslant\infty$, $g\in L^{t_{1}}(\mathbb{R}^{N})$, and $h\in L^{t_{2}}(\mathbb{R}^{N})$. Then, there exists $C > 0$ such that

where

Lemma 3.6. Suppose that all the conditions described in Theorem 1.1 are satisfied. Let $u\in E$ be a nontrivial solution of Eq (1.1). Then,

Proof. In view of Lemma 3.4, we obtain $u\in L^{q}(\mathbb{R}^{N})$ for every $q\in\left[2, \frac{2^{*}N}{\alpha}\right]$. By condition $(F_{1})$, one can infer $F(u)\in L^{\tilde{p}}(\mathbb{R}^{N})$ for every $\tilde{p}\in\left[\frac{2N}{N+\alpha}, \frac{2N^{2}}{\alpha(N+\alpha)}\right]$.

Fixing $\epsilon\in \left(0, \frac{\alpha N}{2N+\alpha}\right)$, $I_{\alpha}$ can be decomposed as

where $I_{\alpha}^{1}\in L^{\frac{N-\epsilon}{N-\alpha}}(\mathbb{R}^{N})$ and $I_{\alpha}^{2}\in L^{\frac{N+\epsilon}{N-\alpha}}(\mathbb{R}^{N})$. Let $s = \infty$ in Lemma 3.5. It follows from $I_{\alpha}^{1}\in L^{\frac{N-\epsilon}{N-\alpha}}(\mathbb{R}^{N})$ that

Similar to (3.15), by $I_{\alpha}^{2}\in L^{\frac{N+\epsilon}{N-\alpha}}(\mathbb{R}^{N})$ we can infer that

In view of $\epsilon\in \left(0, \frac{\alpha N}{2N+\alpha}\right)$, we derive

It follows from (3.15)–(3.17) that

The proof is completed. □

Lemma 3.7. Suppose that all the conditions described in Theorem 1.1 are satisfied. Let $u\in E$ be a nontrivial solution of Eq (1.1). For each $L > 2$, define

For $\tau > 1$, we set $\tilde{u}_{L} = uu_{L}^{2(\tau-1)}$. Then, for any $s\in[2, 2^{*}]$, we have

Proof. Multiplying both sides of Eq (1.1) by $\tilde{u}_L$ and integrating, it follows that

Combining the above relation with Lemma 3.6, this leads to

The proof is completed. □

We now are ready to establish Theorem 1.1.

Proof of Theorem 1.1. (ⅰ) ($L^{\infty}$ estimate) We consider the following two cases separately.

Case 1. $2_{\alpha}^{*}\leqslant2 \Leftrightarrow N\geqslant 4+\alpha$.

In this case, we should keep in mind that $\tau = \frac{2^{*}}{2}$.

Step 1. Clearly, we have

In view of Lemma 3.4, we have $u\in L^{q}(\mathbb{R}^{N})$ for any $q\in\left[2, \frac{2^{*}N}{\alpha}\right]$. That is,

For any $0 < R < \infty$, we set

and

Obviously, we have

and

Clearly, if it holds that $B_{\tau} = B_{\tau}(R)$ or $\tilde{B}_{\tau} = \tilde{B}_{\tau}(R)$, then we have $u\in L^{\infty}(\mathbb{R}^{N})$. This completes our proof. To this end, we just need to consider the following case

Without loss of generality, we set $R = 1$. Then, there exist $0 < C_{1}, \tilde{C}_{1} < \infty$ such that

From $2_{\alpha}^{\sharp} < 2$, we deduce

It follows from (3.18) that

Similarly, one can infer

Combining (3.19), (3.20), and Lemma 3.7, we obtain

Let $L\to\infty$ in the above expression. Then,

By $\tau = \frac{2^{*}}{2}$, we have

Since $s\in[2, 2^{*}]$, we get $u\in L^{p_{1}}(\mathbb{R}^{N})$, where $p_{1}\in \bigl[2, \frac{(2^{*})^{2}}{2}\bigr]$.

Step 2. Obviously, we have

and for $B_{\tau}$ and $\tilde{B}_{\tau}$, we have

Similar to Step 1, we just need to show the case

Moreover, we have

and

In view of (3.19), (3.20), (3.22), and (3.23), it follows that

which implies

Similarly, we have

Taking $L\to\infty$ and making use of Lemma 3.7 again, we get

which implies $u\in L^{p_{2}}(\mathbb{R}^{N})$, where $p_{2}\in \left[2, 2_{s}^{*}\cdot \left(\frac{2_{s}^{*}}{2}\right)^{2} \right]$.

Step 3. Iterating the above procedure, for any $n\in\mathbb{N}^{*}$ we conclude

Let $s = 2^{*}$. Then,

Obviously, we have

This means that the series $\sum\limits_{i = 1} \limits^{n} \frac{1}{2\cdot\left(\frac{2^{*}}{2}\right)^{i}}$ converges absolutely.

Let $n\to\infty$ in (3.24). Then, it holds that

Case 2. $2_{\alpha}^{*} > 2\Leftrightarrow N < 4+\alpha$.

Step 1. Let $\tau_{1}\in \left[ 1+\frac{2-2_{\alpha}^{\sharp}}{2}, 1+\frac{\frac{2^{*}N}{\alpha}-2_{\alpha}^{\sharp}}{2} \right]$. Then, we claim

By the definition of $u_{L}$, we obtain

Let $l > 0$ be chosen later. By the Hölder inequality, we have

By $2_{\alpha}^{*}\in\left[2, \frac{2^{*}N}{\alpha}\right]$, we can choose suitable $l > 0$ such that

It follows from the inequalities and Lemma 3.7 that

Let $L\to\infty$. The above inequality becomes

In view of $2_{\alpha}^{\sharp}+2(\tau_{1}-1)\in \left[2, \frac{2^{*}N}{\alpha} \right]$ and (3.26), we conclude (3.25).

Step 2. Let $\tau_{2} = 1+\frac{2_{\alpha}^{*}}{2}(\tau_{1}-1)$. We claim

We choose $\tau\in\bigl[\tau_{1}, \tau_{2}\bigr]$. Then,

Combining (3.25) and Lemma 3.7, we obtain

Let $\tau = \tau_{2}$ in (3.27). Then,

Making use of the Young inequality, it holds that

where $a = \frac{2_{\alpha}^{*}\bigl(2_{\alpha}^{*}-2_{\alpha}^{\sharp}\bigr)} {2(\tau_{2}-1)}$ and $b = 2_{\alpha}^{\sharp}+2(\tau_{2}-1)-\frac{2_{\alpha}^{*}\bigl(2_{\alpha}^{*}-2_{\alpha}^{\sharp}\bigr)} {2(\tau_{2}-1)}$. Thus, we get

Moreover, by $\frac{2}{2_{\alpha}^{*}} < 1$, it is easy to observe

Then,

which implies

Step 3. We iterate the above procedure and set

Then,

which further gives

This yields that

According to (3.28), we deduce

Taking into account (3.29) and (3.30), it follows that

By a straightforward calculation, we can infer

For the series $\sum\limits^{\infty}_{i = 1} \frac{\ln C} {\tau_{i+1}-1}$, we have

This means $\sum\limits^{\infty}_{i = 1} \frac{\ln C} {\tau_{i+1}-1}$ converges absolutely.

For the series $\sum\limits^{\infty}_{i = 1} \frac{\ln\tau_{i+1}} {\tau_{i+1}-1}$, it follows that

which implies $\sum\limits^{\infty}_{i = 1} \frac{\ln\tau_{i+1}} {\tau_{i+1}-1}$ converges absolutely.

Together with (3.32)–(3.34), we conclude $\prod\limits_{i = 1}\limits^{\infty}\bigl(C\tau_{i+1}\bigr)^{\frac{2}{\tau_{i+1}-1}} < \infty$. Letting $n\to\infty$ in (3.31), we obtain

(ⅱ) (Pohožaev indentity) Observe that, by $u\in L^{\infty}(\mathbb{R}^{N})$, Lemma 3.6, and condition $(F_{1})$, there exists $C > 0$ such that

Set $l(u) = C\bigl(|u|^{2_{\alpha}^{\sharp}-2}u+|u|^{2_{\alpha}^{*}-2}u\bigr)$. By a classical bootstrapping argument for subcritical local problems in ^[33], we infer that $u\in W_{loc}^{2, q}(\mathbb{R}^{N})$ for every $q\geqslant 1$, and hence we have $u\in C_{loc}^{1, \beta}(\mathbb{R}^{N})$ for any $0 < \beta < 1$ by the Sobolev embedding theorem. Under the classical strategy used in [25, Theorem 3], one can show that

This completes the proof of Theorem 1.1. □

4.   Proof of Theorem 1.2

In this section, we prove Theorem 1.2 by virtue of the Pohožaev manifold method and a generalized version of a Lions-type theorem.

Under condition $(F_{2})$, Eq (1.1) turns into a $(p, 2)$-Laplacian equation as follows:

Then, the corresponding energy functional of Eq ($\mathcal{D}$) can be defined as

It is easy to check $J\in C^{1}(E, \mathbb{R})$. Obviously, the critical points of $J$ are weak solutions of Eq ($\mathcal{D}$) and satisfy the following Pohožaev identity:

We define the Pohožaev manifold and its minimum as follows:

Lemma 4.1. Assume that all conditions described in Theorem 1.2 are satisfied. Let $C_{1}, C_{2}, C_{3} > 0$. Define a function $k:\mathbb{R}^{+}\to\mathbb{R}$ as

Then, $k(t)$ has a unique critical point which corresponds to its maximum.

Proof. By the definition of $k(\cdot)$, we have

From the above expression, it is easy to see that $k'(t) > 0$ for $t > 0$ small, and $k'(t) < 0$ for $t > 0$ large. This yields that $k(t)$ possesses at least one maximum point. Next, we claim that the maximun point corresponding to $k(t)$ is unique. Otherwise, we suppose that there exists $t_{1}\not = t_{2} > 0$ such that

and

Combining the above two equalities, it holds that

which further gives $t_{1} = t_{2}$. The proof is complete. □

Lemma 4.2. Assume that all conditions described in Theorem 1.2 are satisfied. Then, for every $u\in E$, there exists a unique $t_{u} > 0$ such that $P(u_{t_{u}}) = 0$, where $u_{t} = u\left(\frac{x}{t}\right)$. Moreover, $J(u_{t_{u}}) = \max\limits_{t\geqslant 0} J(u_{t})$.

Proof. For every $u\in E\backslash\{0\}$, one has

and

Combining the above two formulas, it is easy to see that $P(u_{t}) = tJ'(u_{t}) = 0$. By Lemma 4.1, we complete the proof. □

Lemma 4.3. Suppose that all conditions described in Theorem 1.2 are satisfied. Then, $m > 0$.

Proof. For every $u\in\mathcal{P}$, it follows from Proposition 2.1 that

which implies that $\|u\|_{E}\geqslant C$. Then, it holds that

The proof is complete. □

Lemma 4.4. Assume that all conditions described in Theorem 1.2 are satisfied. Then, we have

Proof. From $\lambda < \Lambda$, we can easily get

where $\Lambda$ is defined in Threorem 1.2.

The extremal function of inequalities (2.1) can be defined as

Let $t_{\sigma} > 0$ satisfy

By the definition of $m$, it is easy to see that

A straightforward calculation shows that

Moreover, we can compute

and

and

and

It follows that

Taking the limit superior as $\sigma\to\infty$ in (4.1), we further obtain

Let $s_{\infty} = \limsup\limits_{\sigma\to\infty}t_{\sigma}$. We can prove $0 < t_{\sigma} < \infty$. Otherwise, we suppose $t_{\sigma} = \infty$. Then,

This yields a contradiction with (4.3).

Now, we show $s_{\infty} > 0$. Arguing by contradiction, we assume $s_{\infty} = 0$. Therefore, there exists $\hat{\sigma} > 0$ large such that $s_{\hat{\sigma}} > 0$ small enough. Then,

which contradicts with (4.2). Hence, we get $0 < s_{\infty} < \infty$.

In view of $0 < s_{\infty} < \infty$ and taking the limit superior as $\sigma\to\infty$ in (4.2) again, it holds that

Then, we have

Applying this for any $\bar{\sigma} > 0$ large, it follows that

Set

Then, we know $h'(s_{\infty}) = 0$ and $s_{\infty}$ is the unique maxium point of $h(\cdot)$. By (4.4), one has

The proof is completed. □

Lemma 4.5. Suppose that all conditions described in Theorem 1.2 hold. Let $\{u_{n}\}$ be a bounded minimizing sequence of $J$ satisfying

Then, we have

Proof. First, we show $\lim\limits_{n\rightarrow \infty} \int_{\mathbb{R}^{N}} |u_{n}|^{2} \mathrm{d}x > 0$. Otherwise, we suppose

Combining Proposition 2.1 and (4.5), we have

and

Combining the above two inequalities, one has

and

Combining (4.6) and (4.7), it follows that

It follows from (2.2) and (4.7) that

This implies

In view of (4.8) and (4.9), we can derive

This yields a contradiction with Lemma 4.4.

Next, we show $\lim\limits_{n\rightarrow \infty} \int_{\mathbb{R}^{N}} |u_{n}|^{2^{*}} \mathrm{d}x > 0$. On the contrary, it suffices to show

From Proposition 2.1 and (4.10), we have

and

Together with the above two expressions, we get

and

From (4.11) and (4.12), we get that

Observe that, by (2.1) and (4.12), we have

This shows

Combining (4.13) and (4.14), we infer that

This contradicts with Lemma 4.5, the proof is complete. □

We now can conclude the proof of Theorem 1.2.

Proof of Theorem 1.2. Let $\{u_{n}\}$ be a minimizing sequence of $J$ satisfying

It is easy to verify $\{u_{n}\}$ is bounded in $E$. Taking into account Lemmas 4.4 and 4.5 and Proposition 2.2, we deduce that $\{u_{n}\}$ converges weakly and a.e. to $u\not\equiv0$ in $L_{loc}^{2}(\mathbb{R}^{N})$. Using the Brézis-Lieb lemma ^[35], we obtain

which implies $J(u) = m$. Moreover, we can choose $u\geqslant0$. Therefore, $u$ is a nonnegative ground state solution of Eq ($\mathcal{D}$).

Let $u$ be a nonnegative ground state solution of Eq ($\mathcal{D}$). In order to show $u$ is radial, it suffices to prove $m = m_{rad}$, where

and

On one hand, by $E_{rad}\subset E$, it follows that $m\leqslant m_{rad}$. On the other hand, for any $v\in\mathcal{P}$, by $J$ being even, we know $|v|\in\mathcal{P}$. Let $|v|^{*}$ be the decreasing rearrangement of $|v|$. In view of [19, Theorem 3.7], one has

and

and

and

and

From (4.15)–(4.19), for any $t > 0$, it follows that

By Lemma 4.2, there exists $t_{v} > 0$ such that $(|v|^{*})_{t_{v}}\in\mathcal{P}$. We have

From the above and $|v|\in\mathcal{P}$, it is easy to observe $t_{v}\in(0, 1]$. In view of (4.20), one has

Therefore, for any $|v|\in\mathcal{P}$, there exists $t_{v} > 0$ such that $t_{v}|v|^{*}\in\mathcal{P}$ and

This implies $m_{rad}\leqslant m$. Hence, we have $m_{rad} = m$. This means $u$ is a radially symmetric ground state solution of Eq ($\mathcal{D}$). The proof is complete. □

Author contributions

Lixiong Wang: Conceptualization, Methodology, Formal analysis, Writing-original draft preparation, Writing-reviewing and editing, Validation, Funding acquisition; Ting Liu: Writing-reviewing and editing, Visualization, Investigation, Supervision. All authors have read and approved the final version of the manuscript for publication.

Acknowledgments

The author desires to convey gratitude to Hunan Institute of Science and Technology and the Outstanding Youth Project of the Education Department of Hunan Province for the financial assistance provided for this work.

Conflict of interest

The authors declare there are no conflicts of interest.