1 Introduction

Let \(\Omega \) be a smooth bounded domain in \({\mathbb {R}}^n\) with \(n\ge 2\) and let \(W_0^{1,n}(\Omega )\) be the usual Sobolev space on \(\Omega \) which is the completion of \(C_0^\infty (\Omega )\) under the Dirichlet norm \(\Vert \nabla u\Vert _{L^n(\Omega )} = \left(\int _\Omega |\nabla u|^ndx\right)^{\frac{1}{n}}\). The classical Trudinger–Moser inequality (see [21, 23, 26, 30]) asserts that

$$\begin{aligned} \sup _{u\in W^{1,n}_0(\Omega ), \Vert \nabla u\Vert _{L^n(\Omega )} \le 1} \int _\Omega e^{\alpha |u|^{\frac{n}{n-1}}} dx < \infty , \end{aligned}$$
(1.1)

for any \(\alpha \le \alpha _n:= n \omega _{n-1}^{\frac{1}{n-1}}\) where \(\omega _{n-1}\) is the surface area of the unit ball in \({\mathbb {R}}^n\). The inequality (1.1) is sharp in the sense that if \(\alpha > \alpha _n\) then the supremum in (1.1) will be infinite though all integrals in (1.1) are still finite. The existence of extremals for the classical Trudinger–Moser inequality (1.1) was settled by Carleson and Chang [9] (see also [14]) when \(\Omega \) is the ball, by Flucher [16] for any bounded domain in \({\mathbb {R}}^2\) and by Lin [19] for any bounded domain in \({\mathbb {R}}^n\) with \(n \ge 3\). The inequality (1.1) was extended to unbounded domains and whole space \({\mathbb {R}}^n\) by Adachi and Tanaka [1], by Ruf [24] for \(n=2\) and by Li and Ruf [18] for any \(n \ge 3\). Some improvements of the Trudinger–Moser inequality (1.1) was established in [2, 22, 25, 27].

In [3], by using a symmetrization argument and a change of variables, Adimurthi and Sandeep proved a singular Trudinger–Moser inequality which generalizes (1.1) to the singular weight case as follows

$$\begin{aligned} \sup _{u\in W^{1,n}_0(\Omega ), \Vert \nabla u\Vert _{L^n(\Omega )} \le 1} \int _\Omega e^{\alpha _n \gamma |u|^{\frac{n}{n-1}}} |x|^{-\beta n} dx< \infty ,\quad 0\le \beta< 1, \; 0< \gamma < 1 -\beta .\nonumber \\ \end{aligned}$$
(1.2)

Again, the inequality (1.2) is sharp in the sense that if \(\gamma >1-\beta \) then the supremum in (1.2) will be infinite though all integrals in (1.2) are still finite. The inequality (1.2) was extended to whole space \({\mathbb {R}}^n\) by Adimurthi and Yang [5]. The problem on the existence of extremals for the singular Trudinger–Moser inequality (1.2) was solved by Csató and Roy [11, 12], Yang and Zhu [28], Iula and Mancini [17] in dimension two, and by Csató, Roy and the author [13] in any dimension \(n \ge 3\). An interesting question is whether or not the inequality (1.2) holds for \(\beta < 0\). Generally, the answers is negative (see the example in the Introduction of [29]). However, if we restrict ourselves to the radial functions only, then the answer is positive. Indeed, it was proved by Yang and Zhu (see [29, Theorem 1.1]) that

$$\begin{aligned} \sup _{u\in W^{1,n}_0(\mathbf{B }),u \text { is radial}, \Vert \nabla u\Vert _{L^n(\mathbf{B })} \le 1} \int _\mathbf{B } e^{\alpha _n \gamma |u|^{\frac{n}{n-1}}} |x|^{\beta n}dx < \infty ,\quad \beta >0, \; \gamma \le 1+ \beta .\nonumber \\ \end{aligned}$$
(1.3)

Again, the inequality (1.3) is sharp in the sense that if \(\gamma >1+ \beta \) then the supremum in (1.3) will be infinite though all integrals in (1.3) are still finite. Furthermore, Yang and Zhu also proved the existence of extremals for (1.3) in the class of radial functions (an earlier result in dimension two was proved in [15, Theorem 1.8]). The inequality (1.3) extends the results of De Figueiredo et al. [15, Theorem 1.8] to higher dimension (see also [6, 8] for more results in dimension two). In fact, De Figueiredo et al. considered more a general weight h(|x|) and fast growth F(u) instead of \(|x|^{2\beta }\) and \(e^{4\pi (1+ \beta ) u^2}\). They presented the precise balance between h and F that guarantees

$$\begin{aligned} s_{F,h}: = \sup _{u\in W^{1,2}_{0}(\mathbf{B }), u \text { is radial}, \Vert \nabla u\Vert _{L^2(\mathbf{B })} \le 1} \int _\mathbf{B } F(u) h(|x|) dx \end{aligned}$$

to be finite. In [15], the following hypothesis on h and F were presented

  1. (H1)

    \(h:[0,1] \rightarrow [0,\infty )\) is continuous, \(h(t) >0\) for \(t>0\) and h is non-decreasing.

  2. (H2)

    \(F: {\mathbb {R}}\rightarrow [0,\infty )\) is \(C^2\), even, strictly increasing and \(F(0) = F'(0) =0\).

  3. (H3)

    There exists \(R >0\) such that \(e^{-4\pi t^2} F(t)\) is non-decreasing on \([R,\infty )\).

  4. (H4)

    \(\limsup _{t\rightarrow \infty } e^{-4\pi t^2} F(t) h(e^{-2\pi t^2}) < \infty \).

  5. (H5)

    \(\limsup _{t\rightarrow \infty } e^{-4\pi t^2} F(t) h(e^{-2\pi t^2}) =0\).

  6. (H6)

    \(0< \limsup _{t\rightarrow \infty } e^{-4\pi t^2} F(t) h(e^{-2\pi t^2}) < \infty \).

The main results in [15] assert that if \(({\mathrm{H}}1){-}({\mathrm{H}}4)\) hold then \(s_{F,h} < \infty \) (see [15, Theorem 1.1]). Conversely, if \(({\mathrm{H}}1){-}({\mathrm{H}}3)\) and the following conditions \(h\in C^1((0,1))\) and \(\limsup _{r\rightarrow 0^+} \frac{h'(r) r}{h(r)} < \infty \) are satisfied then \(s_{F,h} < \infty \) if and only if \(({\mathrm{H}}4)\) holds (see [15, Theorem 1.3]). In other words, \(({\mathrm{H}}4)\) is the precise balance between h and F to show whether or not \(s_{F,h}\) is finite. It is worthy to notice that Theorem 1.1 in [15] includes many cases that have not been considered before, for example \(h(r) =e^{-r^{-\alpha }}\) and \(F(u) = e^{e^{\theta u^2}} -e\) with \(\theta \le 2\pi \alpha \). Concerning to the attainability of \(s_{F,h}\), they proved that if \(({\mathrm{H}}1){-}({\mathrm{H}}3)\) and \(({\mathrm{H}}5)\) are satisfied then \(s_{F,h}\) is attained (see [15, Theorem 1.6]). This corresponds to the sub-criticality of the nonlinearity F. In the critical case of F (i.e., \(({\mathrm{H}}6)\) holds), they proved the attainability of \(s_{F,h}\) for \(h(r) = r^\alpha \) and F satisfies \(({\mathrm{H}}2)\) and

$$\begin{aligned} e^{2\pi (2+ \alpha ) t^2} -1 -\lambda t^2 \le F(t) \le e^{2\pi (2+ \alpha ) t^2} -1,\quad t\ge 0 \end{aligned}$$

for some \(0\le \lambda < 2\pi (2+ \alpha )\) (see [15, Theorem 1.9]). Notice that when \(h(r) =r^{2\beta }\) and \(F(u) = e^{4\pi (1+\beta ) u^2} -1\) we obtain the result of Yang and Zhu in dimension two which solves the open problems in [8, p. 344] and [6, Remark 2.8]. It should be emphasized here that the results in [15] make clear that a strong vanishing of h(|x|) at \(x =0\) allows the nonlinearity F(u) to have very fast growth as u goes to infinity.

The main aim of this paper is to extend the results of De Figueiredo et al. [15] to higher dimensions (also, extend the results of Yang and Zhu to more a general weight h and nonlinearity F). We still denote by \(\mathbf{B }\) the unit ball in \({\mathbb {R}}^n\) and let \(W^{1,n}_{0,rad}(\mathbf{B })\) the set of all radial functions in \(W^{1,n}_0(\mathbf{B })\). Let \(h:[0,1] \rightarrow [0,\infty )\) and \(F: {\mathbb {R}}\rightarrow [0,\infty )\). We look for the balance conditions between h and F such that

$$\begin{aligned} S_{F,h}:= \sup _{u \in W^{1,n}_{0,rad}(\mathbf{B }), \Vert \nabla u\Vert _{L^n(\mathbf{B })} \le 1} \int _\mathbf{B } F(u) h(|x|) dx \end{aligned}$$

is finite. Following [15], we impose the following conditions on h and F:

  1. (A1)

    \(h:[0,1] \rightarrow [0,\infty )\) is continuous, \(h(t) >0\) for \(t>0\) and h is non-decreasing in a neighborhood of 0.

  2. (A2)

    \(F: {\mathbb {R}}\rightarrow [0,\infty )\) is even, continuous, \(F(0) =0\) and \(F(t) >0\) if \(t\not =0\).

  3. (A3)

    \(\limsup _{t\rightarrow \infty } e^{-\alpha _n t^{\frac{n}{n-1}}} F(t) h(e^{-\omega _{n-1}^{\frac{1}{n-1}} t^{\frac{n}{n-1}}}) < \infty \).

Comparing with the conditions \(({\mathrm{H}}1){-}({\mathrm{H}}6)\) in [15], we see that our conditions on h and F are weaker. For example, the condition \(({\mathrm{H}}3)\) is removed. We only need the non-decreasing monotonicity of h near 0, and the continuity of F (in fact, we do not need the differentiability and the monotonicity of F).

Our first main result gives a sufficient condition for the finiteness of \(S_{F,h}\) as follows.

Theorem 1.1

If \((\mathrm{A}1){-}(\mathrm{A}3)\) are satisfied, then \(S_{F,h}\) is finite.

As a consequence of Theorem 1.1, we easily see that \(S_{F,h}\) is finite if F is dominated by a function that satisfies \((\mathrm{A}2){-}(\mathrm{A}3)\). More precisely, if h, F satisfy \((\mathrm{A}1){-}(\mathrm{A}2)\) and there exists a function \(G: {\mathbb {R}}\rightarrow [0,\infty )\) that satisfies \((\mathrm{A}2){-}(\mathrm{A}3)\) and \(\limsup _{t\rightarrow \infty } \frac{F(t)}{G(t)} < \infty \) then \(S_{F,h}\) is finite.

We next show that in some cases the condition \((\mathrm{A}3)\) is necessary for the finiteness of \(S_{F,h}\). In other word, \((\mathrm{A}3)\) is the precise balance between h and F to establish whether or not \(S_{F,h}\) is finite.

Theorem 1.2

Suppose that \((\mathrm{A}1){-}(\mathrm{A}2)\) hold, h is \(C^1\) on (0, 1) and \(\limsup _{r\rightarrow 0^+} {h'(r) r}/{h(r)} <\infty .\) Then \(S_{F,h}\) is finite if and only if \((\mathrm{A}3)\) holds.

Obviously, Theorems 1.1 and 1.2 extend Theorems 1.1 and 1.3 in [15] to higher dimension. Moreover, when \(h(r)= r^{\beta N}\) and \(F(u) = e^{\alpha _n (1+ \beta ) |u|^{\frac{n}{n-1}}} -1\), we obtain the inequality (1.3) of Yang and Zhu. Furthermore, Theorem 1.1 includes cases that have not been studied before. For example, \(h(r) = e^{-r^{-\alpha }}\) and \(F(u) = e^{e^{\theta |u|^{\frac{n}{n-1}}}} -e\) for \(\alpha >0\) and \(\theta \le \omega _{n-1}^{\frac{1}{n-1}} \alpha \). This again emphasizes that strong vanishing of the weight h(r) at zero allows very fast growth of F(u) at infinity (as observed in [15]).

We next move to the question on the attainability of \(S_{F,h}\). First, we have the following definition on the critical or sub-critical nonlinearity of F with respect to h. Following [14, 15], we say that F is h-radially subcritical if

$$\begin{aligned} \qquad \qquad \limsup _{t\rightarrow \infty } e^{-\alpha _n t^{\frac{n}{n-1}}} F(t) h(e^{-\omega _{n-1}^{\frac{1}{n-1}} t^{\frac{n}{n-1}}}) =0 \end{aligned}$$
(1.4)

and F is h-radially critical if

$$\begin{aligned} 0< \limsup _{t\rightarrow \infty } e^{-\alpha _n t^{\frac{n}{n-1}}} F(t) h(e^{-\omega _{n-1}^{\frac{1}{n-1}} t^{\frac{n}{n-1}}}) < \infty . \end{aligned}$$
(1.5)

By normalizing, we always assume that \(\limsup _{t\rightarrow \infty } e^{-\alpha _n t^{\frac{n}{n-1}}} F(t) h(e^{-\omega _{n-1}^{\frac{1}{n-1}} t^{\frac{n}{n-1}}}) =1\) if F is h-radially critical.

We next provide an existence result of extremal for \(S_{F,h}\) in the case that F is h-radially subcritical.

Theorem 1.3

Assume that \((\mathrm{A}1){-}(\mathrm{A}2)\) hold and F is h-radially subcritical. Then \(S_{F,h}\) is attained.

We continue considering the case that F is h-radially critical. In this case, we will need some additional condition on h and F. We first assume that

$$\begin{aligned} \lim _{x\rightarrow 0} h(x) |x|^{-\beta n} =1 \end{aligned}$$
(1.6)

for some \(\beta >0\). Obviously, in this condition, we can replace 1 by any positive number. We have the following existence result in this case

Theorem 1.4

Assume that \((\mathrm{A}1){-}(\mathrm{A}2)\) hold and there exists \(\beta \ge 0\) such that \(h(r) = r^{n\beta }(1 + O(r^\alpha ))\) as \(r \rightarrow 0\) for some \(\alpha >0\), and F is h-radially critical with

$$\begin{aligned} F(t) \ge e^{\alpha _n(1+ \beta ) |t|^{\frac{n}{n-1}}} -1 - \lambda |t|^{\frac{n}{n-1}},\quad t\in {\mathbb {R}}\end{aligned}$$
(1.7)

for some \(0\le \lambda < \alpha _n(1+ \beta )\) and . Then \(S_{F,h}\) is attained.

Theorem 1.3 extends Theorem 1.6 in [15] to higher dimensions and, as well as, weakens the conditions on h and F. Its proof is by simply exploiting the sub-criticality of F, the Trudinger–Moser inequality (1.1) and the concentration–compactness principle of Lions [20]. Theorem 1.4 extends Theorem 1.9 in [15] to higher dimensions. Moreover, Theorem 1.4 is stated for more general weight function h and nonlinearity F. In fact, Theorem 1.9 in [15] is only stated for \(h(r) = r^\alpha \). When \(F(t) =e^{\alpha _n(1+ \beta ) |t|^{\frac{n}{n-1}}} -1\) and \(h(r) = r^{n \beta }\) we obtain the existence result of Yang and Zhu for (1.3) from Theorem 1.4. The proof of Theorem 1.4 follows the ideas in [14] by using the criticality of F, the concentration–compactness principle of Lions [20] and a result of Carleson and Chang [9]. Indeed, using the criticality of F and a result of Carleson and Chang, we obtain an upper bound of the functional \(W^{1,n}_{0,rad}(\mathbf{B }) \ni u \rightarrow \int _\mathbf{B } F(u) h(|x|) dx\) on the sequences \(\{u_k\}_k\subset W^{1,n}_{0,rad}(\mathbf{B })\) with \(\Vert \nabla u_k\Vert _{L^n(\mathbf{B })} \le 1\) and \(u_k \rightharpoonup 0\) weakly in \(W^{1,n}_0(\mathbf{B })\). Using the additional assumptions of F and h, we will show that \(S_{F,h}\) is indeed strictly larger than this upper bound. From this fact, we get that any maximizing sequence for \(S_{F,h}\) has non-zero weak limit. So, we can apply the concentration–compactness of Lions to get the existence of extremals for \(S_{F,h}\). Notice that our proof does not use the blow-up analysis method which is a standard method to study the existence of extremals for the Trudinger–Moser type inequalities.

2 Proofs of the Main Results

In this section, we provide the proofs of our main results. Let us denote

$$\begin{aligned} TM_n := \sup _{u\in W^{1,n}_{0,rad}(\mathbf{B} ), \Vert \nabla u\Vert _{L^n(\mathbf{B })} \le 1} \int _\mathbf{B } e^{\alpha |u|^{\frac{n}{n-1}}} dx. \end{aligned}$$

We start by recalling a sharp pointwise inequality involving functions in \(W^{1,n}_{0,rad}(\mathbf{B })\) (see [4, Corollary 2.2]), namely

$$\begin{aligned} |u(x)| \le \Vert \nabla u\Vert _{L^n(\mathbf{B })} \left(\omega _{n-1}^{-\frac{1}{n-1}} \ln \frac{1}{|x|}\right)^{\frac{n-1}{n}},\quad u\in W_{0,rad}^{1,n}(\mathbf{B }). \end{aligned}$$
(2.1)

2.1 Proof of Theorem 1.1

The proof of Theorem 1.1 follows the one of Theorem 1.1 in [15] with slight modifications due to the weaker assumptions on F and h. By \((\mathrm{A}1)\), there is \(r_0\in (0,1)\) such that h is non-decreasing on \([0,r_0)\). Denote

$$\begin{aligned} C = \limsup _{t\rightarrow \infty } e^{-\alpha _n t^{\frac{n}{n-1}}} F(t) h\Big (e^{-\omega _{n-1}^{\frac{1}{n-1}} t^{\frac{n}{n-1}}}\Big ) < \infty , \end{aligned}$$

by \((\mathrm{A}4)\). Hence, there exists \(t_0 >0\) such that \(e^{-\frac{\alpha _n}{n} t_0^{\frac{n}{n-1}}} \le r_0\) and

$$\begin{aligned} F(t) h\Big (e^{-\omega _{n-1}^{\frac{1}{n-1}} t^{\frac{n}{n-1}}}\Big ) \le (C+1) e^{\alpha _n t^{\frac{n}{n-1}}} \end{aligned}$$

for any \(t \ge t_0\). Let \(u \in W^{1,n}_0(B)\) such that \(\Vert \nabla u\Vert _{L^n(B)} \le 1\), we then have by (2.1) that

$$\begin{aligned} |u(x)| \le \left(-\frac{n}{\alpha _n} \ln |x|\right)^{\frac{n-1}{n}} \Longleftrightarrow |x| \le e^{-\frac{\alpha _n}{n} |u(x)|^{\frac{n}{n-1}}},\quad x \not =0. \end{aligned}$$

Notice that \(|u(x)| \ge t_0\) implies \(|x| \le e^{-\frac{\alpha _n}{n} |u(x)|^{\frac{n}{n-1}}}\le e^{-\frac{\alpha _n}{n} t_0^{\frac{n}{n-1}}} \le r_0\) by the choice of \(t_0\). The monotonicity of h on \([0,r_0)\) implies \(h(|x|) \le h(e^{-\frac{\alpha _n}{n} |u(x)|^{\frac{n}{n-1}}})\). Thus, we have

$$\begin{aligned} F(u(x)) h(|x|) \le (C+1) e^{\alpha _n |u(x)|^{\frac{n}{n-1}}}, \end{aligned}$$

if \(|u(x)| \ge t_0\). Consequently, we get

$$\begin{aligned} \int _\mathbf{B } F(u(x)) h(|x|) dx&= \int _{\{|u| \ge t_0\}} F(u(x)) h(|x|) dx + \int _{\{|u| < t_0\}} F(u(x)) h(|x|) dx\\&\le (C+1) \int _{\{|u| \ge t_0\}} e^{\alpha _n |u(x)|^{\frac{n}{n-1}}} dx + \sup _{r\in [0,1]} h(r) \, \sup _{|t| \le t_0} F(t) |\mathbf{B }|\\&\le (C+1)\int _\mathbf{B } e^{\alpha _n |u(x)|^{\frac{n}{n-1}}} dx + \sup _{r\in [0,1]} h(r) \, \sup _{|t| \le t_0} F(t) |\mathbf{B }|, \end{aligned}$$

for any \(u\in W^{1,n}_{0,rad}(\mathbf{B })\) with \(\Vert \nabla u\Vert _{L^n(\mathbf{B })} \le 1\). This together with the inequality (1.1) in \(\mathbf{B} \) implies

$$\begin{aligned} S_{F,h} \le (C+1)TM_n + \sup _{r\in [0,1]} h(r) \, \sup _{|t| \le t_0} F(t) |\mathbf{B }| < \infty . \end{aligned}$$

2.2 Proof of Theorem 1.2

By Theorem 1.1, it is only to show that if \((\mathrm{A}1){-}(\mathrm{A}2)\) and (1.6) hold, and \(S_{F,h}\) is finite then \((\mathrm{A}3)\) holds. Denote \(\gamma = \limsup _{r\rightarrow 0} h'(r) r/{h(r)} < \infty \) by (1.6). Let \({{\bar{\gamma }}} = \gamma +1\). There exists \(r_0\in (0,1)\) such that \(h'(r) r \le {{\bar{\gamma }}} h(r),\quad \forall \, r\in (0,r_0)\) which is equivalent to \(h(r) \ge \frac{1}{{{\bar{\gamma }}}} h'(r) r,\quad \forall \, r\in (0,r_0).\) Hence, for any \(r \in (0,r_0)\) we have

$$\begin{aligned} \int _0^r h(s) s^{n-1} ds \ge \int _0^r \frac{1}{{{\bar{\gamma }}}} h'(s) s^n ds = \frac{1}{{{\bar{\gamma }}}} h(r)^n r^{n} -\frac{n}{{{\bar{\gamma }}}} \int _0^r h(s) s^{n-1} ds, \end{aligned}$$

here we use integration by parts. Thus, we get

$$\begin{aligned} \int _0^r h(s) s^{n-1} ds \ge \frac{1}{n+ {{\bar{\gamma }}}} h(r) r^n,\quad \forall \, r\in (0,r_0). \end{aligned}$$
(2.2)

For any \(k >0\) such that \(e^{-k} < r_0\), let us consider the Moser function

$$\begin{aligned} u_k(x) ={\left\{ \begin{array}{ll} -\frac{\ln |x|}{( \omega _{n-1} k)^{1/n}} &{}\quad \text{ if }\; e^{-k} \le |x| \le 1,\\ \omega _{n-1}^{-1/n} k^{(n-1)/n}&{}\quad \text{ if }\; 0\le |x| \le e^{-k}. \end{array}\right. } \end{aligned}$$

We have \(u_k \in W_0^{1,n}(\mathbf{B })\) and \(\Vert \nabla u_k\Vert _{L^n(\mathbf{B })} =1\). By the definition of \(S_{F,h}\) and (2.2), it holds

$$\begin{aligned} S_{F,h} \ge \int _\mathbf{B } F(u_k) h(|x|) dx&\ge \int _{\{|x|\le e^{-k}\}} F(u_k) h(|x|) dx\\&\ge F(\omega _{n-1}^{-1/n} k^{(n-1)/n})\omega _{n-1} \int _0^{e^{-k}} h(s) s^{n-1} ds\\&\ge \frac{1}{n+ {{\bar{\gamma }}}}F(\omega _{n-1}^{-1/n} k^{(n-1)/n})\omega _{n-1} h(e^{-k}) e^{-nk}. \end{aligned}$$

Denote \(t = \omega _{n-1}^{-1/n} k^{\frac{n-1}{n}} > (-\frac{n}{\alpha _n} \ln r_0)^{\frac{n-1}{n}}=:t_0\), we then have

$$\begin{aligned} F(t) h(e^{-\frac{\alpha _n}{n} t^{\frac{n}{n-1}}}) e^{-\alpha _n t^{\frac{n}{n-1}}} \le \frac{n+ {{\bar{\gamma }}}}{\omega _{n-1}}S_{F,h},\quad \forall \, t\ge t_0. \end{aligned}$$

Let \(t\rightarrow \infty \) we obtain \((\mathrm{A}3)\). The proof of Theorem 1.2 is then completed.

2.3 Proof of Theorem 1.3

Let \(\{u_k\}_k \subset W_{0,rad}^{1,n}(\mathbf{B })\) be a maximizing sequence for \(S_{F,h}\) i.e. \(\Vert \nabla u_k\Vert _{L^n(\mathbf{B })} \le 1\) and

$$\begin{aligned} \lim _{k\rightarrow \infty } \int _\mathbf{B } F(u_k) h(|x|) dx =S_{F,h}. \end{aligned}$$

By extracting a subsequence, we can assume that there exists a function \(u_0 \in W^{1,n}_{0,rad}(\mathbf{B })\) such that \(u_k\rightharpoonup u_0\) weakly in \(W^{1,n}_0(\mathbf{B })\), \(u_k \rightarrow u_0\) a.e. in \(\mathbf{B} \). We first assume that \(u_0 \equiv 0\). By \((\mathrm{A}1)\) there exists \(r_0\in (0,1)\) such that h is non-decreasing on \((0,r_0)\). For any \(\epsilon >0\), by (1.4), there is \(t_0 >0\) such that \(e^{-\frac{\alpha _n}{n} t_0^{\frac{n}{n-1}}} \le r_0\) and

$$\begin{aligned} F(t) h(e^{-\omega _{n-1}^{\frac{1}{n-1}} t^{\frac{n}{n-1}}}) \le \epsilon e^{\alpha _n t^{\frac{n}{n-1}}} \end{aligned}$$

for any \(t \ge t_0\). By (2.1) and the choice of \(t_0\), the set \(\{|u_k| \ge t_0\}\) is included in \(\{|x| \le r_0\}\). As in the proof of Theorem 1.1, we get

$$\begin{aligned} F(u_k) h(|x|) \le \epsilon e^{\alpha _n |u_k|^{\frac{n}{n-1}}}, \end{aligned}$$
(2.3)

if \(|u_k(x)| \ge t_0\). We claim that

$$\begin{aligned} \lim _{k\rightarrow \infty } \int _{\{|u_k|\le t_0\}} F(u_k) h(|x|) dx =0. \end{aligned}$$
(2.4)

Indeed, the function \(g_k(x):= F(u_k(x)) h(|x|) \chi _{\{|u_k|\le t_0\}}(x)\) is uniformly bounded in \(\mathbf{B }\) and \(g_k \rightarrow 0\) a.e. in \(\mathbf{B }\) since \(u_k \rightarrow 0\) a.e. in \(\mathbf{B }\). The claim (2.4) follows from the Lebesgue dominated convergence theorem. Using (2.3) and (2.4) we get

$$\begin{aligned} S_{F,h} + o(1) = \int _\mathbf{B } F(u_k) h(|x|) dx&= \int _{\{|u_k|\ge t_0\}}F(u_k) h(|x|) dx + \int _{\{|u_k|\le t_0\}}F(u_k) h(|x|) dx\\&\le \epsilon \int _{\{|u_k|\ge t_0\}}e^{\alpha _n |u_k|^{\frac{n}{n-1}}}dx + o(1)\\&\le \epsilon TM_n + o(1). \end{aligned}$$

Letting \(k\rightarrow \infty \) and then \(\epsilon \rightarrow 0\) we get \(S_{F,h} =0\) which is impossible since \(S_{F,h} >0\). Thus, \(u_0\not \equiv 0\).

To proceed, let us recall a concentration–compactness principle due to Lions [20] (which was sharpened by Černy et al. [10]): let \(\{v_k\}_k\) be a sequence in \(W^{1,n}_0(\Omega )\) such that \(v_k \rightharpoonup v_0\) weakly in \(W^{1,n}_0(\Omega )\) then

$$\begin{aligned} \limsup _{k\rightarrow \infty } \int _\Omega e^{\alpha _n p |v_k|^{\frac{n}{n-1}}} dx < \infty , \end{aligned}$$
(2.5)

for any \(p < (1 - \Vert \nabla v_0\Vert _{L^n(\Omega )}^n)^{-\frac{1}{n-1}}\). Evidently, if \(v_0\not \equiv 0\), we obtain a higher order integrability for the sequence \(\{e^{\alpha _n |v_k|^{\frac{n}{n-1}}}\}_k\) along the sequence \(\{v_k\}_k\).

Let \(\epsilon \) and \(t_0\) be chosen as above, we have

$$\begin{aligned} F(u_k) h(|x|) \le C \end{aligned}$$

on \(\{|u_k|\le t_0\}\) for some constant C depending only on F, h and \(t_0\). Combining this estimate together with (2.3), we have

$$\begin{aligned} F(u_k) h(|x|) \le C' e^{\alpha _n |u_k|^{\frac{n}{n-1}}} \end{aligned}$$

for some constant \(C'>0\) depending only on C and \(\epsilon \). By the concentration–compactness of Lions (2.5) above, we get that \(F(u_k) h(x)\) is bounded in \(L^p(\mathbf{B })\) for some \(p >1\) (since \(u_0\not \equiv 0\)). On the other hand, \(F(u_k) h(|x|) \rightarrow F(u_0) h(|x|)\) a.e. in \(\mathbf{B }\), then we have

$$\begin{aligned} S_{F,h} = \lim _{k\rightarrow \infty } \int _\mathbf{B } F(u_k) h(|x|) dx = \int _\mathbf{B } F(u_0) h(|x|) dx. \end{aligned}$$

Notice that \(\Vert \nabla u_0\Vert _{L^n(\mathbf{B })} \le 1\) by the lower semi-continuity of the Dirichlet norm under the weak convergence. Thus \(u_0\) is a maximizer for \(S_{F,h}\). This completes the proof of Theorem 1.3.

2.4 Proof of Theorem 1.4

In this subsection, we prove Theorem 1.4. The following result due to Carleson and Chang [9] (see also [14] for another proof) is crucial in our proof: Let \(\{u_k\}_k\) be a sequence in \(W^{1,n}_{0,rad}(\mathbf{B })\) such that \(u_k\) is non-increasing in |x|, \(\Vert \nabla u_k\Vert _{L^n(\mathbf{B })} \le 1\) for each k and \(|\nabla u_k|^n dx \rightharpoonup \delta _0\) weakly in the measure sense. Then

$$\begin{aligned} \limsup _{k\rightarrow \infty }\int _\mathbf{B } \left(e^{\alpha _n |u_k|^{\frac{n}{n-1}}} - 1\right) dx \le |\mathbf{B }| e^{1+ \frac{1}{2} + \cdots + \frac{1}{n-1}}. \end{aligned}$$
(2.6)

We next extend (2.6) without the non-increasing monotonicity assumption of \(u_k\), i.e., if \(\{u_k\}_k\) be a sequence in \(W^{1,n}_{0,rad}(\mathbf{B })\) such that \(\Vert \nabla u_k\Vert _{L^n(\mathbf{B })} \le 1\) for each k and \(|\nabla u_k|^n dx \rightharpoonup \delta _0\) weakly in the measure sense then (2.6) holds. Indeed, by (2.1) and the fact \(|\nabla u_k|^n dx \rightharpoonup \delta _0\) weakly in the measure sense, we get \(u_k\rightarrow 0\) a.e. in \(\mathbf{B }\). By extracting a subsequence, we assume that

$$\begin{aligned} \lim _{k\rightarrow \infty }\int _\mathbf{B } \left(e^{\alpha _n |u_k|^{\frac{n}{n-1}}} - 1\right) dx = \limsup _{k\rightarrow \infty }\int _\mathbf{B } \left(e^{\alpha _n |u_k|^{\frac{n}{n-1}}} - 1\right) dx. \end{aligned}$$

Let \(u_k^*\) be the non-increasing rearrangement function of \(u_k\) (see [7]). We have that

$$\begin{aligned} \Vert \nabla u_k^*\Vert _{L^n(\mathbf{B })} \le \Vert \nabla u_k\Vert _{L^n(\mathbf{B })} \le 1 \end{aligned}$$

for each k, and

$$\begin{aligned} \int _\mathbf{B } \left(e^{\alpha _n |u_k^*|^{\frac{n}{n-1}}} - 1\right) dx = \int _\mathbf{B } \left(e^{\alpha _n |u_k|^{\frac{n}{n-1}}} - 1\right) dx. \end{aligned}$$
(2.7)

By extracting a subsequence, we can assume \(u_k^* \rightharpoonup v\) weakly in \(W^{1,n}_0(\mathbf{B })\) for some function \(v \in W^{1,n}_0(\mathbf{B} )\) and \(u_k^* \rightarrow v\) a.e. in B. Suppose \(v\not \equiv 0\), by the concentration–compactness principle (2.5), we see that \(e^{\alpha _n |u_k^*|^{\frac{n}{n-1}}}\) is bounded in \(L^p(\mathbf{B })\) for some \(p >1\). So is \(e^{\alpha _n |u_k|^{\frac{n}{n-1}}}\). Since \(u_k\rightarrow 0\) a.e. in \(\mathbf{B }\), hence it holds

$$\begin{aligned} \lim _{k\rightarrow \infty } \int _\mathbf{B } \left(e^{\alpha _n |u_k|^{\frac{n}{n-1}}} - 1\right) dx =0 \le |\mathbf{B }| e^{1+ \frac{1}{2} + \cdots + \frac{1}{n-1}}. \end{aligned}$$

Suppose \(v\equiv 0\), we have two following cases. If \(|\nabla u_k^*|^n dx \rightharpoonup \delta _0\) weakly in the measure sense, then by (2.6) (for sequence \(\{u_k^*\}_k\)) and (2.7), we obtain

$$\begin{aligned} \limsup _{k\rightarrow \infty }\int _\mathbf{B } \left(e^{\alpha _n |u_k|^{\frac{n}{n-1}}} - 1\right) dx \le |\mathbf{B }| e^{1+ \frac{1}{2} + \cdots + \frac{1}{n-1}}. \end{aligned}$$

If weakly in the measure sense, then by extracting a subsequence, there exists \(\eta < 1\) and \(r_0 \in (0,1)\) such that

$$\begin{aligned} \int _\mathbf{B _{r_0}} |\nabla u_k^*|^n dx < \eta , \end{aligned}$$

for any k, here \(\mathbf{B} _{r}\) denotes the ball of radius \(r>0\) with centered at the origin. Define the function \(w_k(r) = u_k^*(r) -u_k^*(r_0)\) if \(r \le r_0\) and \(w_k(r)=0\) if \(r > r_0\). We have \(w_k \in W^{1,n}_{0,rad}(\mathbf{B} )\) and

$$\begin{aligned} \Vert \nabla w_k\Vert _{L^n(\mathbf{B })}^n = \int _{B_{r_0}} |\nabla u_k^*|^n dx< \eta < 1. \end{aligned}$$

Consequently, the sequence \(\{e^{\alpha _n |w_k|^{\frac{n}{n-1}}}\}_k\) is bounded in \(L^p(\mathbf{B })\) for some \(p >1\) by the classical Trudinger–Moser inequality (1.1). For \(r < r_0\), we have \(u_k^*(r) = w_k(r) + u_k^*(r_0)\) and hence by the convexity, we have

$$\begin{aligned} u_k^*(r)^{\frac{n}{n-1}} \le (1+ \epsilon ) w_k(r)^{\frac{n}{n-1}} + C_\epsilon u_k^*(r_0)^{\frac{n}{n-1}}, \end{aligned}$$

with \(C_\epsilon = (1 -(1+ \epsilon )^{1-n})^{-1/(n-1)}\). Recall that \(u_k^*(r_0) \rightarrow 0\) as \(k\rightarrow \infty \). Choosing \(\epsilon >0\) such that \(1+ \epsilon < p\), then the sequence \(\{e^{\alpha _n u_k^*(r)^{\frac{n}{n-1}}}\}_k\) is bounded in \(L^q(\mathbf{B} _{r_0})\) for some \(q >1\). So, we have

$$\begin{aligned} \lim _{k\rightarrow \infty } \int _\mathbf{B _{r_0}} \left(e^{\alpha _n u_k^*(r)^{\frac{n}{n-1}}} -1\right) dx =0. \end{aligned}$$
(2.8)

In \(\mathbf{B }\setminus \mathbf{B} _{r_0}\), the sequence \(\{u_k^*\}\) is uniformly bounded by (2.1). Moreover, \(u_k^*(r) \rightarrow 0\) a.e. in \(\mathbf{B }\). The Lebesgue dominated convergence theorem implies

$$\begin{aligned} \lim _{k\rightarrow \infty } \int _{\mathbf{B }\setminus \mathbf{B} _{r_0}} \left(e^{\alpha _n u_k^*(r)^{\frac{n}{n-1}}} -1\right) dx =0. \end{aligned}$$

Summing the two preceding limits, we get

$$\begin{aligned} \lim _{k\rightarrow \infty } \int _\mathbf{B } \left(e^{\alpha _n |u_k|(r)^{\frac{n}{n-1}}} -1\right) dx= & {} \lim _{k\rightarrow \infty } \int _\mathbf{B } \left(e^{\alpha _n u_k^*(r)^{\frac{n}{n-1}}} -1\right) dx\\= & {} 0 \le |\mathbf{B }| e^{1+ \frac{1}{2} + \cdots + \frac{1}{n-1}}. \end{aligned}$$

In summarization, we obtain the following extension of (2.6)

Lemma 2.1

Let \(\{u_k\}_k\) be a sequence in \(W^{1,n}_{0,rad}(\mathbf{B })\) such that \(\Vert \nabla u_k\Vert _{L^n(\mathbf{B })} \le 1\) for each k and \(|\nabla u_k|^n dx \rightharpoonup \delta _0\) weakly in the measure sense. Then

$$\begin{aligned} \limsup _{k\rightarrow \infty }\int _\mathbf{B } \left(e^{\alpha _n |u_k|^{\frac{n}{n-1}}} - 1\right) dx \le |\mathbf{B }| e^{1+ \frac{1}{2} + \cdots + \frac{1}{n-1}}. \end{aligned}$$

From Lemma 2.1, we easily obtain the following result: Let \(\{u_k\}_k\) be a sequence in \(W^{1,n}_{0,rad}(\mathbf{B })\) such that \(\Vert \nabla u_k\Vert _{L^n(\mathbf{B })} \le 1\) for each k and \(|\nabla u_k|^n dx \rightharpoonup \delta _0\) weakly in the measure sense, then it holds

$$\begin{aligned} \limsup _{k\rightarrow \infty }\int _{B_\delta } \left(e^{\alpha _n |u_k|^{\frac{n}{n-1}}} - 1\right) dx \le |\mathbf{B }| e^{1+ \frac{1}{2} + \cdots + \frac{1}{n-1}}. \end{aligned}$$
(2.9)

Proposition 2.2

Assume \((\mathrm{A}1){-}(\mathrm{A}2)\) and (1.6) with \(\beta =0\). Then for any sequence \(\{u_k\}_k\subset W_{0,rad}^{1,n}(\mathbf{B })\) with \(\Vert \nabla u_k\Vert _{L^n(\mathbf{B })} \le 1\) and \(u_k \rightharpoonup 0\) weakly in \(W_{0}^{1,n}(\mathbf{B })\), we have

$$\begin{aligned} \limsup _{k\rightarrow \infty } \int _\mathbf{B } F(u_k) h(|x|) dx \le |\mathbf{B }| e^{1+ \frac{1}{2} + \cdots + \frac{1}{n-1}}. \end{aligned}$$
(2.10)

Furthermore, if

$$\begin{aligned} F(t) \ge e^{\alpha _n |t|^{\frac{n}{n-1}}} -1 - \lambda |t|^{\frac{n}{n-1}},\quad t\in {\mathbb {R}}\end{aligned}$$
(2.11)

for some \(0\le \lambda < \alpha _n\) and \(h(x) = 1 + O(|x|^\alpha )\) as \(x \rightarrow 0\) for some \(\alpha >0\) then

$$\begin{aligned} S_{F,h} > |\mathbf{B }| e^{1+ \frac{1}{2} + \cdots + \frac{1}{n-1}}. \end{aligned}$$
(2.12)

Proof

By extracting a subsequence, we can assume that \(u_k\rightarrow 0\) a.e. in B and

$$\begin{aligned} \lim _{k\rightarrow \infty } \int _\mathbf{B } F(u_k) h(|x|) dx = \limsup _{k\rightarrow \infty } \int _\mathbf{B } F(u_k) h(|x|) dx. \end{aligned}$$

For any \(0< \epsilon < \frac{1}{2}\), by (1.6), there exists \(r_0\in (0,1)\) such that \(1-\epsilon \le h(r) \le 1+ \epsilon \) for any \(0\le r \le r_0\). From (1.5), there exists \(t_0 >0\) such that \(e^{-\frac{\alpha _n}{n} t_0^{\frac{n}{n-1}}} \le r_0\) and

$$\begin{aligned} F(t) h(e^{-\frac{\alpha _n}{n} t^{\frac{n}{n-1}}}) \le (1+ \epsilon ) \left(e^{\alpha _n t^{\frac{n}{n-1}}}-1\right),\quad t\ge t_0. \end{aligned}$$

So we have

$$\begin{aligned} F(t) \le \frac{1+ \epsilon }{1-\epsilon }\left(e^{\alpha _n t^{\frac{n}{n-1}}}-1\right) \le (1+ 2\epsilon ) \left(e^{\alpha _n t^{\frac{n}{n-1}}}-1\right),\quad t\ge t_0. \end{aligned}$$
(2.13)

We have the two following cases:

Case 1: in the measure sense. By extracting a subsequence, we can find \(0< \eta < 1\) and \(r_1 < r_0\) such that

$$\begin{aligned} \int _\mathbf{B _{r_1}} |\nabla u_k|^n dx < \eta ,\quad \forall \, k. \end{aligned}$$

Repeating the proof of (2.8), we get

$$\begin{aligned} \lim _{k\rightarrow \infty } \int _\mathbf{B _{r_1}} \left(e^{\alpha _n |u_k|^{\frac{n}{n-1}}} -1\right) dx =0. \end{aligned}$$
(2.14)

Denote \(t_1 = (-\frac{n}{\alpha _n} \ln r_1)^{\frac{n-1}{n}} > t_0\). By (2.1), we see that \(\{|u_k|\ge t_1\} \subset \{|x|\le r_1\}\). We next break the integral \(\int _\mathbf{B } F(u_k) h(|x|) dx\) as

$$\begin{aligned} \int _\mathbf{B } F(u_k) h(|x|) dx&= \int _{\{|u_k|\ge t_1\}} F(u_k) h(|x|) dx + \int _{\{|u_k|\ge t_1\}} F(u_k) h(|x|) dx\\&\le (1+ 2\epsilon ) \int _{\{|u_k|\ge t_1\}} \left(e^{\alpha _n |u_k|^{\frac{n}{n-1}}} -1\right) h(|x|) dx\\&\quad + \int _{\{|u_k|\ge t_1\}} F(u_k) h(|x|) dx\\&\le (1+ 2\epsilon ) \int _\mathbf{B _{r_1}} \left(e^{\alpha _n |u_k|^{\frac{n}{n-1}}} -1\right) h(|x|) dx\\&\quad + \int _{\{|u_k|\ge t_1\}} F(u_k) h(|x|) dx\\&\le (1+ 2\epsilon )(1+ \epsilon ) \int _\mathbf{B _{r_1}} \left(e^{\alpha _n |u_k|^{\frac{n}{n-1}}} -1\right) dx\\&\quad + \int _{\{|u_k|\ge t_1\}} F(u_k) h(|x|) dx, \end{aligned}$$

here the second inequality comes from (2.13). Using (2.4) (notice that its proof used only the continuity of F and h) we have

$$\begin{aligned} \lim _{k\rightarrow \infty } \int _{\{|u_k|\ge t_1\}} F(u_k) h(|x|) dx =0. \end{aligned}$$

Combining this limit together with (2.14), we then have

$$\begin{aligned} \limsup _{k\rightarrow \infty } \int _\mathbf{B } F(u_k) h(|x|) dx = 0 \le |\mathbf{B }| e^{1+ \frac{1}{2} + \cdots + \frac{1}{n-1}}. \end{aligned}$$

Case 2: \(|\nabla u_k|^n dx \rightharpoonup \delta _0\) in the measure sense. As in the proof of Case 1, we have

$$\begin{aligned}&\int _\mathbf{B } F(u_k) h(|x|) dx \le (1+ 2\epsilon )(1+ \epsilon ) \int _\mathbf{B _{r_0}} \left(e^{\alpha _n |u_k|^{\frac{n}{n-1}}} -1\right) dx\\&+ \int _{\{|u_k|\ge t_0\}} F(u_k) h(|x|) dx. \end{aligned}$$

Letting \(k\rightarrow \infty \) and using (2.9) and (2.4) again, we get

$$\begin{aligned} \limsup _{k\rightarrow \infty }\int _\mathbf{B } F(u_k) h(|x|) dx \le (1+ 2\epsilon )(1+ \epsilon )|\mathbf{B }| e^{1+ \frac{1}{2} + \cdots + \frac{1}{n-1}} \end{aligned}$$

for any \(\epsilon \in (0,1/2)\). Hence, we have

$$\begin{aligned} \limsup _{k\rightarrow \infty }\int _\mathbf{B } F(u_k) h(|x|) dx \le |\mathbf{B }| e^{1+ \frac{1}{2} + \cdots + \frac{1}{n-1}}. \end{aligned}$$

The proof of (2.10) is completed.

To prove (2.12), we use the following test functions:

$$\begin{aligned} u_k(x) = n^{-\frac{n-1}{n}} \omega _{n-1}^{-\frac{1}{n}} y_k(-n \ln r) \end{aligned}$$

with

$$\begin{aligned} y_k(t) = {\left\{ \begin{array}{ll} \frac{t}{k^{\frac{1}{n}}} (1-\delta _k)^{\frac{n-1}{n}} &{}\quad \text{ if }\; 0\le t \le k,\\ \frac{n-1}{(k(1-\delta _k))^{\frac{1}{n}}} \ln \frac{A_k +1}{A_k + e^{-(t-n)/(N-1)}} + (k(1-\delta _k))^{\frac{n-1}{n}} &{}\quad \hbox {if}\; t\le k \end{array}\right. } \end{aligned}$$

where \(\delta _k = 1 -2 \frac{\ln k}{k}\), and \(A_k\) is chosen such that \(\int _\mathbf{B } |\nabla u_k|^n dx =1\) (or equivalently, \(\int _0^\infty |y_k'(t)|^n dt =1\)). The choice of \(y_k\) is inspired by the test functions in [14]. It was shown in [14] that

$$\begin{aligned} A_k = \frac{1}{k^2} e^{-\sum _{i=1}^{n-1} \frac{1}{i}} + {\left\{ \begin{array}{ll} O(k^{-4}) &{}\,\,\,\hbox { if}\ n =2\\ O(k^{-3} (\ln k)^2)&{}\quad \text{ if }\; n\ge 3. \end{array}\right. } \end{aligned}$$

Let \(r_k = e^{-k/n}\). Since h is bounded in \(\mathbf{B }\), we have

$$\begin{aligned} \int _{\mathbf{B }\setminus \mathbf{B }_{r_k}}|u_k|^{\frac{n}{n-1}} h(|x|) dx&= \frac{\omega _{n-1}^{\frac{n-2}{n-1}}}{n^2}(1-\delta _k) k^{-\frac{1}{n}} \int _0^k t^{n-1} e^{-t} h(e^{-t/n}) dt\\&=\frac{\omega _{n-1}^{\frac{n-2}{n-1}}}{n^2}(1-\delta _k) k^{-\frac{1}{n}} \int _0^\infty t^{n-1} e^{-t} h(e^{-t/n}) dt + O(k e^{-k})\\&= \frac{c}{k^{\frac{1}{n-1}}} -2c \frac{\ln k}{k^{\frac{n}{n-1}}} + O(k e^{-k}), \end{aligned}$$

with

$$\begin{aligned} c = \frac{\omega _{n-1}^{\frac{n-2}{n-1}}}{n^2}\int _0^\infty t^{n-1} e^{-t} h(e^{-t/n}) dt. \end{aligned}$$

It follows from (2.11) that \(F(t) \ge (\alpha _n -\lambda ) |t|^{\frac{n}{n-1}}\) which implies

$$\begin{aligned} \int _{{\mathbf {B}}\setminus \mathbf{B} _{r_k}} F(u_k) h(x)\ge \frac{(\alpha _n-\lambda )c}{k^{\frac{1}{n-1}}} - 2c(\alpha _n -\lambda ) \frac{\ln k}{k^{\frac{n}{n-1}}} + O(k e^{-k}). \end{aligned}$$
(2.15)

There exist \(r_0 \in (0,1)\) and \(C >0\) such that \(h(r) \ge 1 -C r^\alpha \) for any \(r\in (0,r_0)\). For k large enough, we have \(r_k =e^{-k/n} \le r_0\). Using (2.11), we have

$$\begin{aligned} \int _\mathbf{B _{r_k}} F(u_k) h(|x|) dx&\ge (1 -C r_k^\alpha )\int _\mathbf{B _{r_k}} F(u_k) dx \ge \int _\mathbf{B _{r_k}} F(u_k) dx - C r_k^\alpha \int _\mathbf{B } F(u_k) dx \nonumber \\&\ge \int _\mathbf{B _{r_k}} \left(e^{\alpha _n |u_k|^{\frac{n}{n-1}}} -1 -\alpha _n |u_k|^{\frac{n}{n-1}}\right) dx + O(e^{-\alpha k/n}), \end{aligned}$$
(2.16)

here we used Theorem 1.1 for \(h\equiv 1\). From the definition of \(y_k\) and the value of \(A_k\), we have \(y_k(t)^{\frac{n}{n-1}} \le a k\) for some constant \(a>1\) and for any \(t\ge k\), hence

$$\begin{aligned} \int _\mathbf{B _{r_k}} |u_k|^{\frac{n}{n-1}} dx = \frac{\omega _{n-1}^{\frac{n-2}{n-1}}}{n^2} \int _k^\infty y_k(t)^{\frac{n}{n-1}} e^{-t} dt = O(k e^{-k}). \end{aligned}$$
(2.17)

From [14, Formula 2.11], we get

$$\begin{aligned} \int _\mathbf{B _{r_k}} e^{\alpha _n |u_k|^{\frac{n}{n-1}}} dx&= |\mathbf{B }| \int _k^\infty e^{y_k(t)^{\frac{n}{n-1}} -t} dt\ge |\mathbf{B }| e^{\sum _{i=1}^{n-1} \frac{1}{i}} + {\left\{ \begin{array}{ll} O(k^{-2})&{}\quad \text{ if }\; n=2,\\ O(k^{-1} (\ln k)^2)&{}\quad \text{ if }\; n\ge 3. \end{array}\right. } \end{aligned}$$

Inserting this estimate and (2.17) into (2.16), we arrive

$$\begin{aligned} \int _\mathbf{B _{r_k}} F(u_k) h(|x|) dx \ge |\mathbf{B} | e^{\sum _{i=1}^{n-1} \frac{1}{i}} + {\left\{ \begin{array}{ll} O(k^{-2})&{}\quad \text{ if }\;n=2,\\ O(k^{-1} (\ln k)^2)&{}\quad \text{ if }\; n\ge 3. \end{array}\right. } \end{aligned}$$
(2.18)

Combining (2.15) and (2.18), we get

$$\begin{aligned} S_{F,h}\ge \int _\mathbf{B } F(u_k) h(|x|) dx&\ge |\mathbf{B }| e^{\sum _{i=1}^{n-1} \frac{1}{i}}+ \frac{(\alpha _n-\lambda )c}{k^{\frac{1}{n-1}}} - 2c(\alpha _n -\lambda ) \frac{\ln k}{k^{\frac{n}{n-1}}} \\&\quad + {\left\{ \begin{array}{ll} O(k^{-2})&{}\quad \text{ if } \;n=2,\\ O(k^{-1} (\ln k)^2)&{}\quad \text{ if }\;n\ge 3. \end{array}\right. }\\&> |\mathbf{B }| e^{\sum _{i=1}^{n-1} \frac{1}{i}}, \end{aligned}$$

for k large enough since \(\lambda < \alpha _n\). This proves (2.12).

\(\square \)

We are now ready to prove Theorem 1.4. We first consider the case \(\beta =0\). Let \(\{u_k\}_k\) be a maximizing sequence for \(S_{F,h}\). By extracting a subsequence, we can assume that \(u_k \rightharpoonup u_0\) weakly in \(W^{1,n}_0(\mathbf{B })\) and \(u_k \rightarrow u_0\) a.e. in \(\mathbf{B }\). From Proposition 2.2, we have \(u_0 \not \equiv 0\). By (1.5), there is \(t_0>0\) such that

$$\begin{aligned} F(t) \le 2 e^{\alpha _n |t|^{\frac{n}{n-1}}},\quad |t|\ge t_0. \end{aligned}$$

Since F is bounded in \([0,t_0]\), hence there exists \(C >0\) such that

$$\begin{aligned} F(t) \le C e^{\alpha _n |t|^{\frac{n}{n-1}}},\quad \forall t. \end{aligned}$$
(2.19)

Since \(u_0\not \equiv 0\), then the sequence \(\{e^{\alpha _n |u_k|^{\frac{n}{n-1}}}\}_k\) is bounded in \(L^p(\mathbf{B })\) for some \(p >1\) by the concentration-compactness principle (2.5). So is \(F(u_k) h(|x|)\) by (2.19) and h is bounded. Consequently, we get

$$\begin{aligned} S_{F,h} = \lim _{k\rightarrow \infty } \int _\mathbf{B } F(u_k) h(|x|) dx = \int _\mathbf{B } F(u_0) h(|x|) dx. \end{aligned}$$

Notice that \(\Vert \nabla u_0\Vert _{L^n(\mathbf{B })} \le 1\) by the lower semi-continuity of the Dirichlet norm under the weak convergence. Thus \(u_0\) is a maximizer for \(S_{F,h}\).

We next consider \(\beta >0\). To prove this case, we use the change of variables. Let \(u\in W^{1,n}_{0,rad}(\mathbf{B })\) with \(\Vert \nabla u\Vert _{L^n(\mathbf{B })} \le 1\). Define

$$\begin{aligned} v(x) = (1+ \beta )^{(n-1)/n} u(|x|^{-\frac{\beta }{1+ \beta }} x). \end{aligned}$$

By the direct computations, we have

$$\begin{aligned} \int _\mathbf{B }|\nabla v|^{n} dx =\int _\mathbf{B } |\nabla u|^n dx \le 1, \end{aligned}$$

and

$$\begin{aligned} \int _\mathbf{B } F(u) h(x) dx&=\frac{1}{1+ \beta } \int _\mathbf{B } F((1+ \beta )^{-\frac{n-1}{n}} v(y)) |y|^{-\frac{n\beta }{1+\beta }} h(|y|^{\frac{1}{1+\beta }}) dy \nonumber \\&=\frac{1}{1+\beta } \int _\mathbf{B } {{\tilde{F}}}(v) {{\tilde{h}}}(|y|) dy, \end{aligned}$$
(2.20)

where

$$\begin{aligned} {{\tilde{F}}}(t) = F((1+ \beta )^{-\frac{n-1}{n}} t),\quad {{\tilde{h}}}(|y|) = |y|^{-\frac{n\beta }{1+\beta }} h(|y|^{\frac{1}{1+\beta }}) dy. \end{aligned}$$

Thus

$$\begin{aligned} S_{F,h} = \frac{1}{1+\beta } S_{{{\tilde{F}}},{{\tilde{h}}}}. \end{aligned}$$

It is easy to check that \({{\tilde{F}}}\) and \({{\tilde{h}}}\) satisfy \((\mathrm{A}1){-}(\mathrm{A}2)\), \({{\tilde{F}}}\) is \({{\tilde{h}}}\)-radially critical, \({{\tilde{h}}}(y) = 1 + O(|y|^{\frac{\alpha }{1+\beta }})\) as \(y\rightarrow 0\), and

$$\begin{aligned} {{\tilde{F}}}(t) \ge e^{\alpha _n |t|^{\frac{n}{n-1}}} - 1 - {{\tilde{\lambda }}} |t|^{\frac{n}{n-1}},\quad {{\tilde{\lambda }}} = \frac{\lambda }{1+ \beta }. \end{aligned}$$

Note that \(0\le {{\tilde{\lambda }}} < \alpha _n\) by (1.7). Thus the result for \(\beta =0\) implies the existence of an extremal v for \(S_{{{\tilde{F}}}, {{\tilde{h}}}}\). By (2.20), the function \(u(x) = (1+ \beta )^{-(n-1)/n} v(|x|^{\beta } x)\) is an extremal for \(S_{F,h}\).

The proof of Theorem 1.4 is then completed.\(\square \)