1. Introduction and main results

The Hénon equation

(1.1)\begin{equation} -\Delta u = |x|^\ell u^{p-1}, \text{in } B, \quad u=0, \text{on } \partial B, \end{equation}

was introduced in [1.1] as a model for investigating spherically symmetric clusters of stars. In the equation, B is the unit ball of $\mathbb{R}^N$, $N \geq 3$, $\ell \gt 0$ and p > 2. Its mathematical significance grew after Ni’s paper [Reference Ni27], where the existence of positive radial solutions was established for $2 \lt p \lt 2^*+2\ell/(N-2)$. The crucial idea for solutions beyond the critical Sobolev exponent $2^* := 2N/(N-2)$ lies in the radial lemma proved there.

Since Ni’s work, equation (1.1) has been studied in several different perspectives. Since it is impossible to present a complete list, we only cite [Reference Badiale and Serra4–Reference Cao and Peng6, Reference de Figueiredo, dos Santos and Miyagaki8, Reference Guo, Li and Li15, Reference Guo, Guan and Wan16, Reference Liu and Peng21, Reference Luo, Tang and Xie23, Reference Moreira dos Santos and Pacella26, Reference Serra31, Reference Wei and Yan35] and references therein for further exploration. In addition to Ni’s paper, the work of Smets, Su, and Willem [Reference Smets, Willem and Su32] strongly influenced our investigation. They proved that, for large $\ell \gt 0$ and $p \in (2, 2^*)$, the ground state solution of (1.1) is non-radial. This phenomenon was first observed via numerical computations in [Reference Chen, Zhou and Ni7]. For more recent works on symmetry breaking for Hénon type equations we refer to [Reference Asai, Tanaka and Oishi3, Reference He19, Reference Pimenta and Gonzaga28].

In this paper, we split each vector $z\in\mathbb{R}^N$ as $z=(x,y) \in \mathbb{R}^{N_1} \times \mathbb{R}^{N_2}$ and define the Grushin operator

(1.2)\begin{equation} (\mathcal{G}_{\alpha}u) (z) := \Delta_x u(z) + |x|^{2\alpha}\Delta_y u(z), \end{equation}

where $\alpha \geq 0$, $N_1,\,N_2 \in \mathbb{N}$, and $\Delta_x$, $\Delta_y$ denote the usual Laplacian in the variables x, y, respectively. The operator (1.2) was first studied by Grushin in [Reference Grushin13, Reference Grushin14] when $\alpha \in \mathbb{Z}$ and first addressed in [Reference Franchi and Lanconelli10, Reference Franchi and Lanconelli11] for cases where $\alpha \not\in \mathbb{Z}$. Since then, this operator has been the subject of much recent research, see e.g., [Reference Alves and de Holanda1, Reference Duong and Nguyen9, Reference Loiudice22, Reference Monti and Morbidelli25] and references therein. In addition, we remark that this operator belongs to two more generals class of elliptic operators, see e.g. [Reference Gutiérrez and Lanconelli17, Reference Kogoj and Lanconelli20, Reference Luyen and Tri24].

We study the following Hénon type equation

(P)\begin{equation} \begin{cases} \begin{aligned} - \mathcal{G}_{\alpha} u & = |z|^{\ell} f(z,u), & z \in B,\\ u & = 0, & z \in \partial B, \end{aligned} \end{cases} \end{equation}

with $\ell \gt \alpha N_2$ and $f(z,s)$ behaving like $|s|^{p-2}s$ for some suitable values of p > 2. More specifically, we fix $\alpha \geq 0$, define

\begin{equation*} \widetilde{N} := N_1 + (1+\alpha)N_2, \qquad 2^\ast_\alpha := \frac{2\widetilde{N}}{\widetilde{N}-2}, \end{equation*}

and require that the nonlinear term f satisfies the following assumptions:

1. (f1) $f \in C(\overline{B} \times \mathbb{R}, \mathbb{R})$;

1. (f2) there exist

   \begin{equation*} 2 \lt p \lt 2^{\ast}_\alpha + \frac{2(\ell - \alpha N_2)}{\widetilde{N} - 2}, \end{equation*}

   and C > 0 such that

   \begin{equation*} |f(z,s)| \leq C \left( 1 + |s|^{p-1}\right), \quad \forall \,(z,s)\, \in \overline{B} \times \mathbb{R}. \end{equation*}

The natural space to deal with the Grushin operator is

\begin{equation*} H^1_{\alpha}(B) := \Big\{u \in L^2(B) : |\nabla_x u| \in L^2(B),\, |x|^{\alpha} |\nabla_y u|\in L^2(B)\Big\}, \end{equation*}

which becomes a Banach space (cf. [Reference Tri34, Theorem 4]) when endowed with the norm

(1.3)\begin{equation} ||u||_{H^{1}_{\alpha}(B)} := \left( \int_{B} \left[ |\nabla_{\alpha} u|^2 + |u|^2 \right] dz \right)^{1/2}, \end{equation}

where

\begin{equation*} \nabla_{\alpha} u(z) := \left( \nabla_x u(z), |x|^{\alpha} \nabla_y u(z) \right) \in \mathbb{R}^N. \end{equation*}

In line with the preceding studies concerning the Grushin operator, we aim to identify critical points of the Euler-Lagrange functional associated with Problem (P). The main novelty here is that we are able to consider nonlinearities which grows beyond the Grushin critical exponent $2^\ast_\alpha$, since we are assuming $\ell \gt \alpha N_2$. This can be done by a careful estimate of the decay rate of radial functions of our working space together with an embedding result in a wheighted L^p-Lebesgue space. More specifically, we define in $\S$ 2 an appropriated Banach space X consisting of radial functions satisfying

\begin{equation*} |u(z)| \leq C \frac{\|u\|}{|z|^{(\widetilde{N}-2)/2}}, \qquad \mbox{a.e. in } B, \end{equation*}

where $C=C(N) \gt 0$ does not depends on $u \in X$.

After establishing the variational framework, we revisit Problem (P) and impose conditions on f to utilize the full apparatus of Critical Point Theory (see [Reference Rabinowitz30]) for obtaining solutions. The range of problems that can be addressed is extensive. As an illustrative example, we focus on the classical superlinear setting, wherein we assume:

1. (f3) there holds

   \begin{equation*} \lim\limits_{s \to 0} \frac{f(z,s)}{s} = 0, \quad \mbox{uniformly in }\overline{B}; \end{equation*}

1. (f4) there exists µ > 2 and $s_0 \gt 0$ such that

   \begin{equation*} 0 \lt \mu F(z,s) \leq s f(z,s), \quad \forall \, z \in \overline{B},\,s \geq s_0, \end{equation*}

   where $F(z,s) := \int_0^{s} f(z,\tau) d\tau$.

As an application of the variational setting developed in $\S$ 2, we shall prove the following:

To the best of our knowledge, there is limited literature on the Grushin operator with Hénon-type nonlinearities. We first mention the paper by Duong and Nguyen [Reference Duong and Nguyen9], in which a nonexistence result is established for stable weak solutions of the equation

\begin{equation*} -\mathcal{G}_{\alpha} u + \nabla_\alpha w \cdot \nabla_\alpha u = |z|_\alpha^\ell |u|^{p-2} u, \quad z = (x,y) \in \mathbb{R}^{N_1} \times \mathbb{R}^{N_2}, \end{equation*}

with p > 2, $\ell \geq 0$, $|z|_\alpha := (|x|^{2(\alpha +1)} + |y|^2)^{1/(2(\alpha + 1))}$, and where the function w satisfies appropriate decay properties at infinity.

In [Reference Wei, Yang and Yu37], Wei, Yang, and Yu consider a slightly different operator, namely $\nabla_G = (\nabla_x, (1+\alpha)|x|^\alpha \nabla_y)$ and $\operatorname{div}_G u = \operatorname{div}_x u + (1+\alpha)|x|^\alpha \operatorname{div}_y u$. They obtained nontrivial solutions for the Lane–Emden equation

\begin{equation*} \operatorname{div}_G(\omega_1(z) |\nabla_G u|^{p-2} \nabla_G u) = \omega_2(z) |u|^{q-1} u, \quad z \in \mathbb{R}^{N_1} \times \mathbb{R}^{N_2}, \end{equation*}

with $p \geq 2$, $q \gt p-1$, and $\omega_i \in L^1_{\mathrm{loc}}(\mathbb{R}^N)$, $i = 1,2$, nonnegative and behaving like powers of $|z|_\alpha$.

Still for this type of operator and for p = 2, Wei, Chen, and Yang considered in [Reference Wei, Chen and Yang36] a Kirchhoff-type equation and obtained some Liouville-type results. We finally mention the recent paper by Quyet and Thang [Reference Quyet and Thang29], which considers the nonexistence of solutions for

\begin{equation*} -G_\alpha u + c(z) \cdot \nabla_G u = h(z) e^{u}, \quad z \in \mathbb{R}^{N_1} \times \mathbb{R}^{N_2}, \end{equation*}

where $G_\alpha = \Delta_x + (1+\alpha)^2 |x|^{2\alpha} \Delta_y$, the continuous function h behaves like a power of $|z|_G = (|x|^{2(\alpha + 1)} + (\alpha + 1)^2 |y|^2)^{1/(2(\alpha + 1))}$, and c is a smooth vector field satisfying a condition involving a variation of $|z|_G$.

Our first main result differs from and complements all of these findings.

In the homogeneous case where $f(z,s)=|s|^{p-2}s$, it is evident that the solution can be obtained through constrained minimization. When $p \in (2, 2^{\ast}_\alpha)$, it is natural to inquire whether the ground state solutions of the problem are always radial. In our final result, we leverage the action of a semigroup of dilations associated with the Grushin operator to obtain a symmetry-breaking result for large values of $\ell \gt 0$. More specifically, after performing careful estimates of the asymptotic behaviour of the ground state levels of our problem in the whole space and in the space of radial functions, we complement [Reference Smets, Willem and Su32, Theorem 3.1] by proving the following result:

The remainder of the paper is structured as follows: in the subsequent section, we present the variational framework to deal with our problem. Section 3 is dedicated to proving Theorem 1.1, while our final theorem is demonstrated in $\S$ 4.

2. A compact embedding for radial functions

Recall that the points $z\in \mathbb{R}^N$ are written as $z=(x,y) \in \mathbb{R}^{N_1} \times \mathbb{R}^{N_2}$. The ball centred $z \in \mathbb{R}^N$ with radius t > 0 is denoted by $B_t(z)$, as well as $B=B_1(0)$. We denote by $\|u\|_{L^t(B)}$ the L^t-norm of a measurable function $u \in L^t(B)$. Throughout the paper we assume α > 0.

As stated in the introduction, we consider

\begin{equation*} H^1_{\alpha}(B) := \Big\{u \in L^2(B) : |\nabla_x u| \in L^2(B),\, |x|^{\alpha} |\nabla_y u|\in L^2(B)\Big\}, \end{equation*}

with the norm $\|\cdot\|_{H^1_{\alpha}(B)}$ defined in (1.3). We also define the subspace $H_{0,\alpha}^1(B) := \overline{C^\infty_0(B)}^{||\cdot||_{H^1_{\alpha}(B)}}$ with the norm

\begin{equation*} \|u\| := \left( \int_B |\nabla_\alpha u (z)|^2 dz \right)^{1/2}, \end{equation*}

which is equivalent to $\|\cdot\|_{H^1_{\alpha}(B)}$ (cf. [Reference Kogoj and Lanconelli20, (1.8)]). This space is continuously embedded in $L^p(B)$, for $p \in [1, 2^\ast_\alpha]$ (see [Reference Kogoj and Lanconelli20, Theorem 3.3]). In addition, let

\begin{equation*} C^\infty_{0,r}(B) := \{u \in C^\infty_0(B) : u \text{ is radial}\} \end{equation*}

and $X := \overline{C^\infty_{0,r}(B)}^{H_{0,\alpha}^1(B)}$. Finally, for any p > 1 and $m \geq 0$, we set

\begin{equation*} L^{p}_{m}(B) := \left\{u \in L^1_{loc}(B) \,:\, \int_B |u(z)|^p|z|^{mp} dz \lt +\infty \right\}, \end{equation*}

which is a Banach space with the norm

\begin{equation*} \|u\|_{L^p_m} := \left( \int_B |u(z)|^p|z|^{mp} dz \right)^{1/p}. \end{equation*}

The main result of this section reads as:

Theorem 2.1. Let $N_2 \geq 2$, $m \geq 0$ and

\begin{equation*} \widetilde{m} := \begin{cases} \dfrac{2 N}{\widetilde{N}-2-2m}, & \text{if } m \lt (\widetilde{N}-2)/2,\\ +\infty, & \mbox{if } m \geq (\widetilde{N}-2)/2. \end{cases} \end{equation*}

Then, for any $p \in [1, \widetilde{m})$, the embedding $X \hookrightarrow L^{p}_{m}(B)$ is compact.

The key ingredient for the proof is the following radial lemma which is a version, for our setting, of a result presented in [Reference Ni27, p. 802]:

Proposition 2.2. If $N_2 \geq 2$ then, for any $u\in X$, there holds

(2.1)\begin{equation} |u(z)| \leq C \frac{\|u\|}{|z|^{(\widetilde{N}-2)/2}}, \qquad \mbox{a.e. in } B - \{0\}, \end{equation}

where $C=C(N) \gt 0$ depends only on N.

Proof. Let $u \in C^\infty_{0,r}(B)$ and consider $\varphi \in C^\infty_0(-1,1)$ such that $u(z) = \varphi(|z|)$. For any given $z_0 \in B-\{0\}$ and $\widetilde{z} \in \partial B$, we have that

\begin{align*} - u (z_0) = u(\widetilde{z}) - u (z_0) = \int_{|z_0|}^1 \varphi'(t) dt. \end{align*}

Since $\widetilde{N} = N_1 + (1+\alpha)N_2 \gt 2$, it follows from Hölder’s inequality that

\begin{align*} |u(z_0)| & \leq \int_{|z_0|}^1 |\varphi'(t)| dt = \int_{|z_0|}^1 |\varphi'(t)| t^{(\widetilde{N}-1)/2} t^{-(\widetilde{N}-1)/2}dt\\ & \leq \left(\int_{|z_0|}^1 |\varphi'(t)|^2 t^{\widetilde{N} - 1} dt \right)^{1/2} \left( \int_{|z_0|}^1 t^{-(\widetilde{N} - 1)} dt \right)^{1/2}. \end{align*}

After computing the last integral above, we get

\begin{equation*} |u(z_0)|^2 \leq \frac{1}{(\widetilde{N}-2)|z_0|^{\widetilde{N}-2}} \int_{|z_0|}^1 |\varphi'(t)|^2 t^{\widetilde{N} - 1} dt,\qquad \mbox{for a.e. } z_0 \in B. \end{equation*}

We now notice that $N_2 \geq 2$ implies

\begin{equation*} \int_{|z_0|}^1 |\varphi'(t)|^2 t^{\widetilde{N} - 1} dt = \int_{|z_0|}^1 t^{\alpha N_2} |\varphi'(t)|^2 t^{N - 1} dt \leq \int_{|z_0|}^1 t^{2 \alpha} |\varphi'(t)|^2 t^{N - 1} dt. \end{equation*}

We claim that

(2.2)\begin{align} \int_{|z_0|}^1 t^{2 \alpha} |\varphi'(t)|^2 t^{N - 1} dt = C_1 \int_{\{|z_0| \leq |z| \leq 1\}} |x|^{2\alpha} |\nabla_y u(z)|^2 dz, \end{align}

for some $C_1=C_1(N) \gt 0$, independent of u. If this is true, it follows from the inequalities above that

\begin{equation*} |u(z_0)|^2 \leq \frac{C_1}{(\widetilde{N}-2)} \frac{\displaystyle \int_B |x|^{2\alpha} |\nabla_y u(z)|^2 dz}{|z_0|^{\widetilde{N}-2}} \leq C^2 \frac{\|\nabla_\alpha u\|_{L^2(B)}^2}{|z_0|^{(\widetilde{N}-2)}},\qquad \mbox{for a.e. } z_0 \in B. \end{equation*}

with $C := [ C_1/(\widetilde{N}-2)]^{1/2} \gt 0$. So, the proposition holds for smooth functions and the result follows from the density of $C^\infty_{0,r}(B)$ in X.

It remains to prove (2.2). To do this, we denote by $w = (\xi, \eta) \in \partial B$ a general vector in the unit sphere of $\mathbb{R}^N$, with $\xi \in \mathbb{R}^{N_1}$ and $\eta \in \mathbb{R}^{N_2}$. Then, for any $i=1,2,\ldots,N_2$,

\begin{align*} \int_{\{|z_0| \leq |z| \leq 1 \}} |x|^{2\alpha} u_{y_i}^2(z) dz & = \int_{|z_0|}^1 \int_{\partial B_t(0)} |x|^{2\alpha} u_{y_i}^2(z) d\sigma_z dt \\ & = \int_{|z_0|}^1 \int_{\partial B} |t \xi|^{2\alpha} u_{y_i}^2(t w)t^{N-1} d\sigma_w dt. \end{align*}

Thus, since $u_{y_i}(x,y) = \varphi'(|z|)\frac{y_i}{|z|}$, it holds

\begin{align*} \int_{\{|z_0| \leq |z| \leq 1 \}} |x|^{2\alpha} u_{y_i}^2(z) dz & = \int_{|z_0|}^1 \int_{\partial B} t^{2\alpha}|\xi|^{2\alpha} \left[\varphi'(|t w |)\frac{t \eta_i}{|t w|}\right]^2 t^{N-1} d\sigma_w dt \\ & = \int_{\partial B} |\xi|^{2\alpha}\eta_i^2 d\sigma_w\int_{|z_0|}^1 t^{2\alpha} |\varphi'(t)|^2 t^{N-1} dt. \end{align*}

After summing on i, we get

\begin{align*} \int_{\{|z_0| \leq |z| \leq 1 \}} |x|^{2\alpha} |\nabla_y u(z)|^2 dz = C_2 \int_{|z_0|}^1 t^{2\alpha} |\varphi'(t)|^2 t^{N-1} dt, \end{align*}

where

\begin{equation*} C_2 := \int_{\partial B} |\xi|^{2\alpha} |\eta|^2 d\sigma_w = \int_{\partial B} |\xi|^{2\alpha} (1-|\xi|^2) d\sigma_w \gt 0, \end{equation*}

depends only on N, and therefore (2.2) holds true.

Before presenting the proof of Theorem 2.1 we recall that, for any $\gamma \in \mathbb{R}$, there holds

(2.3)\begin{equation} \int_B |z|^\gamma dz = \begin{cases} \dfrac{N\sigma_{N}}{(\gamma+N)}, & \mbox{if } \gamma \gt -N,\\ +\infty, & \mbox{if } \gamma \leq -N, \end{cases} \end{equation}

where σ_N stands for volume of the unit ball $ B \subset \mathbb{R}^N$.

Proof of Theorem 2.1

We first prove the continuity of the embedding for $p \in [1, \widetilde{m})$. To do this, pick $u \in X$ and apply (2.1) to get

\begin{equation*} \|u\|^p_{L^p_m} = \int_B |u|^p|z|^{mp} dz \leq C^p \|u\|^p \int_B |z|^{\gamma} dz, \end{equation*}

where

\begin{equation*} \gamma := \frac{p}{2} \left(2m-\widetilde{N}+2\right). \end{equation*}

If $m \lt (\widetilde{N}-2)/2$, we may use the definition of $\widetilde{m}$ and $p\in [1,\widetilde{m})$ to show that $\gamma+N \gt 0$. Hence, we obtain from (2.3) a constant $C_1=C_1(m,p,\alpha,N_1,N_2) \gt 0$ such that

(2.4)\begin{equation} \|u\|^p_{L^p_m} \leq C_1 \|u\|^p, \quad \forall \, u \in X. \end{equation}

When $m \geq (\widetilde{N}-2)/2$, we have that $\gamma \geq 0$ and therefore the embedding holds for any $p \geq 1$.

For the compactness, we pick $\beta \in (0,1)$ to be chosen later and apply Holder’s inequality with exponents $1/\beta$ and $1/(1-\beta)$, to get

\begin{align*} \int_B |u|^p|z|^{mp} dz & = \displaystyle \int_B |u|^\beta|u|^{p-\beta}|z|^{mp} dz \\ & \leq \|u\|_{L^1(B)}^\beta \left(\displaystyle\int_B |u|^{(p-\beta)/(1-\beta)}|z|^{mp/(1-\beta)}\right)^{1-\beta}. \end{align*}

So, if we set

\begin{equation*} p_\beta :=\frac{p-\beta}{1-\beta}, \quad m_\beta := \frac{mp}{p-\beta}, \end{equation*}

we have that

(2.5)\begin{equation} \|u\|_{L^p_m}^p \leq \|u\|_{L^1(B)}^\beta \|u\|_{L^{p_\beta}_{m_\beta}}^{p-\beta}. \end{equation}

It is clear from the definition of m_β that, if $m \lt (\widetilde{N}-2)/2$, then $m_\beta \lt (\widetilde{N}-2)/2$, for any β > 0 small. So, we can use $p \lt \widetilde{m}$ to get

\begin{equation*} \lim_{\beta \to 0} \left(p_\beta - \widetilde{m_\beta}\right) = \lim_{\beta \to 0} \left(\frac{p-\beta}{1-\beta} - \frac{2N}{\widetilde{N}-2-2m_\beta} \right) = p - \frac{2N}{\widetilde{N}-2-2m} \lt 0, \end{equation*}

and therefore we can pick $\beta_0 \gt 0$ small in such a way that $p_{\beta_0} \in [1,\widetilde{m_{\beta_0}})$. The same holds if $m \geq (\widetilde{N}-2)/2$, because in this case the inequality $m_\beta \gt m$ implies $\widetilde{m_\beta}=+\infty$.

Let $(u_n) \subset X$ be a sequence such that $u_n \rightharpoonup 0$ weakly in X. Since $p_{\beta_0} \in [1,\widetilde{m_{\beta_0}})$, we can use (2.5) and the embedding proved in the first part to get

\begin{equation*} \|u_n\|_{L^p_m}^p \leq C_1 \|u_n\|_{L^1(B)}^{\beta_0} \|u_n\|^{p-\beta_0} \leq C_2 \|u_n\|_{L^1(B)}^{\beta_0}, \end{equation*}

where we also have used the boundedness of $(u_n)$ in X. Since the embedding $X \hookrightarrow L^1(B)$ is compact (see [Reference Kogoj and Lanconelli20, Theorem 3.3]), up to a subsequence $u_n \to 0$ strongly in $L^1(B)$. This and the above expression imply that $u_n \to 0$ strongly in X. This finishes the proof of the theorem.

Given $u \in X$, we define as usual $u^-(x) := \max\{-u(x),0\}$ and $u^+:=u+u^-$. We finish this proving that $u^\pm \in X$, whenever $u \in X$. Since we did not find a clear state of this fact in the literature, we sketch the proof here for the sake of completeness.

3. Existence of solution

Using the compactness result of Theorem 2.1, we can obtain a solution for the Problem (P) in the following way:

Proof of Theorem 1.1

Since we are interested in nonnegative solutions, we can suppose without loss of generality that $f(x,s) = 0$ for s < 0. Formally, the energy functional associated to our problem is

\begin{equation*} I(u) := \frac{1}{2}\|u\|^2 - \int_{B} F(z, u) |z|^{\ell}dz, \qquad u \in X. \end{equation*}

To prove that I is well defined we notice that, for any given ɛ > 0, we may use (f1)-(f3) to obtain

\begin{equation*} |F(z,s)| \leq \frac{\varepsilon}{2}|s|^2 + C_1 |s|^{p}, \quad \forall \,(z,s) \in \overline{B} \times \mathbb{R}. \end{equation*}

Hence,

(3.1)\begin{equation} \int_B F(x,u) |z|^\ell dz \leq \frac{\varepsilon }{2} \|u\|_{L^2(B)}^2 + C_1 \|u\|_{L^p_{\ell/p}}^p. \end{equation}

The last integral above is finite whenever $1 \leq p \lt (\widetilde{\ell/p})$. If $\ell/p \geq (\widetilde{N}-2)/2$, this always happen, since $\widetilde{\ell/p}=+\infty$. In the case $\ell/p \lt (\widetilde{N}-2)/2$, the condition on p reads as

\begin{equation*} p \lt \widetilde{\left(\frac{\ell}{p}\right)} = \frac{2 N}{\widetilde{N} - 2 - 2 \frac{\ell}{p}} \quad \Longleftrightarrow \quad p \lt 2^{\ast}_\alpha + \frac{2(\ell - \alpha N_2)}{\widetilde{N} - 2}. \end{equation*}

So, we conclude that the function is well defined. Moreover, standard arguments shows that $I \in C^1(X,\mathbb{R})$ with

\begin{equation*} I'(u)v = \int_{B} \left(\nabla_\alpha u \cdot \nabla_\alpha v\right) dz - \int_{B} f(z,u) v|z|^{\ell} dz, \quad \forall\,\, u,v \in X, \end{equation*}

and therefore critical points of I are precisely the weak solutions of Problem (P).

Using (2.4) and (3.1) with $\varepsilon = 1/(2C_1)$, we get

\begin{equation*} I(u) \geq \frac{1}{4} \|u\|^2 - C_2 \|u\|^p = \|u\|^2 \left( \frac{1}{4}-C_2\|u\|^{p-2}\right), \end{equation*}

and therefore, since p > 2, we can find $\rho, \,\eta \gt 0$ such that

\begin{equation*} I(u) \geq \eta, \quad \forall\,u \in X,\,\, \|u\|=\rho. \end{equation*}

Moreover, using (f1) and (f4), we obtain $C_2 \gt 0$ such that

\begin{equation*} F(z,s) \geq C_2|s|^{\mu} - C_2 , \quad \forall\, (z,s) \in \overline{B} \times \mathbb{R}. \end{equation*}

So, if we choose a nonnegative function $u_0 \in C^\infty_{0,r}(B) - \{0\}$, we have that

\begin{equation*} I(su_0) \leq \frac{s^2}{2} \|u_0\|^2 - C_2 s^\mu \int_{B} |u_0|^\mu |z|^\ell dz + C_3. \end{equation*}

Once we can assume that $2 \lt \mu \lt 2^\ast_\alpha$, it follows that $I(s u_0) \to -\infty$, as $s\to +\infty$. Thus, there exists $e \in X$ such that $I(e) \leq 0$ and $\|e\| \gt \rho$.

The above considerations show that it is well defined

\begin{equation*} c := \underset{\gamma \in \Gamma}{\inf} \underset{t \in [0,1]}{\max} I(\gamma(t)) \geq \eta \gt 0 , \end{equation*}

where $\Gamma := \left\{\gamma \in C([0,1], X ): \gamma(0) = 0, \gamma(1) = e\right\}$. Hence, according to the Mountain Pass Theorem (cf. [Reference Ambrosetti and Rabinowitz2]), there exists $(u_n) \subset X$ such that

(3.2)\begin{equation} \lim_{n\to +\infty} I(u_n)=c,\qquad \lim_{n\to +\infty} I'(u_n)=0. \end{equation}

We claim that, along a subsequence, $u_n \to u$ strongly in X. Actually, we first use (f1) and (f4) to get

\begin{align*} I(u_n) - \frac{1}{\mu}I'(u_n)u_n & \geq \left(\frac{1}{2} - \frac{1}{\mu}\right) \|u_n\|^2 \\ & + \frac{1}{\mu} \int_{\{0 \leq |u_n| \leq s_0\}} \left[f(z, u_n)u_n - \mu F(z,u_n)\right] |z|^\ell dz\\ & \geq \left(\frac{1}{2} - \frac{1}{\mu}\right) \|u_n\|^2 - C_4. \end{align*}

Since µ > 2, we conclude that $(u_n)$ is bounded in X. Hence, up to a subsequece, we have that $u_n \rightharpoonup u$ weakly in X. Thus,

\begin{align*} o_n(1) = I'(u_n)(u_n - u) &= \|u_n\|^2 - \langle u_n, u \rangle - A_n \\ & = \|u_n\|^2 - \|u\|^2 - A_n + o_n(1), \end{align*}

where $o_n(1)$ denotes a quantity approaching zero as $n\to +\infty$ and

\begin{equation*} A_n := \int_B f(z,u_n)(u_n - u)|z|^\ell dz. \end{equation*}

To get the strong convergence claimed, we need only to check that $A_n \to 0$. Since the embedding $H_{0,\alpha}^1(B) \hookrightarrow L^1(B)$ is compact, we may assume $u_n \to u$ strongly in $L^1(B)$. Hence, it follows from (f1) and Holder’s inequality with exponents p and $p':=p/(p-1)$ that

\begin{align*} |A_n| &\leq C_5 \int_B \left( 1+|u_n|^{p-1}\right)|u_n-u||z|^\ell dz \\ & = o_n(1) + C_5 \int_B |u_n|^{p-1}|z|^{\ell/p'}|u_n-u||z|^{\ell/p} dz \\ & \leq o_n(1) + C_5 \left( \int_B |u_n|^p|z|^\ell dz \right)^{1/p'} \left( \int_B |u_n-u|^p|z|^\ell dz \right)^{1/p} \\ & \leq C_5 \|u_n\|_{L^p_{\ell/p}}^{p/p'} \|u_n-u\|_{L^p_{\ell/p}}. \end{align*}

Recalling that $p \lt (\widetilde{\ell/p})$, we may use the boundedness of $(u_n)$ in X and the compactness of the embedding $X \hookrightarrow L^p_{\ell/p}$, to conclude that the right-hand side above goes to zero, as $n\to +\infty$. This concludes the proof that $u_n \to u$ strongly in X.

From (3.2), the regularity of I and the strong convergence just proved, we obtain $I(u)=c \gt 0$ and $I'(u)=0$. This shows that $u \in X$ is a nonzero solution of Problem (P). Moreover, using Lemma 2.3, we obtain

\begin{align*} 0=I'(u)u^{-} & = \int_B \left[ \nabla_\alpha (u^+ - u^-) \cdot \nabla_\alpha u^-\right] dz - \int_{\{u \geq 0\}} f(z,u) u^- |z|^\ell dz \\ & = -\|u^-\|^2, \end{align*}

which implies $u \geq 0$ a.e. in B and concludes the proof of the theorem.

4. Symmetry breaking

Throughout this section, we assume that $2 \lt p \lt 2^\ast_\alpha$ and consider the ratio

\begin{equation*} R(u) := \frac{\|u\|^2 }{\left( \displaystyle\int_{B} |u|^{p}|z|^\ell dz \right)^{2/p}}, \qquad u \in H_{0,\alpha}^1(B) - \{0\}. \end{equation*}

We also introduce the minimizers

\begin{equation*} S_{\ell,p} := \displaystyle\inf_{\overset{u \in H_{0,\alpha}^1(B)}{u \neq 0}} R(u) = \inf \left\{\|u\|^2 : \int_B |u|^p|z|^{\ell} dz = 1, \,u \in H_{0,\alpha}^1(B) \right\}, \end{equation*}

and

\begin{equation*} S_{\ell,p}^R := \displaystyle\inf_{\overset{u \in X}{u \neq 0}} R(u) = \inf \left\{\|u\|^2 : \int_B |u|^p|z|^{\ell} dz = 1, \,u \in X \right\}. \end{equation*}

The principal ingredients for the proof of our last main theorem are the following estimates, which are versions of [Reference Smets, Willem and Su32, Theorems 4.1 and 4.2]:

Before proving this proposition, let us show how we can use it to get our symmetry breaking result:

Proof of Theorem 1.2

For any $p \in (2, 2^{*}_\alpha)$ fixed, it is sufficient to obtain $\ell^* \gt 0$ such that $S_{\ell, p} \lt S^R_{\ell, p}$, for any $\ell \geq \ell^*$. Suppose, by contradiction, that this is not the case. Then, since $S_{\ell,p} \leq S^{R}_{\ell,p}$ is always true, there exists a sequence $(\ell_n) \subset (0+\infty)$ such that $S_{\ell_n,p} = S^{R}_{\ell_n,p}$ and $\ell_n \to +\infty$. Using Proposition 4.1, we get

\begin{equation*} \mathcal{C}_{rad} \ell_n^{(p+2)/p} \leq S^{R}_{\ell_n,p} = S_{\ell_n,p} \leq \mathcal{C} \ell_n^{2 - \widetilde{N} + (2 \widetilde{N}/p)}, \end{equation*}

for any $n \in \mathbb{N}$ large. This implies

\begin{equation*} 0 \lt \frac{\mathcal{C}_{rad}}{\mathcal{C}} \leq \ell_n^{2 - \widetilde{N} + (2 \widetilde{N}/p)-1-(2/p)} = \ell_n^{(p-2)(1-\widetilde{N})/p}, \end{equation*}

which is impossible since $\ell_n \to +\infty$ and $(p-2)(1-\widetilde{N})/p \lt 0$.

We devote the next two subsections to the proof of Proposition 4.1.

4.1. Estimating $S^R_{\ell,p}$

We start this subsection with a technical lemma which is a consequence of the chain rule:

Lemma 4.2. Let $N \geq 2$ and $\varphi \in C^\infty_0(-1,1)$. Then, for any $\beta \in (0,1]$, the functions

\begin{equation*} v(z) := \varphi(|z|^{\beta}), \qquad w(z) := \varphi(|z|^{1/\beta}), \end{equation*}

belong to $H_{0,r}^1(B)$, the space of radial functions in $H^1_{0}(B)$.

Proof. Of course $v,\,w$ are radial, continuous and null on $\partial B$. Since φ is a C ¹-function with bounded derivatives, we need only to check that the maps $g_1(z) := |z|^{\beta}$ and $g_2(z) := |z|^{1/\beta}$ are in $H^1(B)$. Clearly $g_1,\,g_2 \in L^2(B)$. Moreover, for any z ≠ 0, we have that

\begin{equation*} |\nabla g_1 (z)|^2 = \beta^2 |z|^{2(\beta - 1)}, \qquad |\nabla g_2 (z)|^2 = \frac{1}{\beta^2} |z|^{2(1-\beta)/\beta}, \end{equation*}

and the result follows from (2.3).

Proposition 4.3. Let $u \in C^\infty_{0,r}(B)$ and $\varphi \in C^\infty_0(-1,1)$ be such that $u(z) = \varphi(|z|)$. Given $\beta \in (0,1]$, if we define $v(z) := \varphi(|z|^{\beta})$, then

(4.1)\begin{align} \int_B |\nabla_{\alpha} u|^2 dz \geq \frac{C}{\beta N \sigma_{N}} \int_B |\nabla v |^2 |z|^{(\beta - 1)(N-2)} dz, \end{align}

for some constant $C=C(N) \gt 0$. Moreover, if $\beta = N/(\ell+N)$, then

(4.2)\begin{align} \int_{B} |u|^{p}|z|^{\ell}dz = \beta \int_{B} |v|^p dz. \end{align}

Proof. One can easily check that

\begin{equation*} |\nabla_{\alpha} u (z)|^2 = [\varphi'(|z|)]^2 \left[\frac{|x|^2 + |x|^{2\alpha} |y|^2}{|z|^2} \right]. \end{equation*}

If we call $g_\alpha(z)$ the expression in brackets above and denote $w=(\xi,\eta)\in \partial B$, we get

\begin{align*} \int_B |\nabla_{\alpha} u(z)|^2 dz & = \int_0^1 \int_{\partial B} [\varphi'(|rw|)]^2 g_\alpha (rw) r^{N-1} d\sigma_w dr \\ & = \int_0^1 \int_{\partial B} [\varphi'(r)]^2 \left( |\xi|^2 + |r \xi|^{2\alpha} | \eta|^2 \right) r^{N-1} d\sigma_w dr \\ & \geq \left( \int_{\partial B} |\xi|^2 d\sigma_w \right) \int_0^1 [\varphi'(r)]^2 r^{N-1} dr. \\ & = C \int_0^1 [\varphi'(r)]^2 r^{N-1} dr, \end{align*}

with $C:= \int_{\partial B}|\xi|^2 d\sigma_w$. To check that C > 0, it is sufficient to pick $0 \lt \delta \lt 1$ and notice that the set $A_\delta := \{(\xi,\eta) \in \partial B\,:\,|\xi| \geq \delta\}$ has positive $(N-1)$-dimensional measure.

Using the change of variable $r = \rho^\beta$ in the last integral above and defining $\widetilde{v}(r) := \varphi(r^{\beta})$, we obtain

\begin{align*} \int_B |\nabla_{\alpha} u(z)|^2 dz &\geq \frac{C}{\beta} \int_0^1 [\varphi'(\rho^{\beta}) \beta \rho^{(\beta-1)}]^2 \frac{\rho^{\beta(N-1)}\rho^{\beta-1}}{\rho^{2(\beta-1)}} d\rho \\ & = \frac{C}{\beta} \int_0^1 [\widetilde{v}'(\rho)]^2 \frac{\rho^{\beta(N-1)}\rho^{\beta-1}\rho^{1-N}}{\rho^{2(\beta-1)}} \rho^{N-1} d\rho \\ & = \frac{C}{\beta N \sigma_N} \int_B [\widetilde{v}'(|z|)]^2 |z|^{(\beta-1)(N-2)} dz. \end{align*}

Since $|\nabla v(z)|^2 = [\widetilde{v}'(|z|)]^2$, the above expression yields (4.1).

Using the change of variables $r=\rho^\beta$ again, we get

\begin{align*} \int_{B} |u(z)|^{p} |z|^{\ell} dz &= N \sigma_{N}\int_0^1 |\varphi(r)|^p r^{\ell} r^{N-1}dr \\ &= \beta N \sigma_N \int_0^1 |\varphi(\rho^\beta)|^p \rho^{\left[ \beta(N-1)+\beta \ell + \beta -1+(1-N)\right]} \rho^{N-1} d\rho. \end{align*}

Thus, if $\beta = N/(\ell + N)$, we have that

\begin{equation*} \beta(N-1)+\beta \ell + \beta -1+(1-N) = \beta(N+\ell)-N=0, \end{equation*}

and therefore

\begin{equation*} \int_{B} |u(z)|^{p} |z|^{\ell} dz = \beta N \sigma_{N} \int_0^1 |\widetilde{v}(\rho)|^p \rho^{N-1} d\rho = \beta \int_B |v(z)|^p dz, \end{equation*}

which gives (4.2). This finishes the proof.

We are ready to present the estimation of $S_{\ell,p}^R$.

Proof of item (1) in Proposition 4.1

We first notice that, since the function $g(s) := 2s/(s-2)$ is decreasing in $(2, +\infty)$ and $2 \lt N \leq \widetilde{N}$, we have that $2 \lt p \lt 2^*_\alpha \leq 2^*$.

Let $u \in C^\infty_{0,r}(B) - \{0\}$ and $\varphi \in C^\infty_0(-1,1)$ such that $u(z) = \varphi(|z|)$. We also define $v(z):=\varphi(|z|^\beta)$, with $\beta=N/(\ell+N)$. By Lemma 4.2, we know that $v \in H_{0,r}^1(B)$. Hence, it follows from Proposition 4.3 that

\begin{equation*} R(u) \geq \frac{C}{\beta N \sigma_N} \frac{\displaystyle\int_B |\nabla v|^2|z|^{(\beta-1)(N-2)} dz}{\beta^{2/p} \left(\displaystyle\int_B |v|^p dz \right)^{2/p}} = C_1 \beta^{-(p+2)/p} \frac{\displaystyle\int_B |\nabla v|^2|z|^{(\beta-1)(N-2)} dz}{\left(\displaystyle\int_B |v|^p dz \right)^{2/p}}, \end{equation*}

where $C_1 = C/ (N \sigma_N)$. Let $H_{0,r}^1(B)$ be the space of radial functions of $H_0^1(B)$. As quoted in [Reference Smets, Willem and Su32, p. 473], we have that

\begin{equation*} D_{\beta,p} := \inf_{\overset{\phi \in H_{0,r}^1(B)}{\phi \neq 0}}\left\{\frac{\displaystyle\int_B |\nabla \phi |^2 |z|^{(\beta - 1)(N-2)} dz}{\left(\displaystyle\int_B |\phi|^p dz\right)^{2/p}} \right\} \gt 0, \end{equation*}

for any $\beta \in (0,1]$ and $p\in(2,2^*)$. If $|z| \lt 1$, the number $|z|^{(\beta-1)(N-2)}$ decreases as $\beta \in (0,1]$ increases. Hence, the function $\beta \mapsto D_{\beta,p}$ is non-increasing in $(0,1]$ and we may use that $2 \lt p \lt 2_\alpha^*\leq 2^*$ and $v \in H^1_{0,r}(B)- \{0\}$ (Lemma 4.2) to conclude that

\begin{equation*} R(u) \geq C_1 D_{1,p} \beta^{-(p+2)/p}, \end{equation*}

We now recall that $\beta = N/(\ell+N)$ to obtain

\begin{equation*} R(u) \geq C_1 D_{1,p} \left(\frac{\ell+N}{N}\right)^{(p+2)/p} \geq \mathcal{C}_{rad} \ell^{(p+2)/p},\quad \forall\, u \in C^\infty_{0,r}(B) - \{0\}, \end{equation*}

with

\begin{equation*} \mathcal{C}_{rad} := C_1 D_{1,p}N^{-(p+2)/p}. \end{equation*}

Using the continuous embedding $X \hookrightarrow L^p(B)$ and a density argument, we can prove that the above inequality also holds in X. Thus $ S^R_{\ell, p} \geq \mathcal{C}_{rad} \ell^{(p+2)/p}$, for any $\ell \gt 0$, and the proof is complete.

4.2. Estimating $S_{\ell,p}$

Given $\ell \gt 1$, we introduce the map

\begin{equation*} \delta_\ell(z) := \left(\ell x, \ell^{1+\alpha}y\right), \quad \forall\, z=(x,y) \in \mathbb{R}^{N_1} \times \mathbb{R}^{N_2}. \end{equation*}

It is a linear transformation with linear inverse given by

\begin{equation*} \delta_\ell^{-1}(z) := \left(\frac{x}{\ell},\frac{y}{\ell^{1+\alpha}} \right). \end{equation*}

We now consider

\begin{equation*} z_\ell := \left(0_x,\ell^{\alpha}-\ell^{\alpha-1},0_{y'}\right) \in \mathbb{R}^N, \end{equation*}

where 0_x and $0_{y'}$ are the null vector of $\mathbb{R}^{N_1}$ and $\mathbb{R}^{N_2 - 1}$, respectively. Define the maps $T,\,T^{-1}:\mathbb{R}^N \to \mathbb{R}^N$ by

\begin{equation*} T(z) := \delta_\ell^{-1}(z+\ell z_\ell), \qquad T^{-1}(z) := \delta_\ell(z) - \ell z_\ell. \end{equation*}

A simple computation shows that T ⁻¹ is the inverse of T.

Let $z \in B$ and notice that, since $\ell \gt 1$,

\begin{align*} |T(z)| & \leq |\delta_\ell^{-1}(z)|+|\delta_{\ell}^{-1} (\ell z_ \ell)| = \ell^{-1 }\left( |x|^2 + \frac{|y|^2}{\ell^{2\alpha}} \right)^{1/2} + (1 - \ell^{-1} ) \\ & \leq \ell^{-1} \left( |x|^2 + |y|^2 \right)^{1/2} + (1 - \ell^{-1} ) \lt \ell^{-1} + 1 - \ell^{-1}=1. \end{align*}

Hence,

\begin{equation*} B \subset A_{\ell} := \left\{z \in \mathbb{R}^N : |T(z)| \lt 1 \right\}. \end{equation*}

Another easy consequence of the definition of T and $A_{\ell}$ is that $ T(B) \subset T(A_\ell) = B$. Moreover, if $z \in B \backslash T(B)$, then $T^{-1}(z) \in A_\ell \backslash B$, which implies $|T^{-1}(z)| \geq 1 $ (cf. Figure 1).

Figure 1.

Application T.

We finally notice that, if $w = (\xi, \eta) \in B$ and $z = T(w)$, then

\begin{equation*} |z| \geq \left|\delta_\ell^{-1}(\ell z_\ell)\right| - \left|\delta_\ell^{-1}(\ell z_\ell) - z\right| = 1- \frac{1}{\ell} - \frac{1}{\ell}\sqrt{\xi^2 + \frac{\eta^2}{\ell^{2\alpha}}} \gt 1- \frac{1}{\ell} - \frac{1}{\ell} \sqrt{\xi^2+\eta^2}, \end{equation*}

and therefore

(4.3)\begin{equation} |z| \gt \frac{\ell-2}{\ell}, \quad \forall \, z \in T(B). \end{equation}

The most useful property of the map T is given by the following result:

Proposition 4.4. Let $u \in C^\infty_0(B)$ and $\ell \gt 2$. Then $v := (u \circ T^{-1})\in C_0^\infty(B)$ and

\begin{equation*} \|v\|^2 = \ell^{2 - \widetilde{N}} \|u\|^2, \qquad \int_{B} |v|^{p} |z|^{\ell} dz \gt \left( \frac{\ell-2}{\ell} \right)^{\ell} \ell^{-\widetilde{N}} \|u\|_{L^p(B)}^p. \end{equation*}

Proof. We start proving that $v \in C_0^\infty(B)$. Indeed, let $K \subset B$ be the support of u and notice that, if $z \not\in T(K)$, then $u(T^{-1}(z)) = 0$. Thus, $\textrm{supp}(v) \subset \overline{T(K)} = T(K) \subset T(B) \subset B$. We now compute

\begin{equation*} v_{x_i}(z) = \ell u_{x_i} (T^{-1}z), \qquad v_{y_j}(z) = \ell^{1+\alpha} u_{y_j}(T^{-1}(z)), \end{equation*}

for any $i = 1, \ldots, N_1$ and $j = 1, \ldots, N_2$. Thus,

\begin{align*} |\nabla_{\alpha} v (z)|^2 & = |\nabla_x v (z)|^2 + |x|^{2\alpha} |\nabla_y v (z)|^2 \\ & = \ell^2|\nabla_x u (T^{-1}(z))|^2 + \ell^{2(1 + \alpha)}|x|^{2\alpha} |\nabla_y u (T^{-1}(z))|^2 \\ & = \ell^2 |\nabla_{\alpha} u (T^{-1}(z))|^2, \end{align*}

and therefore, since the Jacobian of the transformation T is $\ell^{-(N_1 + N_2( 1 + \alpha))} = \ell^{- \widetilde{N}}$, we can use $B=T(A_\ell)$ to obtain

\begin{align*} \int_{B} |\nabla_{\alpha} v (z)|^2 dz & = \ell^2 \int_{T(A_\ell)} |\nabla_{\alpha} u (T^{-1}(z))|^2 dz \\ & = \ell^{2 - \widetilde{N}} \int_{A_\ell} |\nabla_{\alpha} u (z)|^2 dz \\ & = \ell^{2 - \widetilde{N}} \int_{B} |\nabla_{\alpha} u (z)|^2 dz + \ell^{2 - \widetilde{N}} \int_{A_\ell \backslash B} |\nabla_{\alpha} u (z)|^2 dz \\ & = \ell^{2 - \widetilde{N}} \int_{B} |\nabla_{\alpha} u (z)|^2 dz, \end{align*}

since the derivatives $u_{x_i}, u_{y_j}$ have support inside B. This proves the first statement of the proposition.

To check the second one, we recall that $|T^{-1}(z)| \geq 1 $, for any $z \in B \backslash T(B)$. Thus,

\begin{align*} \int_{B} |v(z)|^{p} |z|^{\ell}dz & = \int_{T(B)} |u(T^{-1}(z))|^p |z|^{\ell}dz + \int_{B \backslash T(B)} |u(T^{-1}(z))|^p |z|^{\ell}dz \\ &= \int_{T(B)} |u(T^{-1}(z))|^p |z|^{\ell} dz, \end{align*}

and it follows from (4.3) that

\begin{align*} \int_{B} |v(z)|^{p} |z|^{\ell} dz & \gt \left( \frac{\ell-2}{\ell} \right)^{\ell}\int_{T(B)} |u(T^{-1}(z))|^p dz \\ &= \left( \frac{\ell-2}{\ell} \right)^{\ell} \ell^{-\widetilde{N}} \int_{B} |u(z)|^p dz, \end{align*}

which is exactly the second statement of the proposition.

We are ready to prove the last result of this paper.

Proof of item (2) in Proposition 4.1

Let $u \in C^\infty_0(B)$ be fixed and $v := (u \circ T^{-1})$ as in Proposition 4.4. Since it is clear that $v \in H_{0,\alpha}^1(B)$, it follows from Proposition 4.4 that, for any $\ell \gt 2$,

\begin{equation} S_{\ell,p} \leq R(v) \lt \frac{\ell^{2-\widetilde{N}} \displaystyle\int_B |\nabla_\alpha u|^2 dz}{\left[ \left(\frac{\ell-2}{\ell} \right)^\ell \ell^{-\widetilde{N}} \displaystyle\int_B |u|^p dz\right]^{2/p}} \nonumber\\ = \frac{\ell^{2 - \widetilde{N} + (2 \widetilde{N}/p)}}{\left( \frac{\ell-2}{\ell}\right)^{2\ell/p}} \frac{\|u\|^2}{\|u\|_{L^p(B)}^2}. \end{equation}

We now define

\begin{equation*} g(s) := \left( \frac{s-2}{s}\right)^{s}, \qquad \forall \,s \geq \ell_0 \gt 2. \end{equation*}

By a straightforward computation, one obtains

\begin{equation*} g'(s) = g(s) \left[ \ln\left(\frac{s-2}{s}\right) + \frac{2}{s-2}\right]. \end{equation*}

If we call h(s) the expression in the brackets above, we have that

\begin{equation*} h'(s) = -\frac{4}{s(s-2)^2} \lt 0, \quad s \gt \ell_0. \end{equation*}

and therefore h is decreasing. Moreover, $\lim_{s\to +\infty} h(s) = \ln(1)=0$. Hence, we conclude that h is positive, which implies that g is increasing in $[\ell_0,+\infty)$. Thus,

\begin{equation*} 0 \lt \left( \frac{\ell_0-2}{\ell_0} \right)^{2\ell_0/p} \lt \left( \frac{\ell-2}{\ell} \right)^{2\ell/p}, \qquad \forall \,\ell \geq \ell_0. \end{equation*}

Coming back to (4.4), we get

\begin{equation*} S_{\ell,p} \leq \mathcal{C} \ell^{2 - \widetilde{N} + (2 \widetilde{N}/p)}, \quad \forall\, \ell \geq \ell_0, \end{equation*}

where

\begin{equation*} \mathcal{C} := \left( \frac{\ell_0-2}{\ell_0} \right)^{-2\ell_0/p} \frac{\|u\|^2}{\|u\|_{L^p(B)}^2}, \end{equation*}

and the proof is complete.