X_1\oplus X_2 : \|u\|=R\}}} \mathcal J_\lambda(u)<{\displaystyle \inf_{\{u\in X_2\oplus X_3 : \|u\|_{X_0}=\rho\}}\mathcal J_\lambda(u)}\,.$$

First of all, let us show that $$\label{infpositivo}
{\displaystyle \inf_{\{u\in X_2\oplus X_3 : \|u\|_{X_0}=\rho\}}\mathcal J_\lambda(u)}>0\,.$$ For this purpose, let $u$ be a function in $X_2 \oplus X_3=\mathbb P_{k-1}$. By , we get that for any $\varepsilon>0$ $$\label{infJfrac0}
\begin{aligned}
\mathcal J_\lambda(u)& \geq \frac 1 2\int_{{\mathbb R}^n\times {\mathbb R}^n}|u(x)-u(y)|^2 K(x-y)dx\,dy-\frac \lambda 2\int_\Omega |u(x)|^2\,dx\\
& \qquad \qquad \qquad \qquad -\varepsilon\int_\Omega |u(x)|^2dx - C_{\varepsilon}\int_\Omega |u(x)|^q\,dx\,.
\end{aligned}$$ Moreover, by , we get that for any $u\in \mathbb P_{k-1}$ $$\lambda_k\,\int_\Omega |u(x)|^2\,dx \leq \int_{{\mathbb R}^n\times {\mathbb R}^n}|u(x)-u(y)|^2 K(x-y)dx\,dy\,,$$ so that this inequality and give $$\label{infJfrac0bis}
\begin{aligned}
\mathcal J_\lambda(u)& \geq \frac 1 2\left(1-\frac{\lambda}{\lambda_k}\right)\|u\|_{X_0}^2-\varepsilon\|u\|_{L^2(\Omega)}^2
 -C_{\varepsilon}\|u\|_{L^q(\Omega)}^q \\
& \geq \frac 1 2\left(1-\frac{\lambda}{\lambda_k}\right)\|u\|_{X_0}^2-\varepsilon|\Omega|^{(2^*-2)/2^*}
\|u\|_{L^{2^*}(\Omega)}^2 -|\Omega|^{(2^*-q)/2^*}C_{\varepsilon}\|u\|_{L^{2^*}(\Omega)}^q\,.
\end{aligned}$$ Here we used also the fact that $L^{2^*}(\Omega)\hookrightarrow L^\nu(\Omega)$ continuously, being $\Omega$ bounded and $\nu\in [2, 2^*)$ (here $\nu$ takes the values $2$ and $q$).

Using , and [@svmountain Lemma 6], we obtain that for any $\varepsilon>0$, $$\label{infJfrac20}
\begin{aligned}
\mathcal J_\lambda(u) & \geq \frac 1 2\left(1-\frac{\lambda}{\lambda_k}\right)\|u\|_{X_0}^2-\varepsilon c|\Omega|^{(2^*-2)/2^*}
 \int_{{\mathbb R}^n \times {\mathbb R}^n}\frac{|u(x)-u(y)|^2}{|x-y|^{n+2s}}\, dx\,dy \\
& \qquad -C_{\varepsilon}c^{q/2}|\Omega|^{(2^*-q)/2^*} \left( \int_{{\mathbb R}^n \times {\mathbb R}^n}\frac{|u(x)-u(y)|^2}{|x-y|^{n+2s}}\, dx\,dy\right)^{q/2}\\
& \ge \left[\frac 1 2\left(1-\frac{\lambda}{\lambda_k}\right) -\frac{\varepsilon
c|\Omega|^{(2^*-2)/2^*}}{\theta}\right]\int_{{\mathbb R}^n\times {\mathbb R}^n} |u(x)-u(y)|^2 K(x-y)\,dx\,dy\\
& \qquad -\frac{C_{\varepsilon}c^{q/2}|\Omega|^{(2^*-q)/2^*}}{\theta} \left( \int_{{\mathbb R}^n\times {\mathbb 