Extremal values of $L^2$-Pohozaev manifolds and their applications

In this paper, we consider the following Schr\"{o}dinger equation: \begin{equation*} \begin{cases} -\Delta u=\lambda u+\mu|u|^{q-2}u+|u|^{2^*-2}u\quad\text{in }\mathbb{R}^N,\\ \int_{\mathbb{R}^N}|u(x)|^2dx=a,\quad u\in H^1(\mathbb{R}^N),\\ \end{cases} \end{equation*} where $N\ge 3$, $2<q<2+\frac{4}{N}$, $a, \mu>0$, $2^*=\frac{2N}{N-2}$ is the critical Sobolev exponent and $\lambda\in \mathbb{R}$ is one of the unknowns in the above equation which appears as a Lagrange multiplier. By applying the minimization method on the $L^2$-Pohozaev manifold, we prove that if $N\geq3$, $q\in\left(2,2+\frac{4}{N}\right)$, $a>0$ and $0<\mu\leq\mu^{*}_{a}$, then the above equation has two positive solutions which are real valued, radially symmetric and radially decreasing, where \begin{equation*} \mu^*_a=\frac{(2^*-2)(2-q\gamma_q)^{\frac{2-q\gamma_q}{2^*-2}}}{\gamma_q(2^*-q\gamma_q)^{\frac{2^*-q\gamma_q}{2^*-2}}}\inf_{u\in H^1(\mathbb{R}^N), \|u\|_{2}^2=a}\frac{\left(\|\nabla u\|_2^2\right)^\frac{2^*-q\gamma_q}{2^*-2}}{\|u\|_q^q\left(\|u\|_{2^*}^{2^*}\right)^{\frac{2-q\gamma_q}{2^*-2}}}. \end{equation*} Our results improve the conclusions of \cite{JeanjeanLe2021,JeanjeanJendrejLeVisciglia2022,Soave2020-2,WeiWu2022} and we hope that our proofs and discussions in this paper could provide new techniques and lights to understand the structure of the set of positive solutions of the above equations.