New approaches for Schrödinger equations with prescribed mass: The Sobolev subcritical case and The Sobolev critical case with mixed dispersion (2210.14503v4)

Abstract: In this paper, we prove the existence of normalized solutions for the following Schr\"odinger equation \begin{equation*} \left{ \begin{array}{ll} -\Delta u-\lambda u=f(u), & x\in \R^N, \int_{\R^N}u^2\mathrm{d}x=c \end{array} \right. \end{equation*} with $N\ge3$, $c>0$, $\lambda\in \R$ and $f\in \mathcal{C}(\R,\R)$ in the Sobolev subcritical case with weaker $L^2$-supercritical conditions and in the Sobolev critical case when $f(u)=\mu |u|^{q-2}u+|u|^{2^*-2}u$ with $\mu>0$ and $2<q<2^*=\f{2N}{N-2}$ allowing to be $L^2$-subcritical, critical or supercritical. Our approach is based on several new critical point theorems on a manifold, which not only help to weaken the previous $L^2$-supercritical conditions in the Sobolev subcritical case, but present an alternative scheme to construct bounded (PS) sequences on a manifold when $f(u)=\mu |u|^{q-2}u+|u|^{2^*-2}u$ technically simpler than the Ghoussoub minimax principle involving topological arguments, as well as working for all $2<q<2^*$. In particular, we propose new strategies to control the energy level in the Sobolev critical case which allow to treat, in a unified way, the dimensions $N=3$ and $N\ge 4$, and fulfill what were expected by Soave and by Jeanjean-Le . We believe that our approaches and strategies may be adapted and modified to attack more variational problems in the constraint contexts.