Existence and stability of normalized solutions to the mixed dispersion nonlinear Schrödinger equations

We study the existence and orbital stability of normalized solutions of the biharmonic equation with the mixed dispersion and a general nonlinear term

$$\gamma {\mathrm{\Delta }}^{2}u-\beta \mathrm{\Delta }u+\lambda u=f(u),x\in {\mathbb{R}}^N$$

with a priori prescribed $L^2$-norm constraint $S_a:=\left\{u\in H^2({\mathbb{R}}^N):{\int }_{{\mathbb{R}}^N}|u{|}^{2}dx=a\right\},$ where $a>0$, $\gamma >0,\beta \in \mathbb{R}$ and the nonlinear term $f$ satisfies the suitable $L^2$-subcritical assumptions. When $\beta \ge 0$, we prove that there exists a threshold value $a_0\ge 0$ such that the equation above has a ground state solution which is orbitally stable if $a>a_0$ and has no ground state solution if $a<a_0$. However, for $\beta <0$, this case is more involved. Under an additional assumption on $f$, we get the similar results on the existence and orbital stability of ground state. Finally, we consider a specific nonlinearity $f(u)=|u{|}^{p-2}u+\mu |u{|}^{q-2}u,2<q<p<2+8/N,\mu <0$ under the case $\beta <0$, which does not satisfy the additional assumption. And we use the example to show that the energy in the case $\beta <0$ exhibits a more complicated nature than that of the case $\beta \ge 0$.