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

Abstract

<abstract><p>We study the existence and orbital stability of normalized solutions of the biharmonic equation with the mixed dispersion and a general nonlinear term</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \gamma\Delta^2u-\beta\Delta u+\lambda u = f(u), \quad x\in\mathbb{R}^N \end{equation*} $\end{document} </tex-math></disp-formula></p> <p>with a priori prescribed $ L^2 $-norm constraint $ S_a: = \left\{u\in H^2(\mathbb{R}^N):\int_{\mathbb{R}^N}|u|^2dx = a\right\}, $ where $ a &gt; 0 $, $ \gamma &gt; 0, \beta\in\mathbb{R} $ and the nonlinear term $ f $ satisfies the suitable $ L^2 $-subcritical assumptions. When $ \beta\geq0 $, we prove that there exists a threshold value $ a_0\geq0 $ such that the equation above has a ground state solution which is orbitally stable if $ a &gt; a_0 $ and has no ground state solution if $ a &lt; a_0 $. However, for $ \beta &lt; 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 &lt; q &lt; p &lt; 2+8/N, \mu &lt; 0 $ under the case $ \beta &lt; 0 $, which does not satisfy the additional assumption. And we use the example to show that the energy in the case $ \beta &lt; 0 $ exhibits a more complicated nature than that of the case $ \beta\geq0 $.</p></abstract>