Existence of a nontrivial self-similar biharmonic NLS profile

Let $\sigma$ lie in the $L^2$-supercritical and $H^2$-subcritical regime, namely

$$\begin{cases} 4/d<\sigma,&d\leq4,\\ 4/d<\sigma<4/(d-4),&d>4. \end{cases}$$

Consider the radial nonlinear eigenvalue problem for $\tilde S_{\mathrm B}(\rho)$ with $\nu=1$:

$$-\tilde S_{\mathrm B}+i\frac{\kappa^4}{4}\left(\frac{2}{\sigma}\tilde S_{\mathrm B}+\rho\tilde S_{\mathrm B}'\right)-\Delta_\rho^2\tilde S_{\mathrm B}+|\tilde S_{\mathrm B}|^{2\sigma}\tilde S_{\mathrm B}=0,$$

subject to $\tilde S_{\mathrm B}'(0)=\tilde S_{\mathrm B}”'(0)=0$ and $\tilde S_{\mathrm B}(\infty)=0$. Nontrivial profile existence conjecture. There exists a solution pair $\{\tilde S_{\mathrm B}(\rho),\tilde\kappa\}$ with $\tilde S_{\mathrm B}\not\equiv0$ and $\tilde\kappa>0$. The paper presents this as a nonlinear eigenvalue conjecture; no proof of existence is given.