The systems of differential equations whose solutions exactly coincide with Bethe ansatz solutions for generalized Gaudin models are constructed. These equations are called the generalized spectral ${(}^1)$ Riccati equations, because the simplest equation of this class has a standard Riccatian form. The general form of these equations is $R_{n_i}[z_1\left(\lambda \right),...,z_r\left(\lambda \right)]=c_{n_i}\left(\lambda \right)$, i=1,..., r, where $R_{n_i}$ denote some homogeneous polynomials of degrees $n_i$ constructed from functional variables $z_i\left(\lambda \right)$ and their derivatives. It is assumed that $deg{\partial }^k{z}_i\left(\lambda \right)=k+1$. The problem...

We deal with a class on nonlinear Schrödinger equations (NLS) with potentials $V\left(x\right)\sim {\left|x\right|}^{-\alpha }$, $0<\alpha <2$, and $K\left(x\right)\sim {\left|x\right|}^{-\beta }$, $\beta >0$. Working in weighted Sobolev spaces, the existence of ground states $v_{\epsilon }$ belonging to $W^{1,2}\left({ℝ}^N\right)$ is proved under the assumption that $\sigma <p<(N+2)/\left(N-2\right)$ for some $\sigma ={\sigma }_{N,\alpha ,\beta }$. Furthermore, it is shown that $v_{\epsilon }$ are spikes concentrating at a minimum point of $𝒜=V^{\theta }{K}^{-2/\left(p-1\right)}$, where $\theta =(p+1)/\left(p-1\right)-1/2$.