We consider the eigenvalue problem$$\begin{array}{c}-\mathrm{div}\left(a\right(\left|\nabla u\right|\left)\nabla u\right)=\lambda g(x,u)\text{in}\Omega \hfill \\ u=0\text{on}\partial \Omega ,\hfill \end{array}$$in the case where the principal operator has rapid growth. By using a variational approach, we show that under certain conditions, almost all $\lambda \>0$ are eigenvalues.

We consider the eigenvalue problem $$\begin{array}{c}{\displaystyle -\mathrm{div}\left(a\right(\left|\nabla u\right|\left)\nabla u\right)=\lambda g(x,u)\text{in}\Omega u=0\text{on}\partial \Omega ,}\hfill \end{array}$$ in the case where the principal operator has rapid growth. By using a variational approach, we show that under certain conditions, almost all λ > 0 are eigenvalues.

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 $\theta =(p+1)/\left(p-1\right)-1/2$.