Sign-changing saddle point (Q1766545)

The author considers semilinear elliptic bundary value problem \[ -\Delta u= f(x,u)\quad\text{in }\Omega,\qquad u= 0\quad\text{on }\partial\Omega \] and the Schrödinger equation \[ -\Delta u+ V_\lambda(x) u= f(x,u),\;x\in\mathbb R^N,\;u(x)\to 0\quad\text{as }|x|\to \infty, \] where \(\Omega\subset\mathbb R^N\) is a bounded domain with smooth boundary \(\partial\Omega\); the Schrödinger operator \(-\Delta+ V_\lambda(x)\) has both eigenvalues and essential spectrum. The author applies a new abstract result to study the existence of sign-changing solutions.