On existence of nodal solution to elliptic equations with convex-concave nonlinearities

Abstract

In a bounded connected domain $\Omega \subset \mathbb{R}^N$, $N \geqslant 1$, with a smooth boundary, we consider the Dirichlet boundary value problem for elliptic equation with a convex-concave nonlinearity \begin{equation*} \begin{cases} -\Delta u = \lambda |u|^{q-2} u + |u|^{\gamma-2} u, \quad x \in \Omega \\ u|_{\partial \Omega} = 0, \end{cases} \end{equation*} where $1< q< 2< \gamma < 2^*$. As a main result, we prove the existence of a nodal solution to this equation on the nonlocal interval $\lambda \in (-\infty, \lambda_0^*)$, where $\lambda_0^*$ is determined by the variational principle of nonlinear spectral analysis via fibering method.