Least energy nodal solutions for elliptic equations with indefinite nonlinearity

Abstract

We prove the existence of a nodal solution with two nodal domains for the Dirichlet problem with indefinite nonlinearity \begin{equation*} -\Delta_p u = \lambda |u|^{p-2} u + f(x) |u|^{\gamma-2} u \end{equation*} in a bounded domain $\Omega \subset \mathbb{R}^n$, provided $\lambda \in (-\infty, \lambda^*_1)$, where $\lambda^*_1$ is a critical spectral value. The obtained solution has the least energy among all nodal solutions on the interval $(-\infty, \min\{\lambda^*_1, \lambda_2\})$, where $\lambda_2$ is the second Dirichlet eigenvalue of $-\Delta_p$ in $\Omega$. Moreover, the obtained solution forms a branch with continuous energy on $(-\infty, \lambda^*_1)$.

Keywords

• nodal solutions
• indefinite nonlinearity
• $p$-laplacian
• fibering method
• critical spectral parameter