Bifurcation beyond the principal eigenvalues for Neumann problems with indefinite weights

This paper is devoted to the study of the effects of indefinite weights on the following nonlinear Neumann problems
$$$ {(P^\pm)} \begin{cases} -\Delta u &= \lambda \, a(x) u \pm |u|^{p-1}u\ &\quad \hbox{in } \Omega \subset ℝ^N \\ \frac{\partial u}{\partial \nu} &=\ 0 \ & \hbox{on } \partial \Omega \end{cases}$$$
The function $$$ a = a(x)$$$ is assumed to be continuous and sign-changing. Then the linear part has two sequences of eigenvalues. Our results establish a relation between the position of the parameter $$$\lambda$$$ and the number of nontrivial classical solutions of these problems. The proof combines spectral analysis tools, variational methods and the Clark multiplicity theorem.

This paper is devoted to the study of the effects of indefinite weights on the following nonlinear Neumann problems
$$$ {(P^\pm)} \begin{cases} -\Delta u &= \lambda \, a(x) u \pm |u|^{p-1}u\ &\quad \hbox{in } \Omega \subset ℝ^N \\ \frac{\partial u}{\partial \nu} &=\ 0 \ & \hbox{on } \partial \Omega \end{cases}$$$
The function $$$ a = a(x)$$$ is assumed to be continuous and sign-changing. Then the linear part has two sequences of eigenvalues. Our results establish a relation between the position of the parameter $$$\lambda$$$ and the number of nontrivial classical solutions of these problems. The proof combines spectral analysis tools, variational methods and the Clark multiplicity theorem.

eigenvalues indefinite weight Neumann problems bifurcation

eigenvalues indefinite weight Neumann problems bifurcation