Mathematics > Home > Advances in Pure and Applied Mathematics > Issue 2 (Special CSMT 2022) > Article
Marta Calanchi
Università degli Studi di Milano
Italy
Bernhard Ruf
Università degli Studi di Milano
Italy
Published on 7 March 2023 DOI : 10.21494/ISTE.OP.2023.0935
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.