| Contents |
| Non homogeneous superlinear resonant problems were first studied by \cite{Ka-Or}
and more recently by \cite{De-Ya, Cu-De-Sr, Cu-DeC}.
The techniques used in these papers are deeply related to the linear character of the differential operator as well as on the simplicity of the first eigenvalue and the positivity of the corresponding eigenfunction. So the questions we are interested in are the extension of these results to other differential operators such as the $p$-Laplacian problem
\begin{equation}
\label{e}
\left\{
\begin{array}{cl}
-\Delta_p u=\lambda_1 |u|^{p-2}u+ (u^+)^q -f & \hbox{ in } \Omega,\\
u=0 & \hbox{ on }\partial \Omega,
\end{array}
\right.
\end{equation}
where $p>1$ and $-\Delta_pu:= -\mbox{div}\, (|\nabla u|^{p-2}\nabla u)$,
%defined on $W_0^{1,p} (\Omega)$
or to the fourth order problem
\begin{equation}
\label{e4}
\left\{
\begin{array}{cl}
\Delta^2 u=\lambda_1 u+ (u^+)^q -f & \hbox{ in } \Omega,\\
u=0, \quad \frac{\partial u}{\partial \nu}=0 & \hbox{ on }\partial \Omega,
\end{array}
\right.
\end{equation}
for which, depending on the domain, we can have sign changing first eigenfunction and multiple first eigenvalue.\\
\noindent{\bf Presented by M. Cuesta}
\bibitem{Cu-DeC} M. Cuesta and C. De Coster, {\it A resonant-superlinear elliptic problem revisited},
Advanced Nonlinear Studies 13 (2013), 97-114.
\bibitem{Cu-De-Sr} M. Cuesta, D. de Figueiredo and P.N. Srikanth, {\it On a resonant-superlinear elliptic problem},
Calc. Var. Partial Differential Equations 17 (2003), 221-233.
\bibitem{De-Ya} D. de Figueiredo and Yang Jianfu, {\it Critical Superlinear Ambrosetti-Prodi Problems}, Top. Meth. Nonlin. Anal.
14 (1999), 59-80.
\bibitem{Ka-Or} R. Kannan and R. Ortega, {\it Landesman-Lazer conditions for problems with ``one-sided unbounded" nonlinearities},
Nonl. Anal. T.M.A. 9 (1985), 1313-1317. |
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