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\begin{document}

\begin{center}
{\large Sensitivity analysis of the energy functional defined on Sobolev
space}

\bigskip

{\large J\'{u}lia Salamon}

{\large Sapientia University, Department of Mathematics and Informatics}

{\large E-mail: salamonjulia@sapientia.siculorum.ro}
\end{center}

\bigskip

\textbf{Abstract:} Let us consider the following partial differential equation
with a boundary condition:%
\[
\left(  P_{\lambda}\right)
%TCIMACRO{\QATOPD{\{}{.}{-\Delta u=\lambda f\left(  u\right)  ,x\in\Omega
%,}{u|_{\partial\Omega}=0,}}%
%BeginExpansion
\genfrac{\{}{.}{0pt}{}{-\Delta u=\lambda f\left(  u\right)  ,x\in
\Omega,}{u|_{\partial\Omega}=0,}%
%EndExpansion
\]
where $\Omega\in\mathbb{R}^{n}$\ is a compact set.

We assign an energy functional $E_{\lambda}:H_{0}^{1}\left(  \Omega\right)
\rightarrow\mathbb{R}$\ to this problem defined on Sobolev space $H_{0}%
^{1}\left(  \Omega\right)  $\ given by:%
\[
E_{\lambda}\left(  u\right)  =\frac{1}{2}\left\Vert u\right\Vert _{H_{0}^{1}%
}^{2}-\lambda\int_{\Omega}F\left(  u\left(  x\right)  \right)  dx,
\]
where%
\[
F\left(  s\right)  =\int_{0}^{s}f\left(  t\right)  dt.
\]


We know that $E_{\lambda}$ is continuous derivable and the critical points of
$E_{\lambda}$ are the weak solutions of the problem $\left(  P_{\lambda
}\right)  $.

The numerical calculation of the critical points is based on the Ricceri's
theorem: 

\begin{theorem}
[Ricceri]If the function $f$ is sublinear, then there exist an interval
$A\subset\left(  0,\infty\right)  $\ such that, for each $\lambda\in A$ the
functional $E_{\lambda}$ has at least three critical points, two local minimum
and one mountain pass point.
\end{theorem}

In the present work a numerical method is developed for the determination of
the $A\subset\left(  0,\infty\right)  $ interval for concrete problems.


\end{document}