Аннотация:
We consider the following elliptic problem of
exponential-type growth posed in an open bounded
domain with smooth boundary
$B_1(0)\subset \mathbb{R}^4$:
\begin{align*}
( P_\lambda)
\begin{cases}
\Delta^2 u = \lambda (u^{-\delta}+h(u)e^{u^\alpha}),
&\quad u>0\quad\text{in}\;B_1(0) ,
\\
\hphantom{\Delta^2}u=\Delta u = 0,&\quad\text{on}\;\partial B_1(0).
\end{cases}
\end{align*}
\noindent Here
$\Delta^2 (\,\cdot\,) := -\Delta(-\Delta)(\,\cdot\,)$
denotes the biharmonic operator,
$1<\alpha\leq 2$,
$0<\delta<1$,
$\lambda> 0$, and
$h(t)$
is assumed
to be a smooth “perturbation” of
$e^{t^{\alpha}}$
as
$t \to
\infty$
(see
(H1)–(H4)
below).
We employ variational methods in order to show
the existence of at least two distinct (positive) solutions to the
problem $(P_\lambda)$
in
$H^2\cap H^1_0(B_1(0))$.
Список цитирования
