Multiplicity Results for the Biharmonic Equation with Singular Nonlinearity of Super Exponential Growth in $\mathbb{R}^4$

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 $H^2\cap H^1_0(B_1(0))$.