Multiplicity Results for the Biharmonic Equation with Singular Nonlinearity of Super Exponential Growth in ℝ4

Abstract

We consider the following elliptic problem of exponential-type growth posed in an open bounded domain with smooth boundary B₁ (0) ⊂ ℝ⁴: \((P_\lambda)\begin{cases}\Delta^{2}u = \lambda(u^{-\delta}+h(u)e^{u^{\alpha}}),\;\;u>0\;in\;B_{1}(0),\\\;\;\;\;\;u=\Delta{u}=0,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;on\;\partial{B}_{1}(0).\end{cases}\) Here Δ²(.):= −Δ(−Δ)(.) denotes the biharmonic operator, 1 < α < 2, 0 < δ < 1, λ > 0, and h(t) is assumed to be a smooth “perturbation” of \({e^{{t^\alpha }}}\) as t→∞ (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_λ) in \({H^2} \cap H_0^1({B_1}(0))\).