Mathematical Notes

, Volume 105, Issue 3–4, pp 404–424 | Cite as

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

  • K. SaoudiEmail author
  • M. KratouEmail author
  • E. Al ZahraniEmail author


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 (Pλ) in \({H^2} \cap H_0^1({B_1}(0))\).


biharmonic equation multiple solutions super exponential growth Dirichlet boundary conditions 


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© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Department of MathematicsImam Abdulrahman Bin Faisal UniversityDammamKingdom of Saudi Arabia
  2. 2.Basic and Applied Scientific Research CenterImam Abdulrahman Bin Faisal UniversityDammamKingdom of Saudi Arabia

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