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Multiplicity results for the biharmonic equation with singular nonlinearity of super exponential growth in \(\mathbb{R}^4\). (English) Zbl 1422.35082

Summary: We consider the following elliptic problem of exponential-type growth posed in an open bounded domain with smooth boundary \(B_1(0) \subset \mathbb{R}^4\): \[(\mathrm{P}_\lambda) \begin{cases}\Delta^{2}u &= \lambda(u^{-\delta}+h(u)e^{u^{\alpha}}),\quad u>0\text{ in }B_1(0),\\u&=\Delta{u}=0,\quad \text{ on }\partial B_1(0).\end{cases}\] Here \(\Delta^2(\cdot):= -\Delta(-\Delta)(\cdot)\) denotes the biharmonic operator, \(1 < \alpha < 2\), \(0 < \delta < 1\), \(\lambda > 0\), and \(h(t)\) is assumed to be a smooth “perturbation” of \(e^{t^\alpha}\) as \(t\rightarrow\infty\) (see (H1)–(H4) below). We employ variational methods in order to show the existence of at least two distinct (positive) solutions to the problem \((\mathrm{P}_\lambda)\) in \(H^2 \cap H_0^1(B_1(0))\).

MSC:

35J91 Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian
31B30 Biharmonic and polyharmonic equations and functions in higher dimensions
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