Abstract
We are concerned with a class of $(p,q)$-Laplace type biharmonic Kirchhoff equations \[ \begin{cases} M\left( \int_{\Omega} \mathcal{A}(|\Delta u|^{p}) \, dx \right) \Delta(a(|\Delta u|^{p}) |\Delta u|^{p-2} \Delta u) = \lambda f(u) + |u|^{q_{2}^{*}-2} u &\textrm{in $\Omega$}, \\ u = \Delta u = 0 &\textrm{on $\partial \Omega$}, \end{cases} \] where $\Omega$ is a bounded open set in $\mathbb{R}^{N}$ with smooth boundary, $\lambda$ is a positive real parameter, $2 \leq p \lt q \lt q_{2}^{*}$, $q_{2}^{*} = \frac{Nq}{N-2q}$ is the critical exponent, $N \gt 2q$ and $\mathcal{A}(t) = \int_{0}^{t} a(s) \, ds$ for $t \in \mathbb{R}^{+}$. Here, $M \colon \mathbb{R}^{+} \to \mathbb{R}^{+}$ is a Kirchhoff function, $a \colon \mathbb{R}^{+} \to \mathbb{R}^{+}$ is a continuous function satisfying some properties and $f \colon \mathbb{R} \to \mathbb{R}$ is a function which can have an uncountable set of discontinuity points. In this article, we study the existence of a positive weak solution for the problem above involving critical growth and a discontinuous nonlinearity via mountain pass theorem.
Funding Statement
The first author Bae was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education (NRF-2019R1A6A1A10073079 and NRF-2020R1I1A1A01053570). Kim was supported by a National Research Foundation of Korea Grant funded by the Korean Government (MSIT) (NRF-2020R1C1C1A01006521).
Acknowledgments
The authors would like to thank the anonymous referee for her/his careful reading and helpful suggestions.
Citation
Jung-Hyun Bae. Jae-Myoung Kim. "Existence of Weak Solutions for a Class of $(p,q)$-biharmonic Equations with Critical Exponent and Discontinuous Nonlinearity." Taiwanese J. Math. 28 (4) 743 - 765, August, 2024. https://doi.org/10.11650/tjm/240405
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