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Infinitely Many Solutions for a Fourth-Order Semilinear Elliptic Equations Perturbed from Symmetry

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Abstract

In this paper, we study the existence of multiple solutions for the following biharmonic problem

$$\begin{aligned} \Delta ^2 u= & {} f(x,u) + g(x,u)\quad \hbox {in}\quad \Omega ,\\ u= & {} \Delta u =0 \quad \hbox {on}\quad \partial \Omega , \end{aligned}$$

where \(\Omega \subset {\mathbb {R}}^N, (N > 4)\) is a smooth bounded domain and \(f(x,\xi )\) is odd in \(\xi , g(x,\xi )\) is a perturbation term. By using the variant of Rabinowitz’s perturbation method, under some growth conditions on f and g, we show that there are infinitely many weak solutions to the problem.

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Acknowledgements

This research is funded by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant No. 101.02–2020.13. The author warmly thanks the anonymous referees for the careful reading of the manuscript and for their useful and nice comments.

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Authors and Affiliations

  1. Division of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    Duong Trong Luyen

  2. Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    Duong Trong Luyen

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  1. Duong Trong Luyen
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Correspondence to Duong Trong Luyen.

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Communicated by Maria Alessandra Ragusa.

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Luyen, D.T. Infinitely Many Solutions for a Fourth-Order Semilinear Elliptic Equations Perturbed from Symmetry. Bull. Malays. Math. Sci. Soc. 44, 1701–1725 (2021). https://doi.org/10.1007/s40840-020-01031-5

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  • DOI: https://doi.org/10.1007/s40840-020-01031-5

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