Multiple solutions to boundary value problems for semilinear strongly degenerate elliptic differential equations

Abstract

In this paper, we study the existence of multiple solutions for the boundary value problem

$$\begin{aligned} \begin{array}{llll} -\Delta _{\gamma } u&{}= f(x,u) + g(x,u) &{} \text{ in } &{} \Omega , \\ u&{}= 0 &{} \text{ on } &{} \partial \Omega , \end{array} \end{aligned}$$

where \(\Omega\) is a bounded domain with smooth boundary in \(\mathbb {R}^N \ (N \ge 2),\) \(f(x,\xi ), g(x,\xi )\) are Carathéodory functions, \(f(x,\xi )\) is odd in \(\xi\), \(g(x,\xi )\) is perturbation term and \(\Delta _{\gamma }\) is the strongly degenerate elliptic operator of the type

$$\begin{aligned} \Delta _\gamma : =\sum \limits _{j=1}^{N}\partial _{x_j} \left( \gamma _j^2 \partial _{x_j} \right) , \quad \partial _{x_j}: =\frac{\partial }{\partial x_{j}},\quad \gamma : = (\gamma _1, \gamma _2,\ldots , \gamma _N). \end{aligned}$$

We use the minimax method and Rabinowitz’s perturbation method. This result is a generalization of that of Luyen and Tri (Complex Var Elliptic Equ 64(6):1050–1066, 2019).

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Acknowledgements

The author warmly thanks the anonymous referees for the careful reading of the manuscript and for their useful and nice comments. This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.02–2020.13.

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Correspondence to Duong Trong Luyen.

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Luyen, D.T., Van Cuong, P. Multiple solutions to boundary value problems for semilinear strongly degenerate elliptic differential equations. Rend. Circ. Mat. Palermo, II. Ser (2021). https://doi.org/10.1007/s12215-021-00594-x

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Keywords

  • Semilinear strongly degenerate elliptic equations
  • Boundary value problems
  • Critical points
  • Perturbation methods
  • Multiple solutions

Mathematics Subject Classification

  • Primary 35J60
  • Secondary 35B33
  • 35J25
  • 35J70