Electron. J. Differential Equations, Vol. 2021 (2021), No. 48, pp. 1-12.

Multiple solutions to boundary value problems for semilinear elliptic equations

Duong Trong Luyen, Nguyen Minh Tri

In this article, we study the multiplicity of weak solutions to the boundary value problem

where Ω is a bounded domain with smooth boundary in RN (N > 2), f(x,ξ) is odd in ξ and g is a perturbation term. Under some growth conditions on f and g, we show that there are infinitely many solutions. Here we do not require that f be continuous or satisfy the Ambrosetti-Rabinowitz (AR) condition. The conditions assumed here are not implied by the ones in [3,15]. We use the perturbation method by Rabinowitz combined with estimating the asymptotic behavior of eigenvalues for Schrödinger's equations.

Submitted September 18, 2019. Published May 28, 2021.
Math Subject Classifications: 35J60, 35B33, 35J25, 35J70.
Key Words: Semilinear elliptic equations; multiple solutions; critical points; perturbation methods; boundary value problem.

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Duong Trong Luyen
Division of Computational Mathematics and Engineering
Institute for Computational Science, and
Faculty of Mathematics and Statistics
Ton Duc Thang University, Ho Chi Minh City, Vietnam
email: duongtrongluyen@tdtu.edu.vn
  Nguyen Minh Tri
Institute of Mathematics
Vietnam Academy of Science and Technology
18 Hoang Quoc Viet
10307 Cau Giay, Hanoi, Vietnam
email: triminh@math.ac.vn

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