Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 48, pp. 1-12.

Title: Multiple solutions to boundary value problems for semilinear elliptic equations
       
Authors: Duong Trong Luyen (Ton Duc Thang Univ., Ho Chi Minh City, Vietnam)
         Nguyen Minh Tri (Vietnam Academy of Science and Technology, Hanoi, Vietnam)

Abstract: 
 In this article, we study the multiplicity of weak
 solutions to the boundary value problem
 $$\displaylines{
 - \Delta u = f(x,u) + g(x,u) \quad \text{in } \Omega,\cr
 u= 0 \quad \text{on } \partial \Omega,
 }$$
 where &Omega; is a bounded domain with smooth boundary in R<sup>N</sup> (N &gt; 2),
 f(x,&xi;) is odd in &xi; 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 Schr&ouml;dinger'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.