Multiple solutions to boundary value problems for semilinear elliptic equations

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> \((N > 2)\), ](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 Schrodinger's equations.
 For more information see https://ejde.math.txstate.edu/Volumes/2021/48/abstr.html