Existence and multiplicity for a superlinear elliptic problem under a non-quadradicity condition at infinity

Abstract

In this article, we study the existence and multiplicity of solutions of the boundary-value problem $$\displaylines{ -\Delta u = f(x,u), \quad \text{in } \Omega, \cr u = 0, \quad \text{on } \partial\Omega, }$$ where \(\Delta\) denotes the N-dimensional Laplacian, \(\Omega\) is a bounded domain with smooth boundary, \(\partial\Omega\), in \(\mathbb{R}^N\) \((N\geq 3)\), and f is a continuous function having subcritical growth in the second variable. Using infinite-dimensional Morse theory, we extended the results of Furtado and Silva [9] by proving the existence of a second nontrivial solution under a non-quadradicity condition at infinity on the non-linearity. Assuming more regularity on the non-linearity f, we are able to prove the existence of at least three nontrivial solutions.
 For more information see https://ejde.math.txstate.edu/Volumes/2020/60/abstr.html