Nodal ground state solution to a biharmonic equation via dual method

TL;DR

This paper proves the existence of nodal ground state solutions for a class of biharmonic equations using a dual variational method, focusing on boundary conditions and subcritical nonlinearities.

Contribution

It introduces a dual method approach to establish nodal solutions for biharmonic problems with specific boundary conditions, advancing solution existence theory.

Findings

Existence of nodal ground state solutions proved

Applicable to biharmonic equations with subcritical growth

Method extends to different boundary conditions

Abstract

Using dual method we establish the existence of nodal ground state solution for the following class of problems $$ \left\{ \begin{array}{l} \Delta^2 u = f(u), \quad \mbox{in} \quad \Omega, \\ u =Bu=0,\quad\mbox{on} \quad \partial \Omega \end{array} \right. $$ where $\Delta^2$ is the biharmonic operator, $B=\Delta$ or $B=\dfrac{\partial}{\partial \nu}$ and $f$ is a $C^1$-function having subcritical growth.