Nodal solutions for the weighted biharmonic equation with critical exponential growth

TL;DR

This paper establishes the existence of nodal solutions for a weighted biharmonic equation with exponential critical growth in four dimensions using variational methods and degree theory.

Contribution

It introduces a novel approach combining constrained minimization, deformation lemma, and degree theory to find nodal solutions for a critical exponential growth problem.

Findings

Existence of nodal solutions proved for the weighted biharmonic equation.

Application of Nehari set constrained minimization in critical growth context.

Use of degree theory to establish solution existence.

Abstract

In this paper, we deal with the logarithmic weighted fourth order elliptic equation in the unit disk of $B\subset\R^{4}$: $$\displaystyle(P_{\lambda})~~\Delta(w(x) \Delta u) = \lambda\ f(x,u) \quad\mbox{ in }\quad B, \quad u=\frac{\partial u}{\partial n} \quad\mbox{ on } \quad\partial B,$$ where the nonlinearity $f$ is assumed to have exponential critical growth in view of Adam's type inequalities. By using the constrained minimization in Nehari set combined with the quantitative deformation lemma and degree theory, we prove the existence of nodal solutions to the problem $(P_{\lambda})$ .