Existence of solutions to nonlinear, subcritical higher-order elliptic Dirichlet problems

TL;DR

This paper proves the existence of nontrivial solutions for a class of nonlinear higher-order elliptic boundary value problems with superlinear and subcritical growth conditions, using degree theory and a priori estimates.

Contribution

It introduces new a priori estimates and combines them with degree theory to establish solution existence for complex nonlinear elliptic problems.

Findings

Existence of nontrivial solutions under superlinear and subcritical conditions.

Development of new a priori estimates for higher-order elliptic equations.

Application of degree theory to nonlinear boundary value problems.

Abstract

We consider the $2m$-th order elliptic boundary value problem $Lu=f(x,u)$ on a bounded smooth domain $\Omega\subset\R^N$ with Dirichlet boundary conditions on $\partial\Omega$. The operator $L$ is a uniformly elliptic linear operator of order $2m$ whose principle part is of the form $\big(-\sum_{i,j=1}^N a_{ij}(x) \frac{\partial^2}{\partial x_i\partial x_j}\big)^m$. We assume that $f$ is superlinear at the origin and satisfies $\lim \limits_{s\to\infty}\frac{f(x,s)}{s^q}=h(x)$, $\lim \limits_{s\to-\infty}\frac{f(x,s)}{|s|^q}=k(x)$, where $h,k\in C(\overline{\Omega})$ are positive functions and $q>1$ is subcritical. By combining degree theory with new and recently established a priori estimates, we prove the existence of a nontrivial solution.