On the existence of continuous solutions for nonlinear fourth-order elliptic equations with strongly growing lower-order terms

Abstract

In this article, we consider nonlinear elliptic fourth-order equations with the monotone principal part satisfying the common growth and coerciveness conditions for Sobolev space $W^{2,p}(\Omega )$, $\Omega \subset \mathbb {R}^{n}$. It is supposed that the lower-order term of the equations admits arbitrary growth with respect to an unknown function and is arbitrarily close to the growth limit with respect to the derivatives of this function. We assume that the lower-order term satisfies the sign condition with respect to the unknown function. We prove the existence of continuous generalized solutions for the Dirichlet problem in the case $n=2p$.