Existence of Mountain-pass Solutions for $p(\cdot)$-Biharmonic Equations with Rellich-type Term

Abstract

This manuscript discusses the existence of nontrivial weak solution
for a $p(\cdot)$-biharmonic problems involving Rellich-type
term by virtue of Mountain Pass theorem.
Our results are mainly for the case
$$
1<\min_{x\in\overline{\Omega}} p(x)\leq\max_{x\in\overline{\Omega}}
p(x)<\min_{x\in\overline{\Omega}} q(x)\leq
\max_{x\in\overline{\Omega}} q(x)<\min\Big\{\frac{N}{2},\frac{N p(x)}{N-2p(x)}\Big\},
$$
and extend the corresponding result of the reference
\cite{El-Mor-Lag-Tou} for the case
$$
1<\min_{x\in\overline{\Omega}}q(x)\leq \max_{x\in\overline{\Omega}}
q(x)<\min_{x\in\overline{\Omega}} p(x)\leq\max_{x\in\overline{\Omega}}
p(x)<\frac{N}{2}.
$$.