Eigencurves of the <i>p</i>(·)-Biharmonic operator with a Hardy-type term

Abstract

<jats:title>Abstract</jats:title>
<jats:p>This paper is devoted to the study of the homogeneous Dirichlet problem for a singular nonlinear equation which involves the <jats:italic>p</jats:italic>(·)-biharmonic operator and a Hardy-type term that depend on the solution and with a parameter λ. By using a variational approach and min-max argument based on Ljusternik-Schnirelmann theory on <jats:italic>C</jats:italic>
<jats:sup>1</jats:sup>-manifolds [13], we prove that the considered problem admits at least one nondecreasing sequence of positive eigencurves with a characterization of the principal curve <jats:italic>μ</jats:italic>
<jats:sub>1</jats:sub>(λ) and also show that, the smallest curve <jats:italic>μ</jats:italic>
<jats:sub>1</jats:sub>(λ) is positive for all 0 ≤ λ &lt; <jats:italic>C</jats:italic>
<jats:italic>
<jats:sub>H</jats:sub>
</jats:italic>, with <jats:italic>C</jats:italic>
<jats:italic>
<jats:sub>H</jats:sub>
</jats:italic> is the optimal constant of Hardy type inequality.</jats:p>