p⁢(x)p(x)-biharmonic operator involving the p⁢(x)p(x)-Hardy inequality

In this work, we investigate the spectrum denoted by Λ for the {p(x)}-biharmonic operator involving the Hardy term. We prove the existence of at least one non-decreasing sequence of positive eigenvalues of this problem such that {\sup\Lambda=+\infty}. Moreover, we prove that {\inf\Lambda>0} if and only if the domain Ω satisfies the {p(x)}-Hardy inequality.

Infinitely many solutions for critical quasilinear equations involving Hardy potential

In this paper, we consider the following quasilinear elliptic equation with critical growth and a Hardy term: [Formula: see text] where [Formula: see text], [Formula: see text] is a constant, [Formula: see text][Formula: see text] is the Sobolev critical exponent. And [Formula: see text] is an open bounded domain which contains the origin. We will study the existence of infinitely many solutions for (P). To achieve this goal, we first perform various kinds of change of variables to overcome the difficulties caused by the unboundedness of [Formula: see text] ([Formula: see text] for large [Formula: see text]) and the lack of a global monotone condition [Formula: see text] (see below) on [Formula: see text], then combining the idea of regularization approach and subcritical approximation we prove the existence of infinitely many solutions for (P). Our results show that under some suitable assumptions on [Formula: see text], without the perturbation of the lower term [Formula: see text] we can still obtain the existence of infinitely many solutions for (P).