Abstract
In this paper, we consider the nonlinear eigenvalue problem:
$$\begin{array}{}
\displaystyle
\begin{cases}
{\it\Delta}(|{\it\Delta} u|^{p(x)-2}{\it\Delta} u)= \lambda \frac{|u|^{q(x)-2}u}{{\delta(x)}^{2q(x)}} \;\; \mbox{in}\;\; {\it\Omega}, \\
u\in W_0^{2,p(x)}({\it\Omega}),
\end{cases}
\end{array}$$
where Ω is a regular bounded domain of ℝN, δ(x) = dist(x, ∂Ω) the distance function from the boundary ∂Ω, λ is a positive real number, and functions p(⋅), q(⋅) are supposed to be continuous on Ω satisfying
$$\begin{array}{}
\displaystyle
1 \lt \min_{\overline{{\it\Omega} }}\,q\leq \max_{\overline{{\it\Omega}}}\,q \lt \min_{\overline{{\it\Omega} }}\,p \leq \max_{\overline{{\it\Omega}}}\,p \lt \frac{N}{2} \mbox{ and } \max_{\overline{{\it\Omega}}}\,q \lt p_2^*:= \frac{Np(x)}{N-2p(x)}
\end{array}$$
for any x ∈ Ω. We prove the existence of at least one non-decreasing sequence of positive eigenvalues. Moreover, we prove that sup Λ = +∞, where Λ is the spectrum of the problem. Furthermore, we give a proof of positivity of inf Λ > 0 provided that Hardy-Rellich inequality holds.