\end{aligned}$$ $$\label{a3a4}
\begin{aligned}
\mbox{there exist}\,\, & \mbox{two positive constants}\,\, a_3\,\,
\mbox{and}\,\, a_4\,\, \mbox{such that} \\
& F(x,t)\geq a_3|t|^q-a_4\,\, \mbox{a.e.}\,\, x\in \Omega, t\in {\mathbb R};
\end{aligned}$$ $$\label{cond0}
{\displaystyle \lim_{|t|\to 0}\frac{f(x,t)}{|t|}=0}\,\, \mbox{uniformly in}\,\, x\in \Omega\,;$$ $$\label{mu0}
0<q F(x,t)\le tf(x,t)\,\,\mbox{a.e.}\,\, x\in \Omega, t\in {\mathbb R}\setminus\{0\},$$ where $q$ is given in and the function $F$ is the primitive of $f$ with respect to the second variable, that is $$\label{F}
{\displaystyle F(x,t):=\int_0^t f(x,\tau)d\tau}\,.$$

As a model for $f$ we can take the function $f(x,t)=a(x)|t|^{q-2}t$, with $a\in L^\infty(\Omega)$, $\inf_\Omega a>0$ and $q\in (2, 2^*)$. The exponent $2^*$ here plays the role of a fractional critical Sobolev exponent (see, e.g. [@valpal Theorem 6.5]).

As remarked in [@dmnodea2], condition is not a mere consequence of , and must be assumed [*a priori*]{}, unless $f:\bar \Omega \times {\mathbb R}\to {\mathbb R}$ is continuous and holds for every $(x,t)\in \bar \Omega \times {\mathbb R}$.

The main result of the present paper can be stated as follows:

\[thmain\] Let $s\in (0,1)$, $n>2s$ and $\Omega$ be an open bounded subset of ${\mathbb R}^n$ with continuous boundary. Let $K:{\mathbb R}^n\setminus\{0\}\rightarrow(0,+\infty)$ be a function satisfying and and let $f:\Omega\times {\mathbb R}\to {\mathbb R}$ be a function verifying –.

Then, for every eigenvalue $\lambda_k$ of 