In determining some of the spectral properties of the p(X)-Laplace operator, one can use variational methods. So i need to be be guided on the most appropriate technique or principle or theorem applied to prove that the set of all eigenvalues of such operators is discrete.
Perhaps you can use the following known result:
THEOREM: If X is a Banach space and T:X->X is a linear compact operator, then the spectrum of T is countable.
So, spectrum of T is a discrete set and it results that EV(T) - the set of eigenvalues of T- is also discrete.
If the p-Laplacian, under some assumptions, can be compact, then above THEOREM ensures your requirement.
A related article can you find at:
https://www.mat.unical.it/~sciunzi/PDFPub/NonAna-CES.pdf
In physics
https://www.physicsforums.com/threads/discrete-eigenvalues-and-their-eigenfunctions.538437/
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