Consider the eigenvalue problem of the $p$-Laplacian, $$-\Delta _p u=\lambda |u|^{p-2}u,\ u\in W_0^{1,p}(\Omega)$$ In most of the literature I saw, an extra condition is mentioned that $u$ vanish on the boundary of $\Omega$. My question is, if this vanishing condition is not satisfied (this means it can take any values on the boundary), would the results on eigenvalues and eigenfunctions mentioned in the literature differ? I am especially interested in the case when $p>d$, where $d$ is the dimension of the space $\Omega$ ($\Omega$ is a bounded convex subset of $\mathbb{R}^d$).
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$\begingroup$ Is it done to discretize the spectrum? I got to this probable reason by looking at ordinary Laplace operator. $\endgroup$– Rajesh DJul 7, 2017 at 9:45
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