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Fractional \(p\)-eigenvalues. (English) Zbl 1327.35286

This paper is devoted to eigenfunctions of some nonlocal operators of fractional order. Namely, the authors consider weak solutions \(u\) of the equation \[ -{\mathfrak{L}}_K u = \lambda |u|^{p-2}u \eqno{(1)} \] in a domain \(\Omega\subset \mathbb R^n\) with the Dirichlet condition \(u = 0\) on \(\mathbb R^n\setminus\Omega\), where \[ -{\mathfrak{L}}_K u(x) = 2\int_{\mathbb R^n} K(x,y)|u(y)- u(x)|^{p-2}(u(y)-u(x))\, dx \] and \(K\) belongs to a class of singular symmetric kernels modeled on the case \(K(x, y) = |x-y|^{-(n+sp)}\) with \(s\in(0, 1)\), \(p > 1\). The integral is understood in the principal value sense.
The authors prove that, similarly as in the local case, the positive eigenfunctions for the fractional \(p\)-Laplacian uniquely correspond to the first eigenvalue, the one that is obtained by minimizing the Rayleigh quotient \[ {\mathfrak{R}}(\phi) := \frac{\int_{\mathbb R^n}\int_{\mathbb R^n} K(x,y)|\phi(y)- \phi(x)|^{p}\, dxdy}{\int_{\mathbb R^n} K(x,y)|\phi(x)|^{p}\, dx} \eqno{(2)} \] among all smooth functions \(\phi\) compactly supported in a Lipschitz domain \(\Omega\). Moreover, the authors deduce that all the positive fractional \(p\)-eigenfunctions corresponding to the first eigenvalue \(\lambda_1\) are proportional.

MSC:

35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35J60 Nonlinear elliptic equations
35R11 Fractional partial differential equations
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