Abstract
We establish an integration by parts formula in bounded domains for the higher order fractional Laplacian $(-\Delta)^s$ with $s>1$.We also obtain the Pohozaev identity for this operator.Both identities involve local boundary terms, and they extend the identities obtained by the authors in the case $s\in(0,1)$. As an immediate consequence of these results, we obtain a unique continuation property for the eigenfunctions $(-\Delta)^s\phi=\lambda\phi$ in $\Omega$, $\phi\equiv0$ in $\mathbb{R}^n\setminus\Omega$.
Highlights
Introduction and resultsWe consider bounded solutions u ∈ Hs(Rn) to the Dirichlet problem (1.1)(−∆)su = f (x, u) in Ω u=0 in Rn\Ω for the higher order fractional Laplacian (−∆)s, s > 1
(−∆)su(ξ) = |ξ|2su(ξ) for a.e. ξ ∈ Rn. It can be defined inductively by (−∆)s = (−∆)s−1 ◦ (−∆), once (−∆)s is defined for s ∈ (0, 1) —see for example [33, 28] for its definition for s ∈ (0, 1) in terms of an integral formula
The aim of this paper is to establish an integration by parts formula in bounded domains Ω for the operator (−∆)s with s > 1, as well as the Pohozaev identity for problem (1.1)
Summary
The aim of this paper is to establish an integration by parts formula in bounded domains Ω for the operator (−∆)s with s > 1, as well as the Pohozaev identity for problem (1.1). For the Laplacian −∆, the Pohozaev identity follows from integration by parts or the divergence theorem In this nonlocal framework these tools are not available, and our proof is more involved. Applying Theorem 1.4 to solutions of semilinear problems of the form (1.1) we find the Pohozaev identity for the higher order fractional Laplacian. We state this result which, for the sake of simplicity, we state for solutions to (1.5) below instead of (1.1).
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