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Limiting behavior of solutions to an equation with the fractional Laplacian. (English) Zbl 1313.35084

Let \(\Omega \) be the unit ball in \(\mathbb {R}^N\) (\(N\geq 2\)) centered at the origin and let \(\alpha ,\beta >0\) with \(\alpha <2\). Moreover, let \(p\in (2,\frac {2N}{N-\alpha })\). The authors study, as \(p\) tends to the critical exponent \(\frac {2N}{N-\alpha }\), the limiting behavior of the ground state solutions of the following boundary value problem for the Hénon equation involving the fractional Laplacian: \[ \begin{cases} -\Delta ^{\frac {\alpha }{2}}u=| x| ^\beta u^{p-1} & \text{in} \quad \Omega ,\\ u>0 & \text{in} \quad \Omega, \\ u=0 & \text{on} \quad \! \partial \Omega. \end{cases} \qquad \qquad \tag{1} \] The authors prove that, as \(p\rightarrow \frac {2N}{N-\alpha }\), the ground state solutions concentrate at a point of the boundary of the domain \(\Omega \). In particular, they extend to the case of the fractional Laplacian the main result of D. Cao and S. Peng [J. Math. Anal. Appl. 278, No. 1, 1–17 (2003; Zbl 1086.35036)] where the same limiting behavior was proved for the classical Laplacian operator (which corresponds to the case \(\alpha =2\)).
The authors, using the ideas introduced in L. Caffarelli and L. Silvestre [Commun. Partial Differ. Equations 32, No. 8, 1245–1260 (2007; Zbl 1143.26002)], restate the nonlocal problem \((1)\) in a local problem having a variational structure. Then, by the Lions concentration-compactness principle, they prove a concentration phenomenon for the ground state solutions of this local problem which, in turn, gives the limiting behavior of the ground state solutions of \((1)\).

MSC:

35J25 Boundary value problems for second-order elliptic equations
47G30 Pseudodifferential operators
35B44 Blow-up in context of PDEs
35B45 A priori estimates in context of PDEs
35J70 Degenerate elliptic equations
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