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 \((1)\).


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