A uniqueness result for a semilinear elliptic equation involving the critical Sobolev exponent in symmetric domains

Article type: Research Article

Authors: Cerqueti, Katiuscia

Affiliations: Dipartimento di Matematica, II Università degli Studi di Roma “Tor Vergata”, Via della Ricerca Scientifica, I‐00133 Roma, Italy E‐mail: [email protected]

Abstract: We consider the problem \cases{ - \Delta u = N(N-2) u^p +\varepsilon u & \mbox{in} $\varOmega$,\cr u >0 & \mbox{in} $\varOmega$,\cr u =0 & \mbox{on} $\curpartial\varOmega$,\cr} where \varOmega is a bounded smooth domain of \mathbf{R}^N \ (N \geq 5) which is symmetric with respect to the coordinate hyperplanes \{x_i = 0\}, i = 1, \ldots, N , and it is convex in the x_i‐directions for i = 1, \ldots, N; here 0 < \varepsilon < \lambda_1 (\lambda_1 being the first eigenvalue of the Laplace operator in H_0^1 (\varOmega)) and p = ({N+2})/({N-2}). For \varepsilon small, we prove uniqueness and nondegeneracy of the solution u_\varepsilon with the property that \lim_{\varepsilon \rightarrow0} \frac{\int_\varOmega|\nabla u_\varepsilon|^2 \,\mathrm{d} x }{ (\int_\varOmega|u_\varepsilon|^{p+1}\,\mathrm{d} x)^{2/p+1} } = S_N, where S_N is the best Sobolev constant in \mathbf{R}^N.

Journal: Asymptotic Analysis, vol. 21, no. 2, pp. 99-115, 1999