Symmetry results for perturbed problems and related questions - Grossi, Massimo; Pacella, Filomena; Yadava, S. L. (Topological Methods in Nonlinear Analysis, 2003)
In this paper we prove a symmetry result for positive solutions of the Dirichlet problem$$\cases-\Delta u=f(u) & \hbox{in }D,\\u=0 & \hbox{on }\partial D, \endcases\tag{0.1}$$when $f$ satisfies suitable assumptions and $D$ is a small symmetric perturbation of a domain $\Omega$for which the Gidas-Ni-Nirenberg symmetry theorem applies.We consider both the case when $f$ has subcritical growth and $f(s)=s^{(N+2)/(N-2)}+\lambda s$,$N\ge3$, $\lambda$ suitable positive constant.