## Symmetry results for perturbed problems and related questions.(English)Zbl 1274.35021

Summary: In this paper we prove a symmetry result for positive solutions of the Dirichlet problem
\left\{\begin{align*}{-\Delta u=f(u)&\quad\text{in }\; D,\cr u=0\qquad\quad&\quad \text{on }\;\partial D,\cr}\end{align*}\right.
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 $$f(s)=s^{(N+2)/(N-2)}+\lambda s$$, $$N\geq 3$$, $$\lambda$$ suitable positive constant.

### MSC:

 35B38 Critical points of functionals in context of PDEs (e.g., energy functionals) 35B50 Maximum principles in context of PDEs 35J10 Schrödinger operator, Schrödinger equation 35J60 Nonlinear elliptic equations

### Keywords:

elliptic equations; symmetry of solutions
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