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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
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