Positive solutions of the $$p$$-Laplace Emden-Fowler equation in hollow thin symmetric domains.(English)Zbl 1340.35021

Let $$\Omega$$ be a bounded domain in $$\mathbb{R}^N$$ with $$N\geq 2$$ and let $$p$$, $$q\geq 2$$ with $$p<q<p^*$$, where $$p^*=\infty$$ if $$N\leq p$$ and $$p^*=\frac{Np}{N-p}$$ if $$N>p$$ is the critical exponent for the Sobolev embedding. In this paper, the authors study certain symmetry properties of positive solutions to the Dirichlet problem for the Emden-Fowler equation $-\Delta_p u= u^{q-1}\text{ in }\Omega, \, u=0\text{ on }\partial \Omega.{(P)}$ More precisely, they establish the following result: Let $$H,G$$ be two closed subgroups of the orthogonal group $$O(N)$$, let $$U\subset \mathbb{R}^N$$ be a $$G$$-invariant domain, and let $R(u):=\left(\int_\Omega |\nabla u|^pdx\right)\left(\int_\Omega |u|^qdx\right)^{-\frac{p}{q}},\, u\in W_0^{1,p}(\Omega)\setminus \{0\},$ be the Rayleigh quotient. Assume that $$\Omega$$ is a $$G$$-invariant proper subdomain of $$U$$ and that $$\{hx: h\in H\}$$ is a proper subset of $$\{gx: g\in G\}$$ for all $$x\in \overline{U}$$. Then, there exists a positive constant $$C$$ depending only on $$H,G,p,q,U$$ such that $\inf_{u\in W_0^{1,p}(\Omega,H)\setminus \{0\}}R(u)<\inf_{u\in W_0^{1,p}(\Omega,G)\setminus \{0\}}R(u),$ whenever $$\lambda_p(\Omega)>C$$. Here, $$\lambda_p(\Omega)$$ is the first eigenvalue of the $$p$$-Laplacian on $$\Omega$$ and $$W_0^{1,p}(\Omega,G)$$ and $$W_0^{1,p}(\Omega,H)$$ are the subspaces of all $$G$$ invariant and $$H$$ invariant functions of $$W_0^{1,p}(\Omega)$$, respectively. As a consequence of the above inequality, no $$H$$ invariant solution to problem $$(P)$$ which minimizes $$R(u)$$ on $$W_0^{1,p}(\Omega,H)$$ is $$G$$-invariant.
The proof is based on variational methods.

MSC:

 35J25 Boundary value problems for second-order elliptic equations 35J20 Variational methods for second-order elliptic equations 35B06 Symmetries, invariants, etc. in context of PDEs 35B09 Positive solutions to PDEs 35D30 Weak solutions to PDEs
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