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.


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
Full Text: Link