Lyapunov inequality for an elliptic problem with the Robin boundary condition. (December 2015) - Hashizume, Masato; Takahashi, Futoshi
Abstract: In this paper, we study the L q -Lyapunov type inequalities for quasi-linear elliptic problems with the Robin boundary conditions, in which the principal part of the equation is p -Laplacian operator. Similar problems have been considered by several authors for linear elliptic problems with the Dirichlet or Neumann boundary conditions. We show that the critical value of the problem is ( N − 1 ) / ( p − 1 ), where N is the dimension of the domain. We reveal the relation between this critical value and the critical exponent p ∗ of the trace Sobolev embedding. W 1, p ( Ω ) ↪ L p ∗ ( ∂ Ω ) .