Lyapunov inequality for an elliptic problem with the Robin boundary condition

In this paper, we study the LqLq-Lyapunov type inequalities for quasi-linear elliptic problems with the Robin boundary conditions, in which the principal part of the equation is pp-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)(N−1)/(p−1), where NN is the dimension of the domain. We reveal the relation between this critical value and the critical exponent p∗p∗ of the trace Sobolev embedding. W1,p(Ω)↪Lp∗(∂Ω)W1,p(Ω)↪Lp∗(∂Ω).