Lyapunov-type inequalities for quasilinear elliptic equations with Robin boundary condition
The aim of this paper is to prove a Lyapunov-type inequality for the following two-dimensional quasilinear elliptic problem: \[ \begin{cases} -\big(\Phi_p(u_{xy}) \big)_y= r(x,y)\Phi_p(u), & (x,y)\in \Omega,\\ u(a_1,y)=0=u(b_1,y), & a_2\leq y\leq b_2,\\ u_x(x,a_2)=0=u_x(x,b_2), & a_1\leq x\leq b_1, \end{cases} \] where \(\Omega=[a_1,b_1]\times[a_2,b_2],\) \(r\) is a measurable function on \(\Omega,\) and \(\Phi_p(u)=|u|^{p-2}u.\) Also, a lower bound for the first positive eigenvalue of the boundary value problem is obtained.
two dimensional quasilinear elliptic problems
scientific article; zbMATH DE number 6695881
scientific article