ON THE DEGENERATE $(p,q)$-LAPLACE EQUATIONS CORRESPONDING TO AN INVERSE SPECTRAL PROBLEM

The report will discuss two interrelated topics: 1) a new class of applied problems leading to equations with (p,q)-Laplace; 2) the inverse optimal problem method which is a new apparatus that allows to prove the existence, uniqueness and stability of solutions to nonlinear boundary value problems. As a model example, we will consider a boundary value problem for an equation with (p,q)-Laplace and measurable unbounded coefficients of the form:
div(σ(x)|∇u|^ {q−2}∇u) + div(|∇u|^{ p−2}∇u) = λρ(x)|u|^{q−2}u , p > q
As a spectral problem for which the inverse optimal problem method will be applied, we will consider
L_{σ}(ϕ) := −div(σ(x)|∇ϕ|^ {q−2}∇ϕ) = λρ(x)|ϕ|^{q−2}ϕ.