On degenerate $(q,p)$-Laplace equations corresponding to an inverse spectral problem

Abstract

Two main results are presented: 1) a new class of applied problems that lead to equations with $(p,q)$-Laplace is presented; 2) a method for solving nonlinear boundary value problems involving $(p,q)$-Laplace with measurable unbounded coefficients is introduced. In the main result, the existence, uniqueness, and stability of the nonnegative weak solution to the equations of the form $$ -{\rm div}(\rho|\nabla u|^{q-2} \nabla u)-{\rm div}(|\nabla u|^{p-2}\nabla u)=\lambda b |u|^{q-2}u,~~p>q $$ are proven. Additionally, an explicit formula that expresses the solution of the equation through the inverse optimal solution of the spectral problem $$-{\rm div}(\rho|\nabla \phi|^{q-2}\nabla \phi)=\lambda b|\phi|^{q-2}\phi$$ is presented. The advantage of the method is that the inverse optimal problem has a visible geometry and a simple variational structure, which makes it easy to solve it and, as a consequence, find a solution to the associated nonlinear boundary value problem.