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

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 $$-\mathrm{d}\mathrm{i}\mathrm{v}(\rho \mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{q-2}\mathrm{\nabla }u)-\mathrm{d}\mathrm{i}\mathrm{v}(\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{p-2}\mathrm{\nabla }u)=\lambda b\mathrm{\mid }u{\mathrm{\mid }}^{q-2}u,p>q$$ are proven. Additionally, an explicit formula that expresses the solution of the equation through the inverse optimal solution of the spectral problem $$-\mathrm{d}\mathrm{i}\mathrm{v}(\rho \mathrm{\mid }\mathrm{\nabla }\phi {\mathrm{\mid }}^{q-2}\mathrm{\nabla }\phi )=\lambda b\mathrm{\mid }\phi {\mathrm{\mid }}^{q-2}\phi$$ 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.