Partial Differential Equations with Variable Exponents: Variational Methods and Qualitative Analysis provides researchers and graduate students with a thorough introduction to the theory of nonlinear partial differential equations (PDEs) with a variable exponent, particularly those of elliptic type. The book presents the most important variational

In this paper we are concerned with a class of double phase energy functionals arising in the theory of transonic flows. Their main feature is that the associated Euler equation is driven by the Baouendi–Grushin operator with variable coefficient. This partial differential equation is of mixed type and possesses both elliptic and hyperbolic regions. After establishing a weighted inequality for the Baouendi–Grushin operator and a related compactness property, we establish the existence of stationary waves under arbitrary perturbations of the reaction.

We consider a nonlinear eigenvalue problem driven by the sum of and -Laplacians. We show that the problem has a continuous spectrum. Our result reveals a discontinuity property for the spectrum of a parametric ()-differential operator as the parameter goes to .