| Abstract: |
| In this talk, we deal with two novel classes of
quasilinear elliptic equations, each driven by the double phase
operator with variable exponents. The first class features a new
double phase equation where exponents depend on the gradient of the
solution. We delve into giving various properties of the
corresponding Musielak-Orlicz Sobolev spaces, including the
$\Delta_2$ property, uniform convexity, density and compact
embedding. Additionally, we explore the characteristics of the new
double phase operator, such as continuity, strict monotonicity, and
the (S$_+$)-property. Employing both variational and nonvariational
methods, we deal with the existence of solutions for this inaugural
class of double phase equations. In the second category, the
treatment of exponents is dependent on the solution itself. This
class differs from the first one due to the unavailability of
suitable Musielak-Orlicz Sobolev spaces. For this reason, we employ
a perturbation argument that leads to the classical double phase
class. These two new classes highlight how different physical
processes like the movement of special fluids through porous
materials, phase changes, and fluid dynamics interact with each
other. |
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