| Abstract: |
| We present a study on the existence of infinitely many weak solutions for a class of fourth-order elliptic problems involving variable exponents and Navier boundary conditions. The research is characterized by its broad approach, applying the analysis to $W^{2,p(\cdot)}$-extension domains. This category encompasses not only smooth (Lipschitz) domains but also geometrically complex and non-smooth structures, such as fractals (e.g., the Koch snowflake).
The problem is driven by nonhomogeneous Leray--Lions type operators, a choice that allows for the simultaneous treatment of several classical operators, including generalized Laplace, mean curvature, and capillarity operators, as well as $p(\cdot)$-biharmonic operators. Furthermore, the study introduces new non-standard operators involving logarithmic and exponential functions.
Methodologically, the main result is obtained by applying a variational principle for identifying infinitely many critical points. This approach overcomes the limitations of previous literature in two fundamental ways: absence of the Ambrosetti--Rabinowitz condition and absence of symmetry conditions: The existence of infinitely many solutions is guaranteed without imposing parity or symmetry on the nonlinearity. Instead, the nonlinear term only needs to exhibit suitable oscillatory behavior either at infinity or at zero. |
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