Weak solutions for double phase Schrödinger–Kirchhoff problems with Neumann conditions

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This paper addresses a novel class of Schrödinger-Kirchhoff-type problems involving a double-phase structure and subject to Neumann boundary conditions. The model is governed by (η(s),ζ(s))-Laplacian-like operators and includes a convection term that explicitly depends on both the solution and its gradient. By imposing appropriate growth conditions on this nonlinear term, we prove the existence of weak solutions. Our approach combines the Galerkin approximation method with the theory of Young measures, all within the analytical framework of Musielak–Orlicz Sobolev spaces with variable exponents. Our results offer a new contribution to the field, extending existing models by incorporating both non-standard growth and gradient-dependent convection effects.

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