Weak solutions in anisotropic (α→(z),β→(z))-Laplacian Kirchhoff models

Abstract

The focus of this research is to establish the existence of infinitely many weak solutions to the double-phase $\left(\overrightarrow{\alpha }\right(z),\overrightarrow{\text{β}}(z\left)\right)$-Kirchhoff Neumann problem, characterized by the following equation: $\{\begin{array}{l}{\displaystyle -{\mathrm{\Phi }}_1\left(\sum _{i=1}^N{\int }_{\mathrm{\Theta }}\frac{1}{{\alpha }_i\left(z\right)}{\left|\frac{\mathrm{\partial }\phi \left(z\right)}{\mathrm{\partial }{z}_i}\right|}^{{\alpha }_i\left(z\right)}\mathrm{d}z\right){\mathrm{\Delta }}_{\overrightarrow{\alpha }(\cdot )}\phi \left(z\right)}\\ {\displaystyle -{\mathrm{\Phi }}_2\left(\sum _{i=1}^N{\int }_{\mathrm{\Theta }}\frac{1}{{\text{β}}_i\left(z\right)}{\left|\frac{\mathrm{\partial }\phi \left(z\right)}{\mathrm{\partial }{z}_i}\right|}^{{\text{β}}_i\left(z\right)}\mathrm{d}z\right){\mathrm{\Delta }}_{\overrightarrow{\text{β}}(\cdot )}\phi \left(z\right)}\\ {\displaystyle +{\mathrm{\Phi }}_1\left({\int }_{\mathrm{\Theta }}\frac{1}{{\alpha }_0\left(z\right)}|\phi \right(z\left){|}^{{\alpha }_0\left(z\right)}\mathrm{d}z\right)|\phi \left(z\right){|}^{{\alpha }_0\left(z\right)-2}\phi \left(z\right)}\\ {\displaystyle +{\mathrm{\Phi }}_2\left({\int }_{\mathrm{\Theta }}\frac{1}{{\text{β}}_0\left(z\right)}|\phi \right(z\left){|}^{{\text{β}}_0\left(z\right)}\mathrm{d}z\right)|\phi \left(z\right){|}^{{\text{β}}_0\left(z\right)-2}\phi \left(z\right)}\\ {\displaystyle =\psi (z,\phi (z\left)\right)\text{for all }z\text{ in }\mathrm{\Theta },}\\ {\displaystyle \sum _{i=1}^N{\left|\frac{\mathrm{\partial }\phi \left(z\right)}{\mathrm{\partial }{z}_i}\right|}^{{\alpha }_i\left(z\right)-2}\frac{\mathrm{\partial }\phi \left(z\right)}{\mathrm{\partial }{z}_i}{\nu }_i}\\ {\displaystyle =\sum _{i=1}^N{\left|\frac{\mathrm{\partial }\phi \left(z\right)}{\mathrm{\partial }{z}_i}\right|}^{{\text{β}}_i\left(z\right)-2}\frac{\mathrm{\partial }\phi \left(z\right)}{\mathrm{\partial }{z}_i}{\nu }_i=0\text{on}\mathrm{\partial \Theta }.}\end{array}$ By applying a critical point theorem from B. Ricceri, which is derived from a broader variational framework, we demonstrate that this problem admits infinitely many weak solutions.