Multiplicity of weak solutions in double phase Kirchhoff elliptic problems with Neumann conditions

In this paper, we investigate the existence of weak solutions for a class of double phase Kirchhoff elliptic problems under Neumann boundary conditions. The problem is characterized by the equation {-K1(integral Lambda A(y,del zeta)dy)diva(y,del zeta)-K2(integral Lambda B(y,del zeta)dy)divb(y,del zeta)+K1(integral Lambda 1 nu 1(y)|zeta|nu 1(y)dy)|zeta|nu 1(y)-2 zeta+K2(integral Lambda 1 nu 2(y)|zeta|nu 2(y)dy)|zeta|nu 2(y)-2 zeta=theta(y,zeta)in Lambda a(y,del zeta)& sdot;n ->=b(y,del zeta)& sdot;n ->=0,on partial derivative Lambda,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \left \{ \textstyle\begin{array}{l@{\quad}l} -K_{1}\left (\int _{\Lambda} A(y, \nabla \zeta ) \mathrm{d} y\right ) \operatorname{div} a(y, \nabla \zeta ) -K_{2}\left (\int _{\Lambda} B(y, \nabla \zeta ) \mathrm{d} y\right ) \operatorname{div} b(y, \nabla \zeta ) \\ +K_{1}\left (\int _{\Lambda} \frac{1}{\nu _{1}(y)}| \zeta |<^>{\nu _{1}(y)} \mathrm{d} y\right )| \zeta |<^>{\nu _{1}(y)-2} \zeta \\ \quad{} +K_{2}\left ( \int _{\Lambda} \frac{1}{\nu _{2}(y)}| \zeta |<^>{\nu _{2}(y)} \mathrm{d} y\right )| \zeta |<^>{\nu _{2}(y)-2} \zeta =\theta (y, \zeta ) &\text{in } \Lambda \\ a(y, \nabla \zeta )\cdot \vec{n}=b(y, \nabla \zeta )\cdot \vec{n}=0, & \text{on } \partial \Lambda ,\end{array}\displaystyle \right . $$\end{document} where K1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$K_{1} $\end{document} and K2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$K_{2} $\end{document} are Kirchhoff-type functions, and the nonlinearities A(y,del zeta)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$A(y, \nabla \zeta ) $\end{document} and B(y,del zeta)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$B(y, \nabla \zeta ) $\end{document} exhibit double phase behavior. Employing a theorem proposed by B. Ricceri, which extends a more general variational principle, we confirm the existence of countless weak solutions for this complex system. Additionally, we present examples that illustrate the applicability of the theoretical results to specific cases. The findings contribute to the broader understanding of non-standard growth conditions and their implications in the study of Kirchhoff-type elliptic problems.