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

Abstract

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 $$ \left \{ \textstyle\begin{array}{l@{\quad}l} -K_{1}\left (\int _{\Lambda} A(y, \nabla \zeta ) \mathrm{d} y\right ) \operatorname{div} a(y, \nabla \zeta ) -K_{2}\left (\int _{\Lambda} B(y, \nabla \zeta ) \mathrm{d} y\right ) \operatorname{div} b(y, \nabla \zeta ) \\ +K_{1}\left (\int _{\Lambda} \frac{1}{\nu _{1}(y)}| \zeta |^{\nu _{1}(y)} \mathrm{d} y\right )| \zeta |^{\nu _{1}(y)-2} \zeta \\ \quad{} +K_{2}\left ( \int _{\Lambda} \frac{1}{\nu _{2}(y)}| \zeta |^{\nu _{2}(y)} \mathrm{d} y\right )| \zeta |^{\nu _{2}(y)-2} \zeta =\theta (y, \zeta ) &\text{in } \Lambda \\ a(y, \nabla \zeta )\cdot \vec{n}=b(y, \nabla \zeta )\cdot \vec{n}=0, & \text{on } \partial \Lambda ,\end{array}\displaystyle \right . $$ where $K_{1} $ and $K_{2} $ are Kirchhoff-type functions, and the nonlinearities $A(y, \nabla \zeta ) $ and $B(y, \nabla \zeta ) $ 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.

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