On a class of nonlinear elliptic problem of convolution type via topological degree theory

In this manuscript, we investigate a boundary value problem governed by a complex system of partial differential equations given by: \(\begin{aligned} \displaystyle -\text{ div }\Big [{{\mathcal {L}}}_\vartheta \left( z,\vartheta ,\nabla \vartheta \right) \Big ]+{\hat{M}}\vert \vartheta \vert ^{p-2} \vartheta ={\mathcal {L}}_\sigma \Big (z, \hbar *\sigma (\vartheta ), \nabla (\hbar *\sigma (\vartheta ))\Big ), \end{aligned}\) $$\begin{array}{c}\hfill {\displaystyle -\text{div}[{\mathcal{L}}_{\vartheta }\left(z,,,\vartheta ,,,\mathrm{\nabla },\vartheta \right)]+\widehat{M}{|\vartheta |}^{p-2}\vartheta ={\mathcal{L}}_{\sigma }(z,ħ\ast \sigma (\vartheta ),\mathrm{\nabla }(ħ\ast \sigma (\vartheta ))),}\end{array}$$ under the following non-homogeneous Neumann boundary condition: \({\mathcal {L}}_\vartheta \left( z,\vartheta ,\nabla \vartheta \right) .\nu =\Theta (z,\vartheta )\) ${\mathcal{L}}_{\vartheta }\left(z,,,\vartheta ,,,\mathrm{\nabla },\vartheta \right).\nu =\mathrm{\Theta }(z,\vartheta )$ . Here, the functions \({\mathcal {L}}_\vartheta \) ${\mathcal{L}}_{\vartheta }$ , \({\mathcal {L}}_\sigma \) ${\mathcal{L}}_{\sigma }$ , and \(\Theta \) $\mathrm{\Theta }$ exhibit Carathéodory characteristics. The operator \(\sigma : W^{1, p}({\mathcal {B}}) \rightarrow W^{1, p}\left( {\mathbb {R}}^N\right) \) $\sigma :W^{1,p}(\mathcal{B})\to W^{1,p}\left({\mathbb{R}}^N\right)$ operates as an extension operator correlated with the domain \({\mathcal {B}}\) $\mathcal{B}$ , while \(\hbar \) $ħ$ denotes an integrable function defined over \({\mathbb {R}}^N\) ${\mathbb{R}}^N$ . Notably, the problem introduces a nonlocal operator, manifesting as the convolution \(\hbar *\sigma (\vartheta )\) $ħ\ast \sigma (\vartheta )$ of \(\hbar \) $ħ$ with \(\sigma (\vartheta )\) $\sigma (\vartheta )$ , associated with the variable \(\vartheta \) $\vartheta$ . Under conditions necessitating thorough examination, we establish the existence of a weak solution to the mentioned problem by employing the topological degree method.