Existence of renormalized solutions for some noncoercive elliptic problem in a two-component domain with \(L^{1}\) data

In this work, we will focus on studying a specific class of quasilinear elliptic equations with degenerate coercivity in a two-component domain, which is defined as follows: \(\begin{aligned}\left\{ \begin{array}{ll} \displaystyle - \text{ div }(a(x,u_{1},\nabla u_{1})) + \mathcal {K}(x,\nabla u_{1})+\lambda (x)|u_{1}|^{s-1}u_{1} = f(x) \qquad & \text{ in } \Omega _{1},\\ \displaystyle - \text{ div }(a(x,u_{2},\nabla u_{2})) + \mathcal {K}(x,\nabla u_{2})+\lambda (x)|u_{2}|^{s-1}u_{2} = f(x) \qquad & \text{ in } \Omega _{2},\\ \displaystyle u_{1} = 0 & \text{ on } \partial \Omega ,\\ \displaystyle a(x,u_{1},\nabla u_{1})\cdot \nu _{1} = a(x,u_{2},\nabla u_{2})\cdot \nu _{1} & \text{ on } \Gamma ,\\ \displaystyle a(x,u_{1},\nabla u_{1})\cdot \nu _{1} = - h(x)|u_{1}-u_{2}|^{p-2}(u_{1}-u_{2}) & \text{ on } \Gamma , \end{array} \right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\displaystyle -\text{div}(a(x,u_1,\mathrm{\nabla }{u}_1))+\mathcal{K}(x,\mathrm{\nabla }{u}_1)+\lambda (x){|u_1|}^{s-1}{u}_1=f(x)}\hfill & \text{in}{\mathrm{\Omega }}_1,\hfill \\ {\displaystyle -\text{div}(a(x,u_2,\mathrm{\nabla }{u}_2))+\mathcal{K}(x,\mathrm{\nabla }{u}_2)+\lambda (x){|u_2|}^{s-1}{u}_2=f(x)}\hfill & \text{in}{\mathrm{\Omega }}_2,\hfill \\ {\displaystyle u_1=0}\hfill & \text{on}\partial \mathrm{\Omega },\hfill \\ {\displaystyle a(x,u_1,\mathrm{\nabla }{u}_1)·{\nu }_1=a(x,u_2,\mathrm{\nabla }{u}_2)·{\nu }_1}\hfill & \text{on}\mathrm{\Gamma },\hfill \\ {\displaystyle a(x,u_1,\mathrm{\nabla }{u}_1)·{\nu }_1=-h(x){|u_1-u_2|}^{p-2}(u_1-u_2)}\hfill & \text{on}\mathrm{\Gamma },\hfill \end{array}\right)\end{array}$$ where \(\Omega \) $\mathrm{\Omega }$ is a bounded open set of \(\mathbb {R}^N\) ${\mathbb{R}}^N$ ( \(N\ge 2\) $N\ge 2$ ), with \( \Omega = \Omega _{1} \cup \Omega _{2}\cup \Gamma ,\) $\mathrm{\Omega }={\mathrm{\Omega }}_1\cup {\mathrm{\Omega }}_2\cup \mathrm{\Gamma },$ where \(\Omega _{2}\) ${\mathrm{\Omega }}_2$ is an open set such that \(\overline{\Omega _{2}} \subset \Omega \) $\overline{{\mathrm{\Omega }}_2}\subset \mathrm{\Omega }$ with a Lipschitz boundary \(\Gamma \) $\mathrm{\Gamma }$ and \(\displaystyle \Omega _{1}=\Omega {\setminus } \overline{\Omega _{2}},\> \lambda (x)\ge \lambda _{0},\> s\ge 1,\> 0\le \delta <1\> \) ${\displaystyle {\mathrm{\Omega }}_1=\mathrm{\Omega }\backslash \overline{{\mathrm{\Omega }}_2},\lambda (x)\ge {\lambda }_0,s\ge 1,0\le \delta <1}$ and \(f\in L^{1}(\Omega )\) $f\in L^1(\mathrm{\Omega })$ . We show the existence of a renormalized solutions for this class of equation, and we will conclude some regularity results.