Nonlocal Schrödinger–Maxwell System Involving Fractional p -Laplacian With Singular Nonlinearity

Abstract

In this paper, we study the following nonlocal system exhibiting singular nonlinearity and weighted singular terms: $$\{\begin{array}{rlrl}& (-\mathrm{\Delta }{)}_p^{s}u+v{u}^{p-1}=h(x)u^{-\alpha },& u>0\text{in }\mathrm{\Omega };& u=0,\text{in }{\mathbb{R}}^N\setminus \mathrm{\Omega },\\ & (-\mathrm{\Delta }{)}_p^{s}v=u^p,& v>0\text{in }\mathrm{\Omega };& v=0,\text{in }{\mathbb{R}}^N\setminus \mathrm{\Omega },\end{array}$$ where $0<s<1$ , $p>1$ , $\alpha >0$ , and $\mathrm{\Omega }\subset {\mathbb{R}}^N$ , with $N>sp$ , is an open bounded domain with $C^{1,1}$ boundary $\mathrm{\partial }\mathrm{\Omega }$ . The function $h:\mathrm{\Omega }\to {\mathbb{R}}^{+}$ exhibits growth of negative powers of the distance function $d(x):=\text{dist}(x,\mathrm{\partial }\mathrm{\Omega })$ near the boundary, that is, $h(x)\sim d^{-\beta }(x)$ for some $\beta \ge 0$ , when $x$ is close to the boundary $\mathrm{\partial }\mathrm{\Omega }$ . For $\beta <sp$ , we discuss the existence of a positive weak solution $(u,v)\in W_{\text{loc}}^{s,p}(\mathrm{\Omega })\times W_{\text{loc}}^{s,p}(\mathrm{\Omega })$ using the classical method of regularization and the fixed point theorem together. Indeed, we found some essential uniform a priori estimates for the approximating sequence before proceeding to the limits. Moreover, we address the uniqueness of finite energy solutions, that is, $(u,v)\in W_0^{s,p}(\mathrm{\Omega })\times W_0^{s,p}(\mathrm{\Omega })$ , and demonstrate that this solution pair is a saddle point of a suitable functional when $\alpha <1$ . We also provide the boundary behavior of the weak solutions in terms of the distance function. Finally, we establish the nonexistence of a weak solution for the case where $\beta \ge sp$ .