A study of the nonlocal solution of p-Laplacian fractional elliptic problems via approximation methods

In the present paper, we are concerned with the existence of nonnegative solutions for two Kirchhoff-type problems driven by a fractional p-Laplacian operator, ${(-\mathrm{\Delta })}_p^s$ $(-\Delta )_{p}^{s}$ , in a bounded smooth domain Ω of ${\mathbb{R}}^N$ $\mathbb{R}^{N}$ with $N>ps$ $N>ps$ . by employing approximation methods, we establish the existence of nonnegative solutions for each problems under some hypotheses on $f,g:\mathrm{\Omega }\times \mathbb{R}\to \mathbb{R}$ $f,g : \Omega \times \mathbb{R}\to \mathbb{R}$ and weak conditions on the diffusion coefficients $\mathcal{M},\mathcal{N}:(0,+\mathrm{\infty })\to \mathbb{R}$ $\mathcal{M}, \mathcal{N} :(0,+\infty )\to \mathbb{R}$ that we will present. The regularity of finite-energy solutions of both problems is also studied by imposing a few extra hypotheses on Ω, f, and g.