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

Abstract

Abstract 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, ( − Δ ) p s $(-\Delta )_{p}^{s}$ , in a bounded smooth domain Ω of R N $\mathbb{R}^{N}$ with N > p s $N>ps$ . by employing approximation methods, we establish the existence of nonnegative solutions for each problems under some hypotheses on f , g : Ω × R → R $f,g : \Omega \times \mathbb{R}\to \mathbb{R}$ and weak conditions on the diffusion coefficients M , N : ( 0 , + ∞ ) → 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.

Keywords

• Stationary Kirchhoff problems
• Nonlocal p-Laplacian operator
• Nontrivial solution
• Positive solution
• Approximation method
• Regularity