Nehari manifold for non-local elliptic operator with concave–convex nonlinearities and sign-changing weight functions

Abstract

In this article, we study the existence and multiplicity of non-negative solutions of the following p-fractional equation:

$$\quad \left\{ \begin{array}{lr}\displaystyle \!\! - 2{\int}_{\mathbb R^{n}}\!\frac{|u(y)-u(x)|^{p-2}(u(y)\,-\,u(x))}{|x-y|^{n+p\alpha}}\mathrm{d}y = \lambda h(x)|u|^{q-1}u\,+\, b(x)|u|^{r-1} u \; \text{in}\; {\Omega}, \\ \quad \quad\quad \quad\quad u =0\quad\quad \text{in}\;\mathbb{R}^{n}\setminus {\Omega},\quad u\in W^{\alpha,p}(\mathbb R^{n}) \end{array} \right. $$

where Ω is a bounded domain in $\mathbb R^{n}$ with continuous boundary, p ≥ 2, n > p α, α ∈ (0, 1), 0 < q < p −1 < r < p ^∗ − 1 with p ^∗ = np(n − p α)⁻¹, λ > 0 and h, b are sign-changing continuous functions. We show the existence and multiplicity of solutions by minimization on the suitable subset of Nehari manifold using the fibering maps. We find that there exists λ ₀ such that for λ ∈ (0, λ ₀), it has at least two non-negative solutions.