In this article, we study the following p-fractional Laplacian equationwhere Ω is a bounded domain in R n with smooth boundary, n > pα, p ≥ 2, α ∈ (0, 1), λ > 0 and b : Ω ⊂ R n → R is a sign-changing continuous function. We show the existence and multiplicity of non-negative solutions of (P λ ) with respect to the parameter λ, which changes according to whether 1 < β < p or p < β < p * = np n−pα respectively. We discuss both the cases separately. Non-existence results are also obtained.

In this article, we study the existence and multiplicity of non-negative solutions of following p-fractional equation:where Ω is a bounded domain in R n , p ≥ 2, n > pα, α ∈ (0, 1), 0 < q < p − 1 < r < np n−ps − 1, λ > 0 and h, b are sign changing smooth functions. We show the existence 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 solutions.

In this article, we study the existence of non-negative solutions of the class of non-local problem of n-Kirchhoff typewhere ⊂ R n is a bounded domain with smooth boundary, n ≥ 2 and f behaves like e |u| n n−1 as |u| → ∞. Moreover, by minimization on the suitable subset of the Nehari manifold, we study the existence and multiplicity of solutions, when f (x, t) is concave near t = 0 and convex as t → ∞.

In this article, we study the Fučik spectrum of fractional Laplace operator which is defined as the set of all (α, β) ∈ R 2 such thathas a non-trivial solution u, where Ω is a bounded domain in R n with Lipschitz boundary, n > 2s, s ∈ (0, 1). The existence of a first nontrivial curve C of this spectrum, some properties of this curve C, e.g. Lipschitz continuous, strictly decreasing and asymptotic behavior are studied in this article. A variational characterization of second eigenvalue of the fractional eigenvalue problem is also obtained. At the end, we study a nonresonance problem with respect to Fučik spectrum.

In this article, we study the existence and multiplicity of solutions of the singular N-Laplacian equation: [Formula: see text] where N ≥ 2, 0 ≤ q < N - 1 < p + 1, β ∈ [0, N), λ > 0, and h ≥ 0 in ℝN. Using the nature of the Nehari manifold and fibering maps associated with the Euler functional, we prove that there exists λ0 such that for λ ∈ (0, λ0), the problem admits at least two positive solutions. We also show that when h(x) > 0, there exists λ0 such that (Pλ) has no solution for λ > λ0.