In this paper, we investigate a fractional p(·)-Kirchhoff type problem involving variable exponent logarithmic nonlinearity. With the help of the Nehari manifold approach, the existence and multiplicity of nontrivial weak solutions for the above problem are obtained. The main aspect and challenges of this paper are the presence of double non-local terms and logarithmic nonlinearity.

In this paper we prove the existence of multiple nontrivial solutions of the following equation.where Ω ⊂ R N is a smooth bounded domain with N ≥ 3, 1 < q ′ < q < p − 1; λ, and f satisfies certain conditions, µ > 0 is a Radon measure, q ′ = q q−1 is the conjugate of q.

The aim of this paper is to deal with the elliptic pdes involving a nonlinear integrodifferential operator, which are possibly degenerate and covers the case of fractional p-Laplacian operator. We prove the existence of a solution in the weak sense to the problem(p ′ being the conjugate of p), exists in a weak sense, for q ∈ (p, p * s ) under certain condition on λ, where −L Φ is a general nonlocal integrodifferential operator of order s ∈ (0, 1) and p * s is the fractional Sobolev conjugate of p. We further prove the existence of a measure µ * corresponding to which a weak solution exists to the problem −L Φ u = λ|u| q−2 u + µ * in Ω, u = 0 in R N \ Ω depending upon the capacity. keywords: Nonlocal operators, fractional p-laplacian; elliptic PDE; fractional Sobolev space.

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