The Nehari manifold approach for N-Laplace equation with singular and exponential nonlinearities in ℝ^N

Abstract

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 ℝ N . Using the nature of the Nehari manifold and fibering maps associated with the Euler functional, we prove that there exists λ 0 such that for λ ∈ (0, λ 0 ), the problem admits at least two positive solutions. We also show that when h(x) > 0, there exists λ 0 such that (P λ ) has no solution for λ > λ 0 .