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 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.

n-Kirchhoff–Choquard equations with exponential nonlinearity

“…During last few decades, several authors such as in [5,8,11,12,33,34,35] used the Nehari manifold and associated fiber maps approach to study the multiplicity results with polynomial type nonlinearity and sign changing weight functions whereas the n-Laplace problems with exponential type nonlinearity has been addressed in [16,17,18]. In case of Kirchhoff equations with Choquard nonlinearity, we highlight that no result is avalaible in the current literature.…”

Quasilinear Choquard equations involving N-Laplacian and critical exponential nonlinearity

“…al [8]. More recently, authors in [13,14,15] studied the existence of multiple positive solutions for quasilinear equations involving exponential nonlinearities. Unlike as in the case of critical exponential problem involving N -Laplacian, where we generally consider the critical exponential growth as exp(|t| N/(N −1) ), in our problem (P * ), due to the presence of the quasilinear operator, the critical exponential growth becomes exp(|t| 2N/(N −1) ).…”

Lyapunov‐type inequalities for singular elliptic partial differential equations

“…We refer to previous studies 25,26 for Lyapunov‐type inequalities to fractional boundary value problems and other works 27,28 and the references therein for fractional partial differential equations. For the existence and multiplicity questions to N ‐Laplace as well as singular N ‐Laplace equations, one can consult previous studies 16,29–36 and the references therein.…”