Multiplicity of solutions for a class of quasilinear problem in exterior domains with Neumann conditions

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We study the existence and multiplicity of solutions for a class of quasilinear elliptic problem in exterior domain with Neumann boundary conditions.

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Existence and multiplicity of positive solutions to a $p$-Laplacian equation in $\Bbb R^N$ (2006, unknown): “Let vn = un − u, reasoning as in [1] (see also [12]) , we can show that I ′ (vn) → 0 and I (vn) = I (un) − I (u) + on(1) = c − I (u) + on(1) = d + on(1).”
Positive and Nodal Solutions For a Nonlinear Schrödinger Equation with Indefinite Potential (2008, background): “Some of the ideas and calculations are inspired by the papers [10, 9, 1].”
Existence and multiplicity of solutions for Schrödinger–Poisson equations with sign-changing potential (2015, background): “There has been a vast literature on the study of existence and multiplicity of solutions of system (SP )λ under various hypotheses on the potential V (x) and the nonlinearity f(x, u), see [1-3, 5, 9-14, 18, 19, 21, 22, 24-28, 31, 34, 35-37] and the references therein.”
A multiplicity result via Ljusternick-Schnirelmann category and Morse theory for a fractional Schrödinger equation in $${\mathbb R^{N}}$$RN (2015, unknown): “Defining vn := un − u, by [2] we know that ∫ RN F (vn) = ∫ RN F (un) − ∫ RN F (u) + o(1) and arguing as in [18] we have also I ′ ε(vn) → 0.”