Construction of pseudo-gradient vector field and sign-changing multiple solutions involving p-Laplacian

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Multiple solutions for coercive p-Laplacian equations (2006, background): “We refer to [6, 10, 17] for recent work on the existence of three nonzero solutions for coercive elliptic equations.”
Existence of multiple solutions with precise sign information for superlinear Neumann problems (2009, background): “WementiontheworksofBartschandLiu[6],GarciaAzoreroetal.[17],Motreanuetal.[25], Papageorgiou and Papageorgiou [27 ]f orp-superlinear Dirichlet problems and by Carl and Perera [10], Zhang and Li [32], Zhang et al. [ 31 ]f orp-linear Dirichlet problems.”
Existence of five nonzero solutions with exact sign for A p-laplacian equation (2009, methodology): “In [9, 45, 47], the authors considered asymptotically (p− 1)-linear problems (at infinity), while we are dealing with a superlinear (and subcritical) problem at infinity, see the Ambrosetti-Rabinowitz type hypothesis (f3) (or (Hf )(v) below).”
Anisotropic nonlinear Neumann problems (2011, unknown): “We mention the works of Bartsch–Liu [6], Carl–Perera [8], Dancer–Perera [12], Filippakis–Kristaly– Papageorgiou [20], Gasiński–Papageorgiou [23], Liu–Liu [30], Papageorgiou–Papageorgiou [34,35] and Zhang–Chen–Li [38].”