EXISTENCE RESULTS FOR ANISOTROPIC FRACTIONAL (<i>p</i>₁(<i>x</i>, .), <i>p</i>₂(<i>x</i>, .))-KIRCHHOFF TYPE PROBLEMS

Multiple Solutions to a Non-Local Problem of Schrödinger–Kirchhoff Type in ℝN

“…Inspired by the fact illustrated in the remark above, the primary aim of the present paper is devoted to deriving the multiplicity result of solutions to the Kirchhoff-Schrödinger type problem with the fractional p-Laplacian on a class of a non-local Kirchhoff coefficient M that differs slightly from the previous related studies [11,15,18,22,28,[32][33][34][36][37][38]. In particular, for the superlinear p-Laplacian problem: −div(|∇w| p−2 ∇w) + V (y)|w| p−2 w = g(y, w) in R N , the existence result of a ground-state solution is dealt with in the paper [39].…”

“…In particular, to guarantee this compactness condition of an energy functional corresponding to problems of the elliptic type with the nonlinear term satisfying (F2), it is crucial that M(ζ) is non-decreasing for all ζ ≥ 0. Because of this reason, when (M3) is satisfied, many researchers have considered some conditions of the nonlinear term which differ from (F2); see [11,15,18,22,28,[31][32][33][34][36][37][38]45]. From this perspective, one of the novelties of the present paper is to accomplish the existence of a sequence of small energy solutions to (1) without the monotonicity of M when (F2) is assumed.…”

“…To this end, by utilizing the dual fountain theorem as the key tool, we provide the result of the existence of multiple solutions on classes of the Kirchhoff coefficient M and the nonlinear function g, which differ from the previous related studies [11,15,18,22,28,[31][32][33][34][36][37][38]45]. The basic idea of our proof for the existence of a sequence of small energy solutions to problem (1) comes from recent studies [45][46][47].…”

“…A typical model for M satisfying (M1) and (M3) is given by M(ζ) = 1 + aζ ϑ with a ≥ 0 for all ζ ≥ 0; hence, the condition (M3) includes the above classical example, as well as the case that is not monotone. For this reason, the nonlinear elliptic equations with the Kirchhoff coefficient satisfying (M3) (or (M3)) have recently been extensively researched by many authors; see [11,15,18,22,28,[32][33][34][36][37][38].…”