Kirchhoff type elliptic equations with double criticality in Musielak–Sobolev spaces

Abstract

= 0 on 𝜕Ω, where Ω ⊆ R N , N ≥ 2 is a bounded and smooth domain containing two open and connected subsets Ω p and Ω N such that Ωp ∩ ΩN = ∅ and Δin Ω p and to Δ N in Ω N , the nonlinear function 𝑓 ∶ Ω × R → R acts as |t| p * (x)−2 t on Ω p and as e 𝛼|t| N∕(N−1) on Ω N for sufficiently large |t|. To establish the existence results in a Musielak-Sobolev space, we use a variational technique based on the mountain pass theorem.

Multiplicity Results of Solutions to the Double Phase Problems of Schrödinger–Kirchhoff Type with Concave–Convex Nonlinearities

“…The study of elliptic problems with the non-local Kirchhoff term was initially introduced by Kirchhoff [14] in order to study an extension of the classical d'Alembert's wave equation by taking into account the changes to the lengths of strings during vibration. The variational problems of the Kirchhoff type have had influence in various applications in physics and have been intensively investigated by many researchers in recent years; for examples, see [15][16][17][18][19][20][21][22][23][24][25][26][27][28] and the references therein. A detailed discussion about the physical implications based on the fractional Kirchhoff model was initially suggested by the work of Fiscella and Valdinoci [20].…”

“…Under this condition, the authors of [18] obtained multiplicity results for certain classes of double phase problems of the Kirchhoff type with nonlinear boundary conditions; also, see [19] for the Dirichlet boundary condition. For these reasons, the nonlinear elliptic equations with a Kirchhoff coefficient satisfying (M2) have been comprehensively investigated by many researchers in recent years [15,[17][18][19]21,25,27,28].…”

Infinitely many small energy solutions to the $ p $-Laplacian problems of Kirchhoff type with Hardy potential

“…Introduction. Stationary problems involving singular nonlinearities have garnered increasing attention because they can be corroborated as a model for many physical phenomena; see [1,8,18,43,44,48,55,57] for more details. Moreover, motivated by this large interest, singular problems have been investigated more in recent years; see [10,25,27].…”

“…Hence, this condition includes the non-monotonic case as well as the typical example above. For this reason, in recent years, many researchers have studied nonlinear elliptic equations in which the Kirchhoff coefficients satisfy (K1) and (KF1); see [1,8,18,29,35,48,55,57]. For some more recent works, [52] focused on a class of fractional p-Laplacian differential operators with variable exponents, and describes the appropriate function space to deal with.…”

“…Especially, if the nonlinear term satisfies (Ψ2), we can verify this compactness condition of the energy function corresponding to a given problem when K(t) is increasing for all t ≥ 0. For this reason, when (KF2) is assumed, many authors considered a condition of the nonlinear term, which is different from (Ψ2); see [1,6,8,18,29,48,[55][56][57]. Very recently, regarding the Kirchhoff type problem with Hardy potential, the multiplicity result to the nonlocal p 1 &• • •&p m fractional Laplacian problems has been established by the authors of [6] when the Kirchhoff coefficients satisfy (KF1) and a condition on the nonlinear term differs from (Ψ2).…”