Multiple solutions for nonlinear boundary value problems of Kirchhoff type on a double phase setting

Abstract

This paper deals with some classes of Kirchhoff type problems on a double phase setting and with nonlinear boundary conditions. Under general assumptions, we provide multiplicity results for such problems in the case when the perturbations exhibit a suitable behavior in the origin and at infinity, or when they do not necessarily satisfy the Ambrosetti–Rabinowitz condition. To this aim, we combine variational methods, truncation arguments and topological tools.

The Nehari manifold for double‐phase problems with convex and concave nonlinearities

“…[30], Cencelja et l. [7]. For previous work for the p ‐Laplacian equation or double‐phase problem with nonlinear boundary conditions of different type, see [17, 21, 28].…”

On a double phase problem of Kirchhoff-type with variable exponent

“…The same problem was studied in [17] by Fiscella and Pinamonti when p and q are constant, where the authors got a nontrivial solution via the critical point theory. Others recent results for constant case about this topic can be found in [3,16]. The sub-supersolution approach is a power tool to study the existence and multiplicity of solutions for nonlinear problems.…”

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].…”

“…Hence, the condition (M2) includes this classical example as well as cases that are not monotone. 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].…”