Two weak solutions for some Kirchhoff-type problem with Neumann boundary condition

Abstract

In this article we prove the existence of at least two weak solutions for a Kirchhoff-type problem by using the minimum principle, the mountain pass theorem and variational methods in Orlicz–Sobolev spaces.

Negative Energy Solutions for a New Fractionalpx-Kirchhoff Problem without the (AR) Condition

“…Many results concerning the existence and multiplicity of solutions have been appeared. Especially, [24] proved the existence of two weak solutions by using the variational methods in Orlicz-Sobolev spaces, [25] studied a class of Kirchhoff nonlocal fractional equations, and obtained the existence of three solutions. We also mention that [26] discussed a class of p-Kirchhoff equations via the fountain theorem and dual fountain theorem, and [27] studied the existence of nonnegative solutions for a Kirchhoff type problem driven by a nonlocal integrodifferential operator.…”

Multiplicity Results of Solutions to Non-Local Magnetic Schrödinger–Kirchhoff Type Equations in RN

“…which extends the classical D'Alembert's wave equation by taking into account the changes in the length of the strings during the vibrations. In this direction, the non-local problem of Kirchhoff type equations have been investigated in [34][35][36][37]. Now in order to confirm the existence of solutions to the nonlinear elliptic equations, the following Ambrosetti and Rabinowitz condition ((AR)-condition) given in [38] has been widely used;…”

A Sequence of Radially Symmetric Weak Solutions for Some Nonlocal Elliptic Problem in $${\mathbb {R}}^N$$

“…Laplace equation is the prototype for linear elliptic equations, as the most important partial differential equation of the second order. This equation has a non-linear counterpart, the so-called p-Laplace equation (see [1,13,14,18,19,21,22]). There has been a surge of interest in the p-Laplacian in many different contexts from game theory to mechanics and image processing.…”

“…In recent years, p-Kirchhoff type problems have been studied by many researchers, (for example see [19,20]). Beside elliptic problems with boundary conditions on bounded domain of R N which have extensive applications in different parts of scient and have been considered by many authors recently, see [21,22], some elliptic problems arise on unbounded domain R N , see [5,38]. It is worth mentioning that one of the difficulties in studying problems on unbounded domains is that there is no compact embedding for W 1,p (R N ).…”