A uniqueness result for a nonlocal equation of Kirchhoff type and some related open problem

Abstract

We study the uniqueness of solution for the following boundary value problem involving a nonlocal equation of Kirchhoff typeHere, Ω is a bounded open set in R n with smooth boundary, a, b, λ are positive real numbers and f : R → R is a continuous function. In particular, we give an answer to an open problem recently proposed by B. Ricceri.

Existence and multiplicity of positive solutions for a class of singular Kirchhoff type problems with sign‐changing potential

“…The Kirchhoff type problem has been extensively studied. For examples, to our best knowledge, Ma and Muñoz Rivera were the first result by the variational method; a quasilinear elliptic equation of Kirchhoff type was considered in Alves et al; the Kirchhoff type problem with critical exponent was first investigated by Alves et al; for the uniqueness result, see Anello and Liao et al; for the multiplicity of solutions for a superlinear Kirchhoff type equations with critical Sobolev exponent in , see Li and Liao; He and Zou considered the infinitely many solutions of the Kirchhoff type problem; the sign‐changing solution was studied by Mao and Zhang, Tang and Cheng and Zhang and Perera; for bound state solutions, see Xie et al; Naimen was the first that considered the Kirchhoff type problem in dimension four; the Kirchhoff type problems was considered by the Yang index in Perera and Zhang. …”

“…When 0 ≤ |g| ∞ ≤ bA 2 , since  =  + ∪  0 , obviously u * ∈  + . While for the case of |g| ∞ > bA 2 , 0 < |h| 6 5+ < T, according to (15) and (34), it follows that…”

Existence and multiplicity of positive solutions for a class of Kirchhoff type problems with singularity

“…on a smooth bounded domain Ω ⊂ R 3 and f : Ω×R → R a continuous function, has been extensively studied (see [1,3,10,11,[15][16][17]19,18,20,23,24,30]). Particularly, in [24] Sun and Tang have considered the following problem…”

Existence of Multiple Solutions for a p‐Kirchhoff Problem with Nonlinear Boundary Conditions

“…Problem (1) is related to the stationary analogue of problem (2). After Kirchhoff's work, various models of Kirchhoff-type have been studied by many authors: we refer the readers to [2–9]. In [4], by the variational methods, Bensedik and Bouchekif considered the problem $\begin{array}{c}-M\left({{\displaystyle false\int }}_{\Omega }{\left|\nabla u\right|}^{\mathrm{2}}dx\right)\mathrm{\Delta }u=f\left(x,u\right),\hspace{1em}x\in \Omega ,\\ u=\mathrm{0},\hspace{1em}\mathrm{on}\hspace{0.17em}\hspace{0.17em}\partial \Omega ,\end{array}$ where Ω is a bounded domain in ℝ N .…”