On finding solutions of a Kirchhoff type problem

Abstract

Consider the following Kirchhoff type problemwhereN −2 and a, b, λ, µ are positive parameters. By introducing some new ideas and using the well-known results of the problem (P) in the cases of a = µ = 1 and b = 0, we obtain some special kinds of solutions to (P) for all N ≥ 3 with precise expressions on the parameters a, b, λ, µ, which reveals some new phenomenons of the solutions to the problem (P). It is also worth to point out that it seems to be the first time that the solutions of (P) can be expressed precisely on the parameters a, b, λ, µ, and our results in dimension four also give a partial answer to Neimen's open problems

Ground State Solutions for Kirchhoff-type Problems with Critical Nonlinearity

“…Variational methods have been widely used in the last ten more years in studying the Kirchhoff type problems, see, for instance, [4,6,14,15,20,21,26] for results concerning bounded domain, and [1,3,9,12,13,19,22,24] for unbounded domain. Figueiredo [4] studies Kirchhoff type problem on bounded domain with the following version −M Ω |∇u| 2 dx ∆u = λf (x, u) + |u| 2 * −2 u in Ω and u = 0 on ∂Ω, where Ω is a smooth bounded domain in R N .…”

Two solutions for a class of singular Kirchhoff‐type problems with Hardy–Sobolev critical exponent II

“…When s = 0, Lei et al studied the critical case of problem (1.2) with p = 5, = g(x) ≡ 1 and obtained two positive solutions by using the variational method and perturbation method 4 ; the case of p = 3 is considered in Liao et al 5 The first work on the Kirchhoff-type problem with critical Sobolev exponent is Alves et al 6 After that, the Kirchhoff-type equation with critical exponent has been extensively studied, and some important and interesting results have been obtained, for example, previous studies. 4,[7][8][9][10][11][12][13][14][15][16][17][18][19][20] Let A s be the Hardy-Sobolev constant, and S be the best Sobolev constant, namely,…”

“…The first work on the Kirchhoff‐type problem with critical Sobolev exponent is Alves et al 6 After that, the Kirchhoff‐type equation with critical exponent has been extensively studied, and some important and interesting results have been obtained, for example, previous studies 4,7‐20 …”

An Optimal Control Problem Governed by a Kirchhoff-Type Variational Inequality

“…For more details on the physical and mathematical background of Kirchhoff-type problems, we refer the readers to the papers [1][2][3] and the references therein. By variational methods, many interesting results about the existence multiplicity of solutions for Kirchhoff-type problems have been established in the last ten years, see, e.g., [1][2][3][4][5][6][7] and the references therein.…”