Multiplicity and concentration behavior of positive solutions for a Schrödinger–Kirchhoff type problemviapenalization method

Abstract

In this paper we are concerned with questions of multiplicity and concentration behavior of positive solutions of the elliptic problemwhere ε is a small positive parameter, f : R → R is a continuous function, Lε is a nonlocal operator defined byM : IR+ → IR+ and V : IR 3 → IR are continuous functions which verify some hypotheses.

Existence and concentration of bound states for a Kirchhoff type problem with potentials vanishing or unbounded at infinity

“…For λ =1 in , He et al considered the problem under V satisfies and for some open‐bounded set ; Zhang and Zou studied the multiplicity and concentration of the positive solutions to under ; they established the relationship between the number of positive solutions and the profile of the potential V . For other results, we refer to Figueiredo et al, Figueiredo and Santos Júnior, He and Li, Liu and Guo, Li et al, Sun and Wu, Wang and Xiao, and references therein.…”

“…They obtained the existence, concentration, and decay properties of the ground state solution of (5) under the assumption of (4) by minimax theorems. For = 1 in (5), He et al 21 considered the problem (5) under V satisfies inf x∈R 3 V(x) > 0 and inf Ω V < min Ω V for some open-bounded set Ω ⊂ R 3 ; Zhang and Zou 22 studied the multiplicity and concentration of the positive solutions to (5) under inf x∈R 3 V(x) > 0; they established the relationship between the number of positive solutions and the profile of the potential V. For other results, we refer to Figueiredo et al, 23 Figueiredo and Santos Júnior, 24,25 He and Li, 26 Liu and Guo, 27 Li et al, 28 Sun and Wu, 29 Wang and Xiao, 30 and references therein.…”

Existence and concentration of positive ground states for a Kirchhoff equation involving critical Sobolev exponent

“…He et al [22] not only obtained that bound states concentrate around a local minimum point of V in Λ as → 0 + , but also obtained multiple solutions by employing the topology construct of the set where the potential V (z) attains its local minimums. We also notice that Figueiredo and Júnior [18] employed penalization method to study the following Kirchhoff type problem…”

“…Figueiredo and Júnior [18] also investigated the multiplicity and concentration behavior of positive solutions for the above problem. For more related existence results on (1.3), we refer to [5,17,18,24,[27][28][29][30]36]. Motivated by the above fact, we will consider (SK ) involving critical Sobolev exponent and study the existence and concentration of positive ground state solutions in the case where V (x) satisfies the assumption (V ).…”

On a Kirchhoff type problem inRN

“…The unique positive solution is radial symmetric and is also the unique least energy solution to (P 0,1 ), while the energy values of the sign-changing solutions go to infinity. On the other hand, to the best of our knowledge, for the Kirchhoff type problem (P α,β ), only the existence result of one positive solution with N = 1, 2, 3 and p ∈ (4, 2 * ) was established in [3] by Alves and Figueiredo. Simultaneously, in recent years, the Kirchhoff type problems in the whole space R N with N = 1, 2, 3 have been studied widely by the variational methods since then the nice work [8], and various existence results of the solutions to such problems were established, see for example [1,3,6,7,[9][10][11][12]14] and the references therein. Inspired by the above facts, the purpose of the current paper is to make a detailed description on the solutions of (P α,β ).…”