Bounded state solutions of Kirchhoff type problems with a critical exponent in high dimension

Abstract

In the present paper, we consider the following Kirchhoff type problemwhere a is a positive constant, λ is a positive parameter, V ∈ L N 2 (R N ) is a given nonnegative function and 2 * is the critical exponent. The existence of bounded state solutions for Kirchhoff type problem with critical exponents in the whole R N (N ≥ 5) has never been considered so far. We obtain sufficient conditions on the existence of bounded state solutions in high dimension N ≥ 4, and especially it is the fist time to consider the case when N ≥ 5 in the literature. 129 130 QILIN XIE AND JIANSHE YU (i) N = 4 and λ > S −2 ; (ii) N ≥ 5 and λ > Λ 0 ; then problem (SK * ) has no nontrivial solution.Remark 1.6. Now, we have partially solved the existence of nontrivial solutions for the Kirchhoff type problem (SK * ). When N = 4, it follows from Theorem 1.1 and Theorem 1.5 that problem has at least one nontrivial solution for 0 < λ < S −2 ; and problem has no nontrivial solution for λ > S −2 . We do not know what will happen for λ = S −2 . So does the case when N ≥ 5 and λ = Λ 0 . Because of the limitation of our methods and technique, Theorem 1.3 only obtains nontrivial solutions for problem (SK * ) with Λ 0 − µ < λ < Λ 0 . A natural and interesting question is whether we can establish multiplicity theorems for problem (SK * ) with small positive parameter λ. We guess that the answer to the above problem is positive. The main difficulties when we investigate problem (SK * ) are that for Kirchhoff type problems with critical exponents, it is not easy to verify a range where Palais-Smale condition holds, especially when the energy level may be negative. This paper is organized as follows. In Section 2, we give the variational setting for our problem and investigate the solutions for the limit equation of (SK * ). After that, we prove that problem (SK * ) can not be solved by minimization on Nehari manifold. At the end of Section 2, we prove our Theorem 1.5 indirectly. In Section 3, by a description of Palais-Smale sequences, we obtain a local compactness result Proposition 3.2 with N = 4 and a global compactness result Proposition 3.3 with N ≥ 5. By a standard argument, we obtain the existence of the bounded state solutions in Section 4. In Section 5, we prove some necessary lemmas, which have been used in our main context. 2.Preliminaries. In this section, we give the variational setting for problem (SK * ) and some basic information on the limit equation of (SK * ). The main works in this paper are considered in the Hilbert space D 1,2 (R N ) = {u ∈ L 2 * (R N ) : QILIN XIE AND JIANSHE YU

A study on the critical Kirchhoff problem in high-dimensional space

“…Little results have been obtained for the high-dimensional cases N ≥ 4. The authors in Xie et al [45,46] studied the following critical Kirchhoff problem in D 1,2 (R N ), With some suitable assumptions on b and the potential V , they obtained a bound stated solution for N = 3, 4 and two solutions for N ≥ 5 by recovering a local (P S) c condition with c ∈ (c N , 2c N ) for N = 3 or 4 and c = c N,± for N ≥ 5, (…”

Existence and concentration of solutions for a class of biharmonic Kirchhoff equations with discontinuous nonlinearity

“…2,3 For more mathematical and physical background on Kirchhoff type problems, we refer the reads to previous studies. [4][5][6][7][8][9][10][11][12][13] Recently, many authors have studied the existence of solutions to these problems by different methods. For example, in their work, 14 Liu et al discussed problem (1.3) with the term V(x) and 𝑓 (u) = |u| p−1 u; by variational methods and some properties of Hessian matrix, the existence of ground state and nodal solutions are obtained.…”

“…In one and two dimensions, problem () is employed to demonstrate some phenomena appearing in different engineering, physical, and other sciences since it is regarded as a good approximation for describing nonlinear vibrations of plates or beams 2,3 . For more mathematical and physical background on Kirchhoff type problems, we refer the reads to previous studies 4–13 . Recently, many authors have studied the existence of solutions to these problems by different methods.…”