Zhong, Xiao-Jing; Tang, Chun-Lei Nodal solutions for a critical Kirchhoff type problem in \(\mathbb{R}^N\). (English) Zbl 1484.35227 Topol. Methods Nonlinear Anal. 58, No. 2, 549-568 (2021). Summary: In the present paper, we concentrate on the following critical Kirchhoff type problem \[ -\bigg( a+b\int_{\mathbb{R}^N}|\nabla u|^2 dx\bigg) \triangle u+u=|u|^{2^{\ast}-2}u+\mu |u|^{p-2}u,\quad \text{in } \mathbb{R}^N, \] where \(N\geq 3\), \(a, b> 0\), \(p\in (2, 2^{\ast})\) and \(\mu\) is an arbitrary positive parameter. With the help of an equivalent transformation, we first obtain at least one ground state nodal solution with precisely two nodal domains for \(N=3\), all \(b> 0\) and \(N\geq 4\), \(b> 0\) small enough. Moreover, we give a convergence property of ground state nodal solutions as \(b\searrow 0\). Besides, we attain infinitely many nodal solutions for \(N=3\), \(p\in (4, 6)\), all \(b> 0\) and \(N\geq 4\), \(p\in (2, 2^{\ast})\), \(b> 0\) sufficiently small, and also establish nonexistence results of nodal solutions for \(N\geq 4\) and \(b\) large enough. Cited in 2 Documents MSC: 35J62 Quasilinear elliptic equations 35B33 Critical exponents in context of PDEs 35A01 Existence problems for PDEs: global existence, local existence, non-existence Keywords:Kirchhoff-type problem; critical growth; existence; non-existence × Cite Format Result Cite Review PDF Full Text: DOI References: [1] T. Bartsch and M. Willem, Infinitely many radial solutions of a semilinear elliptic problem on \(\mathbb R^N\), Arch. Rational Mech. Anal. 124 (1993), 261-276. · Zbl 0790.35020 [2] H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure. Appl. Math. 36 (1983), 437-477. · Zbl 0541.35029 [3] D.M. Cao and X.P. 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