Least-energy nodal solutions of critical Kirchhoff problems with logarithmic nonlinearity
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Least-energy nodal solutions of critical Kirchhoff problems with logarithmic nonlinearity 3,4 Sihua Liang1,2 · Vicen¸tiu D. Radulescu ˘
Received: 2 July 2020 / Accepted: 20 August 2020 © The Author(s) 2020
Abstract In this paper, we are concerned with the existence of least energy sign-changing solutions for the following fractional Kirchhoff problem with logarithmic and critical nonlinearity: ∗ p a + b[u]s, p (−)sp u = λ|u|q−2 u ln |u|2 + |u| ps −2 u u=0
in , in R N \,
where N > sp with s ∈ (0, 1), p > 1, and p [u]s, p
=
R2N
|u(x) − u(y)| p d xd y, |x − y| N + ps
ps∗ = N p/(N − ps) is the fractional critical Sobolev exponent, ⊂ R N (N ≥ 3) is a bounded domain with Lipschitz boundary and λ is a positive parameter. By using constraint variational methods, topological degree theory and quantitative deformation arguments, we prove that the above problem has one least energy sign-changing solution u b . Moreover, for any λ > 0, we show that the energy of u b is strictly larger than two times the ground state energy. Finally, we consider b as a parameter and study the convergence property of the least energy sign-changing solution as b → 0. Keywords Fractional p-Laplacian · Critical problem · Logarithmic nonlinearity · Variational methods · Nodal solution Mathematics Subject Classification 35A15 · 35J60 · 47G20
B
Vicen¸tiu D. R˘adulescu [email protected]
Extended author information available on the last page of the article 0123456789().: V,-vol
45
Page 2 of 31
S. Liang et al.
1 Introduction In this paper, we are interested in the existence, energy estimates and the convergence property of the least energy sign-changing solution for the following fractional Kirchhoff problems with logarithmic and critical nonlinearity: ∗ p a + b[u]s, p (−)sp u = λ|u|q−2 u ln |u|2 + |u| ps −2 u in , u=0 in R N \,
(1.1)
where N > sp with s ∈ (0, 1), p > 1, and p [u]s, p
=
R2N
|u(x) − u(y)| p d xd y, |x − y| N + ps
ps∗ = N p/(N − ps) is the fractional critical Sobolev exponent, ⊂ R N (N ≥ 3) is a bounded domain with Lipshcitz boundary and λ is a positive parameter. We denote by (−)sp the fractional p-Laplace operator which, up to a normalization constant, is defined as (−)sp ϕ(x) = 2 lim
ε→0+ R N \Bε (x)
|ϕ(x) − ϕ(y)| p−2 (ϕ(x) − ϕ(y)) dy, x ∈ R N , |x − y| N + ps
for all ϕ ∈ C0∞ (R N ). Henceforward, Bε (x) denotes the open ball of R N centered at x ∈ R N and radius ε > 0. One of the classical topics in the qualitative analysis of PDEs is the study of existence and multiplicity properties of solutions for both the Kirchhoff problems and the fractional Kirchhoff problems under various hypotheses on the nonlinearity. In the recent past there is a vast literature concerning the existence and multiplicity of solutions for the following Dirichlet problem of Kirchhoff type ⎧ ⎨ ⎩
2 |∇u| d x u = f (x, u), x ∈ , − a+b u|∂ = 0.
(1.2)
Problem (1.2) is a generalization of a model introduced by Kirchhoff [24]. More precisely, Kirchhoff proposed a model given by the equation ∂ 2u ρ 2
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