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

2019 ◽  
Vol 18 (1) ◽  
pp. 129-158 ◽  
Author(s):  
Qilin Xie ◽  
◽  
Jianshe Yu
Keyword(s):  
Critical Exponent ◽  
High Dimension ◽  
Kirchhoff Type ◽  
Bounded State ◽  
Kirchhoff Type Problems

scholarly journals Superlinear Schrödinger–Kirchhoff type problems involving the fractional p–Laplacian and critical exponent

2019 ◽  
Vol 9 (1) ◽  
pp. 690-709 ◽  
Author(s):  
Mingqi Xiang ◽  
Binlin Zhang ◽  
Vicenţiu D. Rădulescu
Keyword(s):  
Critical Exponent ◽  
Nonlocal Problem ◽  
Growth Conditions ◽  
Infinitely Many Solutions ◽  
Kirchhoff Type ◽  
Existence And Multiplicity ◽  
Concentration Compactness Principle ◽  
Superlinear Growth ◽  
Sobolev Exponent ◽  
Kirchhoff Type Problems

Abstract This paper concerns the existence and multiplicity of solutions for the Schrődinger–Kirchhoff type problems involving the fractional p–Laplacian and critical exponent. As a particular case, we study the following degenerate Kirchhoff-type nonlocal problem: $$\begin{align}& \left\| u \right\|_{\lambda }^{\left( \theta -1 \right)p}\left[ \lambda \left( -\Delta \right)_{p}^{s}u+V\left( x \right){{\left| u \right|}^{p-2}}u \right]={{\left| u \right|}^{p_{s}^{\star }-2}}u+f\left( x,u \right)\,in\,{{\mathbb{R}}^{N}}, \\ & {{\left\| u \right\|}_{\lambda }}={{\left( \lambda \int\limits_{\mathbb{R}}{\int\limits_{2N}{\frac{{{\left| u\left( x \right)-u\left( y \right) \right|}^{p}}}{{{\left| x-y \right|}^{N+ps}}}}dxdy+\int\limits_{{{\mathbb{R}}^{N}}}{V\left( x \right){{\left| u \right|}^{p}}dx}} \right)}^{{1}/{p}\;}} \\ \end{align}$$ where $\left( -\Delta \right)_{p}^{s}$is the fractional p–Laplacian with 0 < s < 1 < p < N/s, $p_{s}^{\star }={Np}/{\left( N-ps \right)}\;$is the critical fractional Sobolev exponent, λ > 0 is a real parameter, $1<\theta \le {p_{s}^{\star }}/{p}\;,$and f : ℝN × ℝ → ℝ is a Carathéodory function satisfying superlinear growth conditions. For $\theta \in \left( 1,{p_{s}^{\star }}/{p}\; \right),$by using the concentration compactness principle in fractional Sobolev spaces, we show that if f(x, t) is odd with respect to t, for any m ∈ ℕ+ there exists a Λm > 0 such that the above problem has m pairs of solutions for all λ ∈ (0, Λm]. For $\theta ={p_{s}^{\star }}/{p}\;,$by using Krasnoselskii’s genus theory, we get the existence of infinitely many solutions for the above problem for λ large enough. The main features, as well as the main difficulties, of this paper are the facts that the Kirchhoff function is zero at zero and the potential function satisfies the critical frequency infx∈ℝ V(x) = 0. In particular, we also consider that the Kirchhoff term satisfies the critical assumption and the nonlinear term satisfies critical and superlinear growth conditions. To the best of our knowledge, our results are new even in p–Laplacian case.

scholarly journals Existence and Multiplicity of Positive Solutions for Kirchhoff-Type Equations with the Critical Sobolev Exponent

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Junjun Zhou ◽  
Xiangyun Hu ◽  
Tiaojie Xiao
Keyword(s):  
Critical Exponent ◽  
Positive Solutions ◽  
Mountain Pass Theorem ◽  
Critical Sobolev Exponent ◽  
Mountain Pass ◽  
Kirchhoff Type ◽  
Existence And Multiplicity ◽  
Sobolev Exponent ◽  
Multiplicity Of Positive Solutions ◽  
Kirchhoff Type Problems

In this paper, we consider the following Kirchhoff-type problems involving critical exponent −a+b∫Ω∇u2dxΔu+Vxu=μu2∗−1+λgx,u, x∈Ωu>0, x∈Ωu=0, x∈∂Ω. The existence and multiplicity of positive solutions for Kirchhoff-type equations with a nonlinearity in the critical growth are studied under some suitable assumptions on Vx and gx,u. By using the mountain pass theorem and Brézis–Lieb lemma, the existence and multiplicity of positive solutions are obtained.

scholarly journals Multiple solutions for critical Choquard-Kirchhoff type equations

2020 ◽  
Vol 10 (1) ◽  
pp. 400-419 ◽  
Author(s):  
Sihua Liang ◽  
Patrizia Pucci ◽  
Binlin Zhang
Keyword(s):  
Critical Exponent ◽  
Critical Exponents ◽  
Sobolev Inequality ◽  
Multiple Solutions ◽  
Concentration Compactness ◽  
Positive Real ◽  
Multiplicity Results ◽  
Kirchhoff Type ◽  
Concentration Compactness Principle ◽  
Compactness Principle

Abstract In this article, we investigate multiplicity results for Choquard-Kirchhoff type equations, with Hardy-Littlewood-Sobolev critical exponents, $$\begin{array}{} \displaystyle -\left(a + b\int\limits_{\mathbb{R}^N} |\nabla u|^2 dx\right){\it\Delta} u = \alpha k(x)|u|^{q-2}u + \beta\left(\,\,\displaystyle\int\limits_{\mathbb{R}^N}\frac{|u(y)|^{2^*_{\mu}}}{|x-y|^{\mu}}dy\right)|u|^{2^*_{\mu}-2}u, \quad x \in \mathbb{R}^N, \end{array}$$ where a > 0, b ≥ 0, 0 < μ < N, N ≥ 3, α and β are positive real parameters, $\begin{array}{} 2^*_{\mu} = (2N-\mu)/(N-2) \end{array}$ is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality, k ∈ Lr(ℝN), with r = 2∗/(2∗ − q) if 1 < q < 2* and r = ∞ if q ≥ 2∗. According to the different range of q, we discuss the multiplicity of solutions to the above equation, using variational methods under suitable conditions. In order to overcome the lack of compactness, we appeal to the concentration compactness principle in the Choquard-type setting.

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