In this paper, we investigate the multiplicity of solutions for a p-Kirchhoff system driven by a nonlocal integro-differential operator with zero Dirichlet boundary data. As a special case, we ...consider the following fractional p-Kirchhoff system {(∑i=1kuis,pp)θ−1(−Δ)psuj(x)=λj|uj|q−2uj+∑i≠jβij|ui|m|uj|m−2ujin Ω,uj=0in RN\Ω, where ujs,p=(∫∫R2N|uj(x)−uj( y)|p|x−y|N+psdxdy)1/p, j=1,2,...,k, k 2, θ 1, Ω is an open bounded subset of RN with Lipschitz boundary ∂Ω, N > ps with s∈(0,1), (−Δ)ps is the fractional p-Laplacian, λj>0 and βij=βji for i≠j, j=1,2,⋯,k. When 1<q<θp<2m<ps∗ and βij>0 for all 1 i<j k, two distinct solutions are obtained by using the Nehari manifold method. When 1<θp<2m q<ps∗ and βij∈R for all 1 i<j k or 1<θp<q<2m<ps∗ and βij>0 for all 1 i<j k, the existence of infinitely many solutions is obtained by applying the symmetric mountain pass theorem. To our best knowledge, our results for the above system are new in the study of Kirchhoff problems.
The Kirchhoff model is derived from the vibration problem of stretchable strings. This paper focuses on the longtime dynamics of a higher-order
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-coupled Kirchhoff system with ...higher-order rotational inertia and nonlocal damping. We first obtain the state of the model’s solutions in different spaces through prior estimation. After that, we immediately prove the existence and uniqueness of their solutions in different spaces through the Faedo-Galerkin method. Subsequently, we prove their family of global attractors using the compactness theorem. Finally, we reflect on the subsequent research of the model and point out relevant directions for further research on the model. In this way, we systematically study the longtime dynamics of the higher-order
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-coupled Kirchhoff model with higher-order rotational inertia, thus enriching the relevant findings of higher-order coupled Kirchhoff models and laying a theoretical foundation for future practical applications.
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This paper examines a class of fractional p-Kirchhoff systems driven by a nonlocal integro-differential operator with singular nonlinearity. By making use of Nehari manifold techniques, the existence ...of two nontrivial solutions is established. Our results extend those in Xiang et al. Nonlinearity 29(2016), 3186–3205 for the corresponding subcritical case.
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We study the nonlinear coupled Kirchhoff system with purely Sobolev critical exponent. By using appropriate transformation, we get one equivalent system involving a critical Schrödinger system and an ...algebraic system. Through solving the critical Schrödinger system with a corresponding algebraic system, under suitable conditions we obtain the existence and classification of positive ground states for the Kirchhoff system in dimensions 3 and 4. Furthermore, for the degenerate case, we give a complete classification of positive ground states for the Kirchhoff system in any dimension. To the best of our knowledge, this paper is the first to give classification results for the ground states of Kirchhoff systems. The results in this paper partially extend and complement the main results established by Lü and Peng Existence and asymptotic behavior of vector solutions for coupled nonlinear Kirchhoff-type system, J. Differ. Equ. 263 (2017) 8947–8978 considering the linearly coupled Kirchhoff system with subcritical exponent and some partial results established by Chen and Zou Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent, Arch. Ration. Mech. Anal. 205 (2012) 515–551; Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent: higher dimensional case, Calc. Var. Partial Differ. Equ. 52 (2015) 423–467, where the authors considered the coupled purely critical Schrödinger system.
The paper is concerned with existence, multiplicity and asymptotic behavior of nonnegative solutions for a fractional Schrödinger–Poisson–Kirchhoff type system. As a consequence, the results can be ...applied to the special case (a+b‖u‖2)(−Δ)su+V(x)u+ϕk(x)|u|p−2u=λh(x)|u|q−2u+|u|2s∗−2uinR3,(−Δ)tϕ=k(x)|u|pinR3,‖u‖=∫R3∫R3|u(x)−u(y)|2|x−y|3+2sdxdy+∫R3V(x)|u|2dx1∕2,where a,b≥0 are two numbers, with a+b>0, 1<p<2s,t∗=3+2t3−2s, λ>0 is a parameter, s,t∈(0,1), (−Δ)s is the fractional Laplacian, k∈L63+2t−p(3−2s)(R3) may change sign, V:R3→0,∞) is a potential function, 2s∗=6∕(3−2s) is the critical Sobolev exponent, 1<q<2s∗ and h∈L2s∗2s∗−q(R3). First, when θ<p<2s∗∕2, 2p≤q<2s∗ and λ is large enough, existence of nonnegative solutions is obtained by the mountain pass theorem. Moreover, we obtain that limλ→∞‖uλ‖=0. Then, via the Ekeland variational principle, existence of nonnegative solutions is investigated when θ<p<2s∗∕2, 1<q<2 and λ is small enough, and we obtain that limλ→0‖uλ‖=0. Finally, we consider the system with double critical exponents, that is, p=2s,t∗, and obtain two nontrivial and nonnegative solutions in which one is least energy solution and another is mountain pass solution. The paper covers a novel feature of Kirchhoff problems, which is the fact that the parameter a can be zero. Hence the results of the paper are new even for the standard stationary Schrödinger–Poisson–Kirchhoff system.
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The Kirchhoff model is derived from the vibration problem of stretchable strings. This article focuses on the long-time dynamics of a class of higher-order coupled Kirchhoff systems with nonlinear ...strong damping. The existence and uniqueness of the solutions of these equations in different spaces are proved by prior estimation and the Faedo-Galerkin method. Subsequently, the family of global attractors of these problems is proved using the compactness theorem. In this article, we systematically propose the definition and proof process of the family of global attractors and enrich the related conclusions of higher-order coupled Kirchhoff models. The conclusions lay a theoretical foundation for future practical applications.
In this paper, we investigate the multiplicity of solutions for Kirchhoff fractional p-Laplacian system in bounded domains:
By using the Nehari manifold method, together with Ekeland's variational ...principle, we show that there exist two distinct solutions under suitable conditions on weight functions
and h. Our results extend and generalize the main results in Xiang et al. Multiplicity of solutions for a class of quasilinear Kirchhoff system involving the fractional p-Laplacian. Nonlinearity. 2016;29:3186-3205 in Nonlinearity 2016, in which
are constants.
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