Search

This work discusses a class of double phase variable exponent problems of Kirchhoff-type. By utilizing sub-supersolutions method in certain appropriate Musielak-Orlicz Sobolev space, we establish the existence of at least one nonnegative solution. Furthermore, with the help of additional assumption on the nonlinearity, a second nonnegative solution is obtained by employing variational arguments. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence of infinitely many solutions for a class of quasilinear elliptic m-polyharmonic Kirchhoff equations where the nonlinear function has a quasicritical growth at infinity and without assuming the Ambrosetti and Rabinowitz type condition. The new aspect consists in employing the notion of a Schauder basis to verify the geometry of the symmetric mountain pass theorem. Furthermore, we introduce a positive quantity $ \lambda _M $ λ M similar to the first eigenvalue of the m-polyharmonic operator to find a mountain pass solution, and also to discuss the sublinear case under large growth conditions at infinity and at zero. Our results are an improvement and generalization of the corresponding results obtained by Colasuonno-Pucci (Nonlinear Analysis: Theory, Methods and Applications, 2011) and Bae-Kim (Mathematical Methods in the Applied Sciences, 2020). [ABSTRACT FROM AUTHOR]

In this paper, we deal with fractional [Formula: see text]-Laplace Kirchhoff equations with exponential growth of the form 𝜀ps(a + b[u] s,pp)(−Δ) psu + Z(x)|u|p−2u = h(u)in ℝN, where [Formula: see text] is a positive parameter, [Formula: see text], [Formula: see text] and [Formula: see text]. Under some appropriate conditions for the nonlinear function [Formula: see text] and potential function [Formula: see text], and with the help of penalization method and Lyusternik–Schnirelmann theory, we establish the existence, multiplicity and concentration of solutions. To some extent, we fill in the gaps in [W. Chen and H. Pan, Multiplicity and concentration of solutions for a fractional [Formula: see text]-Kirchhoff type equation, Discrete Contin. Dyn. Syst. 43 (2023) 2576–2607; G. Figueiredo and J. Santos, Multiplicity and concentration behavior of positive solutions for a Schrödinger–Kirchhoff type problem via penalization method, ESAIM Control Optim. Calc. Var. 20 (2014) 389–415; X. He and W. Zou, Existence and concentration behavior of positive solutions for a Kirchhoff equation in [Formula: see text] J. Differential Equations 252 (2012) 1813–1834; J. Wang, L. Tian, J. Xu and F. Zhang, Multiplicity and concentration of positive solutions for a Kirchhoff type problem with critical growth, J. Differential Equations 253 (2012) 2314–2351].

This paper dealt with a class of Kirchhoff type equations involving singular nonlinearity in a closed Riemannian manifold $ (M, g) $ of dimension $ n\ge3 $. Existence and uniqueness of a positive weak solution were obtained under certain assumptions with the help of the variation methods and some analysis techniques.

We consider the following Kirchhoff equation in the Sobolev critical case with combined power nonlinearities having prescribed mass ∫ R 3 | u | 2 = c 2 , where a , c , μ > 0 are positive constants, b > 0 is a positive parameter, 2 < q < p ¯ : = 2 + 8 3 which is L 2 -critical exponent. For the L 2 -subcritical case 2 < q < 10 3 and Sobolev critical case, Li et al. (2021) proved that (K) has a solution which is ground state solution and corresponds to local minima of the associated energy functional. Here we extend the result in Li et al. (2021) by proving that (K) has the second solution which is not a ground state and is located at a mountain-pass level of the energy functional. Meanwhile, let u b are normalized solutions of mountain-pass type to (K) , then u b → u in H 1 (R 3) as b → 0 up to a subsequence, where u ∈ H 1 (R 3) is a normalized solution of mountain-pass type to - a ▵ u = λ u + μ | u | q - 2 u + | u | 4 u in R 3. Our results also extend the results of Soave (J Differ Equ 269:6941–6987, 2020; J Funct Anal 279:108610, 2020). [ABSTRACT FROM AUTHOR]

In this article, we consider the following Kirchhoff equation involving fractional Laplacian a + b [ u ] s 2 (- Δ) s u + u = | u | p - 2 u in Ω , u = 0 on R 3 \ Ω , where a , b > 0 are constants, 3 4 < s < 1 , [ u ] s is the so-called Gagliardo (semi)norm of u, 4 < p < 2 s ∗ = 6 3 - 2 s and Ω ⊂ R 3 is an exterior domain with smooth boundary ∂ Ω ≠ ∅. By establishing a global compactness lemma of the fractional Kirchhoff equation in exterior domains, we verify the compactness of Palais–Smale sequences corresponding to above problem at higher energy level interval. Then combining some crucial estimates and barycentric function, we determine the existence of positive bound state solutions provided that R 3 \ Ω is contained in a small ball. In addition, we point out that the main result can be extended to fractional Sobolev critical case with small parameter. [ABSTRACT FROM AUTHOR]

We deal with the following singular perturbation Kirchhoff equation: − ϵ 2 a + ϵ b ∫ R 3 | ∇ u | 2 d y Δ u + Q (y) u = | u | p − 1 u , u ∈ H 1 (R 3) , where constants a , b , ϵ > 0 and 1 < p < 5 . In this paper, we prove the uniqueness of the concentrated solutions under some suitable assumptions on asymptotic behaviors of Q (y) and its first derivatives by using a type of Pohozaev identity for a small enough ϵ. To some extent, our result exhibits a new phenomenon for a kind of Q (x) which allows for different orders in different directions. [ABSTRACT FROM AUTHOR]

The results of model numerical calculations are presented, showing the effect of the TRT vacuum vessel on the amplitudes and phases of the magnetic sensors signals, which are located on the inner and outer vacuum vessel surfaces. It is shown that the characteristic times of loop voltage sensors considerably depend on their position on the TRT vacuum vessel. Therefore, their accurate mutual matching is required, especially in the dynamic stage of the discharge, when high eddy currents are induced in the vacuum vessel. The results of numerical calculations for the case of periodic disturbances in the plasma column are presented. They showed that the vacuum vessel almost completely shields the signals of the magnetic sensors located on the outer surface of the vacuum vessel. Moreover, it affects not only the amplitudes of magnetic sensors signals, but also their phases. Numerical studies brought us to conclusion that it is of priority to install the magnetic sensors just on the inner surface of the TRT vacuum vessel. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following Schrödinger–Kirchhoff equation - a + b ∫ R N | ∇ u | 2 d x Δ u + V (x) u = f (x , u) , in R N , where N ≥ 3 , a and b are positive parameters, V(x) is a positive and continuous potential. Under some suitable assumptions on the nonlinearity f(x, u) which allow it is linearly bounded at infinity, the existence of least energy solutions and their asymptotic behavior as b → 0 are established via variational methods. The nonexistence of nontrivial solutions is also established for large b. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence of solutions for a class of integrodifferential Kirchhoff equations. These equations involve a nonlocal operator with a measurable kernel that satisfies "structural properties" that are more general than the standard kernel of the fractional Laplacian operator. Additionally, the potential can be periodic or asymptotically periodic, and the nonlinear term exhibits critical exponential growth in the sense of Trudinger–Moser inequality. To guarantee the existence of solutions, we employ variational methods, specifically the mountain-pass theorem. In this context, it is important to emphasize that we have additional difficulties due to the lack of compactness in our problem, because we deal with critical growth nonlinearities in unbounded domains. Moreover, the Kirchhoff term adds complexity to the problem, as it requires suitable calculations for control the estimate the minimax level, representing the main challenge in this work. Finally, we consider two different approaches to estimate the minimax level. The first approach is based on a hypothesis proposed by D. M. Cao, while the second one involves a slightly weaker assumption addressed by Adimurthi and Miyagaki. [ABSTRACT FROM AUTHOR]

Given a Musielak–Orlicz function φ (x , s) : Ω × [ 0 , ∞) → R on a bounded regular domain Ω ⊂ R n and a continuous function M : [ 0 , ∞) → (0 , ∞) , we show that the eigenvalue problem for the elliptic Kirchhoff's equation - M ∫ Ω φ (x , | ∇ u (x) |) d x div ∂ φ ∂ s (x , | ∇ u (x) |) ∇ u (x) | ∇ u (x) | = λ ∂ φ ∂ s (x , | u (x) |) u (x) | u (x) | has infinitely many solutions in the Sobolev space W 0 1 , φ (Ω) . No conditions on φ are required beyond those that guarantee the compactness of the Sobolev embedding theorem. [ABSTRACT FROM AUTHOR]

In this paper we are concerned with Kirchhoff equations with strong damping and quasi-periodic external forcing on arbitrary flat tori. Such models arise from nonlinear forced vibrations of multidimensional bodies in which the dependence of the tension on the deformation cannot be neglected. We establish the existence of quasi-periodic traveling waves whose frequency is consistent with that of the forcing term. Due to strong damping, we don't impose any non-resonance conditions. Our arguments provide a general framework for studying singular perturbation problems with strong damping. [ABSTRACT FROM AUTHOR]

In this paper, we mainly investigate the following nonlinear Kirchhoff equation ... where a, b > 0 are constants, 2 < q < 6, and ε > 0 is a parameter. Under some suitable assumptions on the function Q(x), we obtain that the equation above has positive multi-peak solutions concentrating at a critical point of Q(x) for ε > 0 sufficiently small, by using the finite dimensional reduction method. Different from the local Schrödinger problem, here the corresponding limit problem is a system. Moreover, the nonlocal term brings some new difficulties which involve some technical and complicated estimates. [ABSTRACT FROM AUTHOR]

In this paper, we study the Kirchhoff elliptic equations of the form - M(∥∇λu∥²)Δλu = w(x)f(u) in RN, where M is a smooth monotone function, w is a weight function and f(u) is of the form up, eu or -u-p. The operator Δλ is strongly degenerate and given by .... We shall prove some classifications of stable solutions to the equation above under general assumptions on M and λj, j = 1, ..., N. [ABSTRACT FROM AUTHOR]

In this paper, we use variational methods to study the existence and multiplicity of non negative solutions for a p-Kirchhoff equation involving critical Sobolev exponent. [ABSTRACT FROM AUTHOR]

In this paper, we study the least energy sign-changing solutions to the following nonlinear Kirchhoff equation − (a + b ∫ V | ∇ u | 2 d μ) Δ u + c (x) u = f (u) on a locally finite graph G = (V , E) , where a, b are positive constants. We use the constrained variational method to prove the existence of a least energy sign-changing solution u b of the above equation if c (x) and f satisfy certain assumptions, and to show the energy of u b is strictly larger than twice that of the least energy solutions. Moreover, if we regard b as a parameter, as b → 0 + , the solution u b converges to a least energy sign-changing solution of a local equation − a Δ u + c (x) u = f (u). [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence of sign-changing solutions for the following class of fractional Kirchhoff type equations with potential (1+b[u]α2)((-Δx)αu-Δyu)+V(x,y)u=f(x,y,u),(x,y)∈ℝN=ℝn×ℝm,\left( {1 + b\left[ u \right]_\alpha ^2} \right)\left( {{{\left( { - {\Delta _x}} \right)}^\alpha }u - {\Delta _y}u} \right) + V\left( {x,y} \right)u = f\left( {x,y,u} \right),\left( {x,y} \right) \in {\mathbb{R}^N} = {\mathbb{R}^n} \times {\mathbb{R}^m}, where [u]α=(∫ℝN(|(-Δx)α2u|2+|∇yu|2)dxdy)12{\left[ u \right]_\alpha } = {\left( {\int {_{{\mathbb{R}^N}}\left( {{{\left| {{{\left( { - {\Delta _x}} \right)}^{{\alpha \over 2}}}u} \right|}^2} + {{\left| {{\nabla _y}u} \right|}^2}} \right)dxdy} } \right)^{{1 \over 2}}}. Based on variational approach and a variant of the quantitative strain lemma, for each b > 0, we show the existence of a least energy nodal solution ub. In addition, a convergence property of ub as b ↘ 0 is established.

This article is dedicated to the following Kirchhoff-type problem with Hartree-type nonlinearity: - (ε 2 a + ε b ∫ R 3 | ∇ u | 2 d x) ▵ u + V (x) u = ε μ - 3 K (x) (W μ (x) ∗ | u | p ) | u | p - 2 u , u ∈ H 1 (R 3) , where a , b ≥ 0 are constants, ε > 0 is a parameter, 0 < μ < 3 , p ∈ [ 2 , 6 - μ) , W μ (x) is convolution kernel, V(x) is nonnegative continuous external potential and K (x) ∈ C 1 (R 3) . Under some suitable assumptions on V(x) and K(x), we prove the existence and concentration of a positive ground state solution as ε → 0 via Poho z ˇ aev–Nehari manifold method. If K(x) is a positive constant, as its supplementary results, we also obtain the nonexistence of nontrivial solutions for p ≥ 6 - μ . [ABSTRACT FROM AUTHOR]

Copyright of Rad HAZU: Matematicke Znanosti is the property of Croatian Academy of Sciences & Arts (HAZU) and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In connection with the start-up of the T-15MD tokamak, testing of numerous magnetic probes of the electromagnetic diagnostic system is an important problem, because measurements from magnetic probes are used to control the plasma discharge. The article analyzes the first experiments on measuring the magnetic field performed on the T-15MD tokamak. A program has been created in which Kirchhoff equations for currents in poloidal coils and eddy currents in a vacuum chamber and passive stabilization conductors are solved. The distribution of the magnetic field and voltage in the magnetic probes are calculated and compared with experimentally measured values. As a result, it is found that some probes have incorrect connection polarity and that the signals from two probes are mixed up during recording to the database. It is shown that large currents are induced in the vacuum chamber, which will influence the plasma equilibrium, especially at the initial stage of current rise. An algorithm is proposed to improve the 2D model of the T-15MD chamber, which increases the accuracy of calculating the eddy currents and thereby improves plasma discharge control. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the existence of nontrivial solutions for the following Kirchhoff-type equation where $0 < s < 1 < p < \infty$, $sp < N$, $\lambda > 0$ is a real parameter, $(-\Delta)_{p}^{s}$ is the fractional $p$-Laplacian operator, $V:\mathbb{R}^N\rightarrow\mathbb{R}^N$ is a potential function, $M$ is a Kirchhoff function, the nonlinearity $f:\mathbb{R}^N\times\mathbb{R}\rightarrow\mathbb{R}$ is a continuous function and just super-linear in a neighborhood of $u = 0$. By using an appropriate truncation argument and the mountain pass theorem, we prove the existence of nontrivial solutions for the above equation, provided that $\lambda$ is sufficiently large. Our results extend and improve the previous ones in the literature.

In this paper we consider four-superlinear Kirchhoff equations. In contrast to most studies, we allowed the potential V is indefinite, precisely, the operator − Δ + V possesses a finite-dimensional negative space. When the working space loss compactness property, we obtain nontrivial solutions via Morse theory. [ABSTRACT FROM AUTHOR]

We consider a Kirchhoff-type equation with critical and supercritical nonlinear terms in a ball. By providing a method of decomposition of energy functional and subtle analysis, we show that every Palais–Smale sequence at a level below a certain energy threshold admits a subsequence that converges strongly to a nontrivial critical point of the variational functional. [ABSTRACT FROM AUTHOR]

Abstract In this paper, using variational method, we study the existence of an infinite number of solutions (some are positive, some are negative, and others are sign-changing) for a non-homogeneous elliptic Kirchhoff equation with a nonlinear reaction term.

In this paper, we use the genus properties in critical point theory to prove the existence of infinitely many solutions for fractional p -Laplacian equations of Schrödinger-Kirchhoff type. [ABSTRACT FROM AUTHOR]

In this paper, we consider the existence of nodal solutions for a class of Kirchhoff-type problem - a ∫ R N | ∇ u | 2 d x + b Δ u + u = f (x) | u | p - 2 u , in R N , where a , b > 0 , N ≥ 4 , 2 < p < 2 ∗ = 2 N N - 2 and the positive continuous function f (x) ∈ L ∞ (R N) . Combining a novel constraint manifold method with the detailed energy estimates, we prove that the above problem admits a nodal solution when N = 4 while two nodal solutions with opposite energy level for corresponding action functional are achieved when N ≥ 5 by controlling the parameter a sufficiently small. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we consider the following Kirchhoff type equation: { − ( a + b ∫ Ω | ∇ u | 2 d x ) Δ u = f ( x , u ) in Ω , u = 0 on ∂ Ω , $$ \textstyle\begin{cases} - (a+b \int _{\varOmega } \vert \nabla u \vert ^{2}\,dx ) \Delta u= f(x,u) &\text{in } \varOmega , \\ u=0 &\text{on } \partial \varOmega , \end{cases} $$ where a , b > 0 $a,b>0$ are constants and Ω ⊂ R N $\varOmega \subset \mathbb{R}^{N}$ ( N = 1 , 2 , 3 $N=1,2,3$ ) is a bounded domain with smooth boundary ∂Ω. By applying Morse theory, we obtain some existence and multiplicity results of nontrivial solutions for either a or b being sufficiently small.

In this paper, by Morse theory we will study the Kirchhoff type equation with an additional critical nonlinear term, and the main results are to compute the critical groups including the cases where zero is a mountain pass solution and the nonlinearity is resonant at zero. As an application, the multiplicity of nontrivial solutions for this equation with the parameter across the first eigenvalue is investigated under appropriate assumptions. To our best knowledge, estimates of our critical groups are new even for the Kirchhoff type equations with subcritical nonlinearities. [ABSTRACT FROM AUTHOR]

The paper deals with the following Kirchhoff equations: − a + b ∫ R N | ∇ u | 2 d x △ u + V (x) u = K (x) f (u) i n R N , u ∈ H 1 (R N) , where N ≥ 3 , f (u) is C 1 real function satisfying quasicritical growth at infinity, and V (x) , K (x) are positive and continuous functions. Combining Mountain Pass Theorem and compact embeddings in weighted Sobolev spaces, we establish the existence of at least a positive and a negative solution. Moreover, using a quantitative deformation lemma, we prove that the problem possesses one least energy sign-changing solution u 0 with two nodal domains. Finally, we show that the energy of u 0 is strictly larger than the ground state energy. [ABSTRACT FROM AUTHOR]

The paper deals with the following Kirchhoff equations: − a + b ∫ R N | ∇ u | 2 d x △ u + V (x) u = K (x) f (u) i n R N , u ∈ H 1 (R N) , where N ≥ 3 , f (u) is C 1 real function satisfying quasicritical growth at infinity, and V (x) , K (x) are positive and continuous functions. Combining Mountain Pass Theorem and compact embeddings in weighted Sobolev spaces, we establish the existence of at least a positive and a negative solution. Moreover, using a quantitative deformation lemma, we prove that the problem possesses one least energy sign-changing solution u 0 with two nodal domains. Finally, we show that the energy of u 0 is strictly larger than the ground state energy. [ABSTRACT FROM AUTHOR]

In this paper we consider 4-superlinear Schrödinger–Poisson systems. The potential V here is indefinite so that the Schrödinger operator − Δ + V possesses a finite-dimensional negative space. By Morse theory, we obtain the existence of nontrivial solutions for this problem. Similar result for Kirchhoff equations on R 3 is also presented. [ABSTRACT FROM AUTHOR]

Abstract The aim of this paper is to prove the Liouville-type theorem of the following weighted Kirchhoff equations: 0.1 − M ( ∫ R N ω ( z ) | ∇ G u | 2 d z ) div G ( ω ( z ) ∇ G u ) = f ( z ) e u , z = ( x , y ) ∈ R N = R N 1 × R N 2 $$\begin{aligned} \begin{aligned}[b] & -M \biggl( \int _{{\mathbb{R}} ^{N}}\omega (z) \vert \nabla _{G}u \vert ^{2}\,dz \biggr) \operatorname{div}_{G} \bigl(\omega (z) \nabla _{G}u \bigr)=f(z)e^{u}, \\ &\quad z=(x,y) \in R^{N}=R^{N_{1}}\times R^{N_{2}} \end{aligned} \end{aligned}$$ and 0.2 M ( ∫ R N ω ( z ) | ∇ G u | 2 d z ) div G ( ω ( z ) ∇ G u ) = f ( z ) u − q , z = ( x , y ) ∈ R N = R N 1 × R N 2 , $$\begin{aligned} \begin{aligned}[b] & M \biggl( \int _{\mathbb{R}^{N}}\omega (z) \vert \nabla _{G}u \vert ^{2}\,dz \biggr) \operatorname{div}_{G} \bigl(\omega (z) \nabla _{G}u \bigr)=f(z)u^{-q}, \\ &\quad z=(x,y) \in {\mathbb{R}} ^{N}={\mathbb{R}} ^{N_{1}}\times {\mathbb{R}} ^{N_{2}}, \end{aligned} \end{aligned}$$ where M ( t ) = a + b t k $M(t)=a+bt^{k}$ , t ≥ 0 $t\geq 0$ , with a > 0 $a>0$ , b , k ≥ 0 $b, k\geq 0$ , k = 0 $k=0$ if and only if b = 0 $b=0$ . q > 0 $q>0$ and ω ( z ) , f ( z ) ∈ L loc 1 ( R N ) $\omega (z), f(z)\in L^{1}_{\mathrm{loc}}({\mathbb{R}} ^{N})$ are nonnegative functions satisfying ω ( z ) ≤ C 1 ∥ z ∥ G θ $\omega (z)\leq C_{1}\|z \|_{G}^{\theta }$ and f ( z ) ≥ C 2 ∥ z ∥ G d $f(z)\geq C_{2}\|z\|_{G}^{d}$ as ∥ z ∥ G ≥ R 0 $\|z\|_{G} \geq R_{0}$ with d > θ − 2 $d>\theta -2$ , R 0 $R_{0}$ , C i $C_{i}$ ( i = 1 , 2 $i=1,2$ ) are some positive constants, here α ≥ 0 $\alpha \geq 0$ and ∥ z ∥ G = ( | x | 2 ( 1 + α ) + | y | 2 ) 1 2 ( 1 + α ) $\|z\|_{G}=(|x|^{2(1+ \alpha )}+|y|^{2})^{\frac{1}{2(1+\alpha )}}$ is the norm corresponding to the Grushin distance. N α = N 1 + ( 1 + α ) N 2 $N_{\alpha }=N_{1}+(1+\alpha )N_{2}$ is the homogeneous dimension of R N ${\mathbb{R}} ^{N}$ . div G $\operatorname{div}_{G}$ (resp., ∇ G $\nabla _{G}$ ) is Grushin divergence (resp., Grushin gradient). Under suitable assumptions on k, θ, d, and N α $N_{\alpha }$ , the nonexistence of stable weak solutions to equations (0.1) and (0.2) is investigated.

In this paper, we study Kirchhoff equations with logarithmic nonlinearity: \begin{equation*} \begin{cases} -(a+b\int_\Omega|\nabla u|^2)\Delta u+ V(x)u=|u|^{p-2}u\ln u^2, & \mbox{in}\ \Omega,\\ u=0,& \mbox{on}\ \partial\Omega, \end{cases} \end{equation*} where $a,b>0$ are constants, $4

In this paper, using variational method, we study the existence of an infinite number of solutions (some are positive, some are negative, and others are sign-changing) for a non-homogeneous elliptic Kirchhoff equation with a nonlinear reaction term. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following Kirchhoff type equation: { − (a + b ∫ Ω | ∇ u | 2 d x) Δ u = f (x , u) in Ω , u = 0 on ∂ Ω , where a , b > 0 are constants and Ω ⊂ R N (N = 1 , 2 , 3 ) is a bounded domain with smooth boundary ∂Ω. By applying Morse theory, we obtain some existence and multiplicity results of nontrivial solutions for either a or b being sufficiently small. [ABSTRACT FROM AUTHOR]

The aim of this paper is to prove the Liouville-type theorem of the following weighted Kirchhoff equations: 0.1 − M (∫ R N ω (z) | ∇ G u | 2 d z) div G (ω (z) ∇ G u) = f (z) e u , z = (x , y) ∈ R N = R N 1 × R N 2 and 0.2 M (∫ R N ω (z) | ∇ G u | 2 d z) div G (ω (z) ∇ G u) = f (z) u − q , z = (x , y) ∈ R N = R N 1 × R N 2 , where M (t) = a + b t k , t ≥ 0 , with a > 0 , b , k ≥ 0 , k = 0 if and only if b = 0 . q > 0 and ω (z) , f (z) ∈ L loc 1 (R N) are nonnegative functions satisfying ω (z) ≤ C 1 ∥ z ∥ G θ and f (z) ≥ C 2 ∥ z ∥ G d as ∥ z ∥ G ≥ R 0 with d > θ − 2 , R 0 , C i (i = 1 , 2 ) are some positive constants, here α ≥ 0 and ∥ z ∥ G = (| x | 2 (1 + α) + | y | 2) 1 2 (1 + α) is the norm corresponding to the Grushin distance. N α = N 1 + (1 + α) N 2 is the homogeneous dimension of R N . div G (resp., ∇ G ) is Grushin divergence (resp., Grushin gradient). Under suitable assumptions on k, θ, d, and N α , the nonexistence of stable weak solutions to equations (0.1) and (0.2) is investigated. [ABSTRACT FROM AUTHOR]

In this paper, we study the Kirchhoff equations with perturbation in R3. Applying the finite reduction method, we prove that the equation has one‐bump positive solutions under some suitable conditions that are given in Section 1. [ABSTRACT FROM AUTHOR]

In this paper, we revisit the singularly perturbation problem (0.1) − (ϵ 2 a + ϵ b ∫ R 3 | ∇ u | 2) Δ u + V (x) u = | u | p − 1 u in R 3 , where a , b , ϵ > 0 , 1 < p < 5 are constants and V is a potential function. First we establish the uniqueness and nondegeneracy of positive solutions to the limiting Kirchhoff problem − (a + b ∫ R 3 | ∇ u | 2) Δ u + u = | u | p − 1 u in R 3. Then, combining this nondegeneracy result and Lyapunov-Schmidt reduction method, we derive the existence of solutions to (0.1) for ϵ > 0 sufficiently small. Finally, we establish a local uniqueness result for such derived solutions using this nondegeneracy result and a type of local Pohozaev identity. [ABSTRACT FROM AUTHOR]

In this paper, we study the nonexistence and existence of positive solutions for the Kirchhoff type equation with nonlinearity having prescribed asymptotic behavior. Because of the structure of our equation, our nonexistence results have some special properties, and our conditions for positive solutions are also different from the existing results. [ABSTRACT FROM AUTHOR]

We are concerned with sign-changing solutions to nonlocal perturbation of the nonlinear Schrödinger equation in the whole R 3. By using the method of invariant sets of descending flow, we prove the existence of multiple sign-changing solutions provided the perturbation is sufficiently small. [ABSTRACT FROM AUTHOR]

We investigate the stability of ground states to a nonlinear focusing Schrödinger equation in presence of a Kirchhoff term. Through a spectral analysis of the linearized operator about ground states, we show a modulation stability estimate of ground states in the spirit of one due to Weinstein (1985) [17]. [ABSTRACT FROM AUTHOR]

In the present paper, we consider the nonlocal Kirchhoff problem $$-\left(\epsilon^2a+\epsilon b\int_{{\open R}^{3}}\vert \nabla u \vert^{2}\right)\Delta u+V(x)u=u^{p}, \quad u \gt 0 \quad {\rm in} {\open R}^{3},$$ where a, b >0, 1< p <5 are constants, ϵ>0 is a parameter. Under some mild assumptions on the function V , we obtain multi-peak solutions for ϵ sufficiently small by Lyapunov–Schmidt reduction method. Even though many results on single peak solutions to singularly perturbed Kirchhoff problems have been derived in the literature by various methods, there exist no results on multi-peak solutions before this paper, due to some difficulties caused by the nonlocal term $\left(\int_{{\open R}^{3}} \vert \nabla u \vert^{2}\right)\Delta u$. A remarkable new feature of this problem is that the corresponding unperturbed problem turns out to be a system of partial differential equations, but not a single Kirchhoff equation, which is quite different from most of the elliptic singular perturbation problems. [ABSTRACT FROM AUTHOR]

In this paper, we study the initial boundary value problem for Kirchhoff equation with linear strong damping term, nonlinear weak damping term and power-type logarithmic source term at three different initial energy levels, i.e. subcritical energy E (0) < d , critical initial energy E (0) = d and the arbitrary high energy E (0) > 0. By potential well method, we prove global existence, finite time blow up and asymptotic behavior of solutions in cases of E (0) < d and E (0) = d. We also prove the finite time blow up of solutions to the problem with linear weak and strong damping term in the case of E (0) > 0. [ABSTRACT FROM AUTHOR]

The paper is focused on the analysis of the drift of tabular iceberg observed in 2009 in the marginal ice zone of the North-West Barents Sea. Momentum balance equations are derived from the Kirchhoff equations describing plane motion of solid body in an ideal fluid. Field works performed on the drift iceberg and on the drift icenear the iceberg are described. Results of the field works and numerical simulations of the iceberg drift and rotation are performed and discussed. It is shown that acceleration of water flow around the iceberg has visible influence on the iceberg drift. Kinetic energy balance of drift iceberg is used to estimate the forces applied to the iceberg by the drift ice. [ABSTRACT FROM AUTHOR]

In this paper, we show that the Kirchhoff equations are derived from the Schrödinger equation by assuming the wave function to be a polynomial like solution. These Kirchhoff equations describe the evolution of n point vortices in hydrodynamics. In two dimensions, Kirchhoff equations are used to demonstrate the solution to single particle Laughlin wave function as complex Hermite polynomials. We also show that the equation for optical vortices, a two dimentional system, is derived from Kirchhoff equation by using paraxial wave approximation. These Kirchhoff equations satisfy a Poisson bracket relationship in phase space which is identical to the Heisenberg uncertainty relationship. Therefore, we conclude that being classical equations, the Kirchhoff equations, describe both a particle and a wave nature of single particle quantum mechanics in two dimensions. [ABSTRACT FROM AUTHOR]