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A perturbation approach to studying sign-changing solutions of Kirchhoff equations with a general nonlinearity

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 Added by Jianjun Zhang
 Publication date 2018
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and research's language is English




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By employing a novel perturbation approach and the method of invariant sets of descending flow, this manuscript investigates the existence and multiplicity of sign-changing solutions to a class of semilinear Kirchhoff equations in the following form $$ -left(a+ bint_{R^3}| abla u|^2right)triangle {u}+V(x)u=f(u),,,xinR^3, $$ where $a,b>0$ are constants, $Vin C(R^3,R)$, $fin C(R,R)$. The methodology proposed in the current paper is robust, in the sense that, the monotonicity condition for the nonlinearity $f$ and the coercivity condition of $V$ are not required. Our result improves the study made by Y. Deng, S. Peng and W. Shuai ({it J. Functional Analysis}, 3500-3527(2015)), in the sense that, in the present paper, the nonlinearities include the power-type case $f(u)=|u|^{p-2}u$ for $pin(2,4)$, in which case, it remains open in the existing literature that whether there exist infinitely many sign-changing solutions to the problem above without the coercivity condition of $V$. Moreover, {it energy doubling} is established, i.e., the energy of sign-changing solutions is strictly large than two times that of the ground state solutions for small $b>0$.



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In this paper, we consider the following Kirchhoff type equation $$ -left(a+ bint_{R^3}| abla u|^2right)triangle {u}+V(x)u=f(u),,,xinR^3, $$ where $a,b>0$ and $fin C(R,R)$, and the potential $Vin C^1(R^3,R)$ is positive, bounded and satisfies suitable decay assumptions. By using a new perturbation approach together with a new version of global compactness lemma of Kirchhoff type, we prove the existence and multiplicity of bound state solutions for the above problem with a general nonlinearity. We especially point out that neither the corresponding Ambrosetti-Rabinowitz condition nor any monotonicity assumption is required for $f$. Moreover, the potential $V$ may not be radially symmetry or coercive. As a prototype, the nonlinear term involves the power-type nonlinearity $f(u) = |u|^{p-2}u$ for $pin (2, 6)$. In particular, our results generalize and improve the results by Li and Ye (J.Differential Equations, 257(2014): 566-600), in the sense that the case $pin(2,3]$ is left open there.
157 - Gongbao Li , Xiao Luo , Tao Yang 2021
In this paper, we consider the existence and asymptotic properties of solutions to the following Kirchhoff equation begin{equation}label{1} onumber - Bigl(a+bint_{{R^3}} {{{left| { abla u} right|}^2}}Bigl) Delta u =lambda u+ {| u |^{p - 2}}u+mu {| u |^{q - 2}}u text { in } mathbb{R}^{3} end{equation} under the normalized constraint $int_{{mathbb{R}^3}} {{u}^2}=c^2$, where $a!>!0$, $b!>!0$, $c!>!0$, $2!<!q!<!frac{14}{3}!<! p!leq!6$ or $frac{14}{3}!<!q!<! p!leq! 6$, $mu!>!0$ and $lambda!in!R$ appears as a Lagrange multiplier. In both cases for the range of $p$ and $q$, the Sobolev critical exponent $p!=!6$ is involved and the corresponding energy functional is unbounded from below on $S_c=Big{ u in H^{1}({mathbb{R}^3}): int_{{mathbb{R}^3}} {{u}^2}=c^2 Big}$. If $2!<!q!<!frac{10}{3}$ and $frac{14}{3}!<! p!<!6$, we obtain a multiplicity result to the equation. If $2!<!q!<!frac{10}{3}!<! p!=!6$ or $frac{14}{3}!<!q!<! p!leq! 6$, we get a ground state solution to the equation. Furthermore, we derive several asymptotic results on the obtained normalized solutions. Our results extend the results of N. Soave (J. Differential Equations 2020 $&$ J. Funct. Anal. 2020), which studied the nonlinear Schr{o}dinger equations with combined nonlinearities, to the Kirchhoff equations. To deal with the special difficulties created by the nonlocal term $({int_{{R^3}} {left| { abla u} right|} ^2}) Delta u$ appearing in Kirchhoff type equations, we develop a perturbed Pohozaev constraint approach and we find a way to get a clear picture of the profile of the fiber map via careful analysis. In the meantime, we need some subtle energy estimates under the $L^2$-constraint to recover compactness in the Sobolev critical case.
We study the nonlinear eigenvalue problem for the p-Laplacian, and more general problem constituting the Fucik spectrum. We are interested in some vanishing properties of sign changing solutions to these problems. Our method is applicable in the plane.
115 - Lintao Liu , Haibo Chen , Jie Yang 2021
In this paper, we study the existence and asymptotic properties of solutions to the following fractional Kirchhoff equation begin{equation*} left(a+bint_{mathbb{R}^{3}}|(-Delta)^{frac{s}{2}}u|^{2}dxright)(-Delta)^{s}u=lambda u+mu|u|^{q-2}u+|u|^{p-2}u quad hbox{in $mathbb{R}^3$,} end{equation*} with a prescribed mass begin{equation*} int_{mathbb{R}^{3}}|u|^{2}dx=c^{2}, end{equation*} where $sin(0, 1)$, $a, b, c>0$, $2<q<p<2_{s}^{ast}=frac{6}{3-2s}$, $mu>0$ and $lambdainmathbb{R}$ as a Lagrange multiplier. Under different assumptions on $q<p$, $c>0$ and $mu>0$, we prove some existence results about the normalized solutions. Our results extend the results of Luo and Zhang (Calc. Var. Partial Differential Equations 59, 1-35, 2020) to the fractional Kirchhoff equations. Moreover, we give some results about the behavior of the normalized solutions obtained above as $murightarrow0^{+}$.
145 - Penghui Zhang , Zhiqing Han 2021
This paper is concerned with the existence of ground states for a class of Kirchhoff type equation with combined power nonlinearities begin{equation*} -left(a+bint_{mathbb{R}^{3}}| abla u(x)|^{2}right) Delta u =lambda u+|u|^{p-2}u+u^{5}quad text{for some} lambdainmathbb{R},quad xinmathbb{R}^{3}, end{equation*} with prescribed $L^{2}$-norm mass begin{equation*} int_{mathbb{R}^{3}}u^{2}=c^{2} end{equation*} in Sobolev critical case and proves that the equation has a couple of solutions $(u_{c},lambda_{c})in S(c)times mathbb{R}$ for any $c>0$, $a,b >0$ and $frac{14}{3}leq p< 6,$ where $S(c)={uin H^{1}(mathbb{R}^{3}):int_{mathbb{R}^{3}}u^{2}=c^{2}}.$ textbf{Keywords:} Kirchhoff type equation; Critical nonlinearity; Normalized ground states oindent{AMS Subject Classification:, 37L05; 35B40; 35B41.}
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