ترغب بنشر مسار تعليمي؟ اضغط هنا

Normalized solutions to a class of Kirchhoff equations with Sobolev critical exponent

158   0   0.0 ( 0 )
 نشر من قبل Yang Tao
 تاريخ النشر 2021
  مجال البحث
والبحث باللغة English




اسأل ChatGPT حول البحث

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.



قيم البحث

اقرأ أيضاً

200 - Juncheng Wei , Yuanze Wu 2021
In this paper, we consider the following nonlinear Schr{o}dinger equations with mixed nonlinearities: begin{eqnarray*} left{aligned &-Delta u=lambda u+mu |u|^{q-2}u+|u|^{2^*-2}uquadtext{in }mathbb{R}^N, &uin H^1(bbr^N),quadint_{bbr^N}u^2=a^2, endalig nedright. end{eqnarray*} where $Ngeq3$, $mu>0$, $lambdainmathbb{R}$ and $2<q<2^*$. We prove in this paper begin{enumerate} item[$(1)$]quad Existence of solutions of mountain-pass type for $N=3$ and $2<q<2+frac{4}{N} $. item[$(2)$]quad Existence and nonexistence of ground states for $2+frac{4}{N}leq q<2^*$ with $mu>0$ large. item[$(3)$]quad Precisely asymptotic behaviors of ground states and mountain-pass solutions as $muto0$ and $mu$ goes to its upper bound. end{enumerate} Our studies answer some questions proposed by Soave in cite[Remarks~1.1, 1.2 and 8.1]{S20}.
In this paper, we consider the existence and asymptotic behavior on mass of the positive solutions to the following system: begin{equation}label{eqA0.1} onumber begin{cases} -Delta u+lambda_1u=mu_1u^3+alpha_1|u|^{p-2}u+beta v^2uquad&hbox{in}~R^4, -De lta v+lambda_2v=mu_2v^3+alpha_2|v|^{p-2}v+beta u^2vquad&hbox{in}~R^4, end{cases} end{equation} under the mass constraint $$int_{R^4}u^2=a_1^2quadtext{and}quadint_{R^4}v^2=a_2^2,$$ where $a_1,a_2$ are prescribed, $mu_1,mu_2,beta>0$; $alpha_1,alpha_2in R$, $p!in! (2,4)$ and $lambda_1,lambda_2!in!R$ appear as Lagrange multipliers. Firstly, we establish a non-existence result for the repulsive interaction case, i.e., $alpha_i<0(i=1,2)$. Then turning to the case of $alpha_i>0 (i=1,2)$, if $2<p<3$, we show that the problem admits a ground state and an excited state, which are characterized respectively by a local minimizer and a mountain-pass critical point of the corresponding energy functional. Moreover, we give a precise asymptotic behavior of these two solutions as $(a_1,a_2)to (0,0)$ and $a_1sim a_2$. This seems to be the first contribution regarding the multiplicity as well as the synchronized mass collapse behavior of the normalized solutions to Schr{o}dinger systems with Sobolev critical exponent. When $3leq p<4$, we prove an existence as well as non-existence ($p=3$) results of the ground states, which are characterized by constrained mountain-pass critical points of the corresponding energy functional. Furthermore, precise asymptotic behaviors of the ground states are obtained when the masses of whose two components vanish and cluster to a upper bound (or infinity), respectively.
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.}
128 - Xiaoyu Zeng 2016
We study the following nonlinear critical curl-curl equation begin{equation}label{eq0.1} ablatimes ablatimes U +V(x)U=|U|^{p-2}U+ |U|^4U,quad xin mathbb{R}^3,end{equation} where $V(x)=V(r, x_3)$ with $r=sqrt{x_1^2+x_2^2}$ is 1-periodic in $x_3$ dire ction and belongs to $L^infty(R^3)$. When $0 otin sigma(-Delta+frac{1}{r^2}+V)$ and $pin(4,6)$, we prove the existence of nontrivial solution for (ref{eq0.1}), which is indeed a ground state solution in a suitable cylindrically symmetric space. Especially, if $ sigma(-Delta+frac{1}{r^2}+V)>0$, a ground state solution is obtained for any $pin(2,6)$.
التعليقات
جاري جلب التعليقات جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا