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Register a new userNormalized solutions to a class of Kirchhoff equations with Sobolev critical exponent
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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.
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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, endalignedright. 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, -Delta 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.
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 $murightarrow0^{+}$.
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.}
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 $x_3$ direction 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 $pin(2,6)$.
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