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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.
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
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
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
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
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