اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدNormalized solutions for Schrodinger equations with critical Sobolev exponent and mixed nonlinearities
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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}.
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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 {|
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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
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