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Register a new userNormalized solutions for a fourth-order Schr{o}dinger equation with positive second-order dispersion coefficient
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We are concerned with the existence and asymptotic properties of solutions to the following fourth-order Schr{o}dinger equation begin{equation}label{1} {Delta}^{2}u+mu Delta u-{lambda}u={|u|}^{p-2}u, ~~~~x in R^{N} end{equation} under the normalized constraint $$int_{{mathbb{R}^N}} {{u}^2}=a^2,$$ where $N!geq!2$, $a,mu!>!0$, $2+frac{8}{N}!<!p!<! 4^{*}!=!frac{2N}{(N-4)^{+}}$ and $lambdainR$ appears as a Lagrange multiplier. Since the second-order dispersion term affects the structure of the corresponding energy functional $$ E_{mu}(u)=frac{1}{2}{||Delta u||}_2^2-frac{mu}{2}{|| abla u||}_2^2-frac{1}{p}{||u||}_p^p $$ we could find at least two normalized solutions to (ref{1}) if $2!+!frac{8}{N}!<! p!<!{ 4^{*} }$ and $mu^{pgamma_p-2}a^{p-2}!<!C$ for some explicit constant $C!=!C(N,p)!>!0$ and $gamma_p!=!frac{N(p!-!2)}{4p}$. Furthermore, we give some asymptotic properties of the normalized solutions to (ref{1}) as $muto0^+$ and $ato0^+$, respectively. In conclusion, we mainly extend the results in cite{DBon,dbJB}, which deal with (ref{1}), from $muleq0$ to the case of $mu>0$, and also extend the results in cite{TJLu,Nbal}, which deal with (ref{1}), from $L^2$-subcritical and $L^2$-critical setting to $L^2$-supercritical setting.
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We investigate normalized solutions for the Schr{o}dinger equation with combined Hartree type and power nonlinearities, namely begin{equation*} left{ begin{array}{ll} -Delta u+lambda u=gamma (I_{alpha }ast leftvert urightvert ^{p})|u|^{p-2}u+mu |u|^{q-2}u & quad text{in}quad mathbb{R}^{N}, int_{mathbb{R}^{N}}|u|^{2}dx=c, & end{array}% right. end{equation*} where $Ngeq 2$ and $c>0$ is a given real number. Under different assumptions on $gamma ,mu ,p$ and $q$, we prove several nonexistence, existence and multiplicity results. In particular, we are more interested in the cases when the competing effect of Hartree type and power nonlinearities happens, i.e. $gamma mu <0,$ including the cases $gamma <0,mu >0$ and $% gamma >0,mu <0.$ Due to the different strength of two types of nonlinearities, we find some differences in results and in the geometry of the corresponding functionals between these two cases.
In this paper, we give a complete study on the existence and non-existence of normalized solutions for Schr{o}dinger system with quadratic and cubic interactions. In the one dimension case, the energy functional is bounded from below on the product of $L^2$-spheres, normalized ground states exist and are obtained as global minimizers. When $N=2$, the energy functional is not always bounded on the product of $L^2$-spheres. We give a classification of the existence and nonexistence of global minimizers. Then under suitable conditions on $b_1$ and $b_2$, we prove the existence of normalized solutions. When $N=3$, the energy functional is always unbounded on the product of $L^2$-spheres. We show that under suitable conditions on $b_1$ and $b_2$, at least two normalized solutions exist, one is a ground state and the other is an excited state. Furthermore, by refining the upper bound of the ground state energy, we provide a precise mass collapse behavior of the ground state and a precise limit behavior of the excited state as $betarightarrow 0$. Finally, we deal with the high dimensional cases $Ngeq 4$. Several non-existence results are obtained if $beta<0$. When $N=4$, $beta>0$, the system is a mass-energy double critical problem, we obtain the existence of a normalized ground state and its synchronized mass collapse behavior. Comparing with the well studied homogeneous case $beta=0$, our main results indicate that the quadratic interaction term not only enriches the set of solutions to the above Schr{o}dinger system but also leads to a stabilization of the related evolution system.
In this paper, we study a class of Schr{o}dinger-Poisson (SP) systems with general nonlinearity where the nonlinearity does not require Ambrosetti-Rabinowitz and Nehari monotonic conditions. We establish new estimates and explore the associated energy functional which is coercive and bounded below on Sobolev space. Together with Ekeland variational principle, we prove the existence of ground state solutions. Furthermore, when the `charge function is greater than a fixed positive number, the (SP) system possesses only zero solutions. In particular, when `charge function is radially symmetric, we establish the existence of three positive solutions and the symmetry breaking of ground state solutions.
In this paper, we study important Schr{o}dinger systems with linear and nonlinear couplings begin{equation}label{eq:diricichlet} begin{cases} -Delta u_1-lambda_1 u_1=mu_1 |u_1|^{p_1-2}u_1+r_1beta |u_1|^{r_1-2}u_1|u_2|^{r_2}+kappa (x)u_2~hbox{in}~mathbb{R}^N, -Delta u_2-lambda_2 u_2=mu_2 |u_2|^{p_2-2}u_2+r_2beta |u_1|^{r_1}|u_2|^{r_2-2}u_2+kappa (x)u_1~ hbox{in}~mathbb{R}^N, u_1in H^1(mathbb{R}^N), u_2in H^1(mathbb{R}^N), onumber end{cases} end{equation} with the condition $$int_{mathbb{R}^N} u_1^2=a_1^2, int_{mathbb{R}^N} u_2^2=a_2^2,$$ where $Ngeq 2$, $mu_1,mu_2,a_1,a_2>0$, $betainmathbb{R}$, $2<p_1,p_2<2^*$, $2<r_1+r_2<2^*$, $kappa(x)in L^{infty}(mathbb{R}^N)$ with fixed sign and $lambda_1,lambda_2$ are Lagrangian multipliers. We use Ekland variational principle to prove this system has a normalized radially symmetric solution for $L^2-$subcritical case when $Ngeq 2$, and use minimax method to prove this system has a normalized radially symmetric positive solution for $L^2-$supercritical case when $N=3$, $p_1=p_2=4, r_1=r_2=2$.
We analyze dynamical properties of the logarithmic Schr{o}dinger equation under a quadratic potential. The sign of the nonlinearity is such that it is known that in the absence of external potential, every solution is dispersive, with a universal asymptotic profile. The introduction of a harmonic potential generates solitary waves, corresponding to generalized Gaussons. We prove that they are orbitally stable, using an inequality related to relative entropy, which may be thought of as dual to the classical logarithmic Sobolev inequality. In the case of a partial confinement, we show a universal dispersive behavior for suitable marginals. For repulsive harmonic potentials, the dispersive rate is dictated by the potential, and no universal behavior must be expected.
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