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Register a new userBound state nodal solutions for the non-autonomous Schr{o}dinger--Poisson system in $mathbb{R}^{3}$
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In this paper, we study the existence of nodal solutions for the non-autonomous Schr{o}dinger--Poisson system: begin{equation*} left{ begin{array}{ll} -Delta u+u+lambda K(x) phi u=f(x) |u|^{p-2}u & text{ in }mathbb{R}^{3}, -Delta phi =K(x)u^{2} & text{ in }mathbb{R}^{3},% end{array}% right. end{equation*}% where $lambda >0$ is a parameter and $2<p<4$. Under some proper assumptions on the nonnegative functions $K(x)$ and $f(x)$, but not requiring any symmetry property, when $lambda$ is sufficiently small, we find a bounded nodal solution for the above problem by proposing a new approach, which changes sign exactly once in $mathbb{R}^{3}$. In particular, the existence of a least energy nodal solution is concerned as well.
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We investigate the structure of nodal solutions for coupled nonlinear Schr{o}dinger equations in the repulsive coupling regime. Among other results, for the following coupled system of $N$ equations, we prove the existence of infinitely many nodal solutions which share the same componentwise-prescribed nodal numbers begin{equation}label{ab} left{ begin{array}{lr} -{Delta}u_{j}+lambda u_{j}=mu u^{3}_{j}+sum_{i eq j}beta u_{j}u_{i}^{2} ,,,,,,, in W , u_{j}in H_{0,r}^{1}(W), ,,,,,,,,j=1,dots,N, end{array} right. end{equation} where $W$ is a radial domain in $mathbb R^n$ for $nleq 3$, $lambda>0$, $mu>0$, and $beta <0$. More precisely, let $p$ be a prime factor of $N$ and write $N=pB$. Suppose $betaleq-frac{mu}{p-1}$. Then for any given non-negative integers $P_{1},P_{2},dots,P_{B}$, (ref{ab}) has infinitely many solutions $(u_{1},dots,u_{N})$ such that each of these solutions satisfies the same property: for $b=1,...,B$, $u_{pb-p+i}$ changes sign precisely $P_b$ times for $i=1,...,p$. The result reveals the complex nature of the solution structure in the repulsive coupling regime due to componentwise segregation of solutions. Our method is to combine a heat flow approach as deformation with a minimax construction of the symmetric mountain pass theorem using a $mathbb Z_p$ group action index. Our method is robust, also allowing to give the existence of one solution without assuming any symmetry of the coupling.
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 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.
We prove that the derivative nonlinear Schr{o}dinger equation is globally well-posed in $H^{frac 12} (mathbb{R})$ when the mass of initial data is strictly less than $4pi$.
In the first part of the paper we continue the study of solutions to Schrodinger equations with a time singularity in the dispersive relation and in the periodic setting. In the second we show that if the Schrodinger operator involves a Laplace operator with variable coefficients with a particular dependence on the space variables, then one can prove Strichartz estimates at the same regularity as that needed for constant coefficients. Our work presents a two dimensional analysis, but we expect that with the obvious adjustments similar results are available in higher dimensions.
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