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Register a new userGround state normalized solution to Schrodinger systems with general nonlinearities and potentials
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In present paper, we prove the existence of solutions $(lambda_1,lambda_2, u_1,u_2)in R^2times H^1(R^N, R^2)$ to systems of nonlinear Schrodinger equations with potentials $$begin{cases} -Delta u_1+V_1(x)u_1+lambda_1 u_1=partial_1 G(u_1,u_2);quad&hbox{in};R^N -Delta u_2+V_2(x)u_2+lambda_2 u_2=partial_2G(u_1,u_2);quad&hbox{in};R^N 0<u_1,u_2in H^1(R^N), Ngeq 1 end{cases}$$ satisfying the normalization constraints $int_{R^N}u_1^2dx=a_1$ and $int_{R^N}u_2^2dx=a_2$, which appear in mean-field models for binary mixtures of Bose-Einstein condensates or models for binary mixtures of ultracold quantum gases of fermion atoms. The potentials $V_iota(x) (iota=1,2)$ are given functions. The nonlinearities $G(u_1,u_2)$ are considered of the form $$ begin{cases} G(u_1, u_2):=sum_{i=1}^{ell}frac{mu_i}{p_i}|u_1|^{p_i}+sum_{j=1}^{m}frac{ u_j}{q_j}|u_2|^{q_j}+sum_{k=1}^{n}beta_k |u_1|^{r_{1,k}}|u_2|^{r_{2,k}}, mu_i, u_j,beta_k>0, ~ p_i, q_j>2, ~r_{1,k}, r_{2,k}>1. end{cases} $$ Under some assumptions on $V_iota$ and the parameters, we can prove the strict binding inequality for the mass sub-critical problem and obtain the existence of ground state normalized solutions for any given $a_1>0,a_2>0$.
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It is established existence of bound and ground state solutions for quasilinear elliptic systems driven by (phi 1, phi 2)-Laplacian operator. The main feature here is to consider quasilinear elliptic systems involving both nonsingular nonlinearities combined with indefinite potentials and singular cases perturbed by superlinear and subcritical couple terms. These prevent us to use arguments based on Ambrosetti-Rabinowitz condition and variational methods for differentiable functionals. By exploring the Nehari method and doing a fine analysis on the fibering map associated, we get estimates that allow us unify the arguments to show multiplicity of semi-trivial solutions in both cases.
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 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 $murightarrow0^{+}$.
We consider a phase-field model where the internal energy depends on the order parameter in a nonlocal way. Therefore, the resulting system consists of the energy balance equation coupled with a nonlinear and nonlocal ODE for the order parameter. Such system has been analyzed by several authors, in particular when the configuration potential is a smooth double-well function. More recently, in the case of a potential defined on (-1,1) and singular at the endpoints, the existence of a finite-dimensional global attractor has been proven. Here we examine both the case of smooth potentials as well as the case of physically realistic (e.g., logarithmic) singular potentials. We prove well-posedness results and the eventual global boundedness of solutions uniformly with respect to the initial data. In addition, we show that the separation property holds in the case of singular potentials. Thanks to these results, we are able to demonstrate the existence of a finite-dimensional attractors in the present cases as well.
We prove the existence of ground state in a multidimensional nonlinear Schrodinger model of paraxial beam propagation in isotropic local media with saturable nonlinearity. Such ground states exist in the form of bright counterpropagating solitons. From the proof, a general threshold condition on the beam coupling constant for the existence of such fundamental solitons follows.
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