Existence and asymptotic behavior of ground states for a nonlocal magnetic system

In this paper, we are concerned with the following fractional Choquard equation with critical growth: $$(-\Delta)^s u+\lambda V(x)u=(|x|^{-\mu} \ast F(u))f(u)+|u|^{2^*_s-2}u ~\hbox{in}~\mathbb{R}^N,$$ where $s\in (0,1)$, $N>2s$, $\mu\in (0,N)$, $2^*_s=\frac{2N}{N-2s}$ is the fractional critical exponent, $V$ is a steep well potential, $F(t)=\int_0^tf(s)ds$. Under some assumptions on $f$, the existence and asymptotic behavior of the positive ground states are established. In particular, if $f(u)=|u|^{p-2}u$, we obtain the range of $p$ when the equation has the positive ground states for three cases $2s<N<4s$ or $N=4s$ or $N>4s$.

Asymptotic behavior of ground states of quasilinear elliptic problems with two vanishing parameters, Part III

We are interested in the asymptotic behavior of ground states for a class of quasilinear elliptic equations when the nonlinear term has $$H^1$$ -critical growth. By applying the blow-up analysis and the variational method, we are able to obtain the precise asymptotic behavior of ground states.

Existence and asymptotic behavior of ground states for Schrödinger systems with Hardy potential

Asymptotic behavior of ground states of quasilinear elliptic problems with two vanishing parameters, Part II

We consider the semiclassical asymptotic behaviors of ground state solution for the following two‐component Hartree system: urn:x-wiley:mma:media:mma5820:mma5820-math-0001 which is originated from the study on cold atoms of boson and fermion system with long‐range interaction. Under the assumption urn:x-wiley:mma:media:mma5820:mma5820-math-0002 by detailed compactness analysis, we prove that there is a β0&gt;0 such that if β&lt;β0, the system has a ground state solution. For this solution, the energy estimates and the decay rates are presented, and the asymptotic profiles as ε→0 are displayed in details for β&lt;0 and β&gt;0, respectively. Furthermore, we show that for β&lt;0, the phase separation phenomenon may occur.

Existence and asymptotic behavior of ground states for quasilinear singular equations involving Hardy–Sobolev exponents

The purpose of this article is twofold. First, we study the existence and asymptotic behavior of ground states of a fractional Schrodinger system with quadratic interaction. Second, we give some conditions, in terms of the mass and energy of the ground states, under which the solutions of the associated initial value problem have the uniform bound or may blow up in finite time. As a corollary, we show the strong instability of the ground states.

Asymptotic behavior of ground states of quasilinear elliptic problems with two vanishing parameters

With appropriate hypotheses on the nonlinearity f, we prove the existence of a ground state solution u for the problem [Formula: see text] where V is a bounded potential, not necessarily continuous, and F the primitive of f. We also show that any of this problem is a classical solution. Furthermore, we prove that the ground state solution has exponential decay.

The asymptotic behavior of ground states of two-electron atoms is investigated. Suppose ψ(x1,x2) is the ground state of helium, ρ(x1) = ℱψ2(x1,x2) dx2 the corresponding electron density, and Φ(x2) the ground state of He+. We show that in the L2(dx2)-sense, lim‖x1‖→∞ ψ(x1,x2)[ρ(x1)]−1/2 = Φ(x2), and that ψ[ρ(x1)]−1/2 solves for large ‖x1‖ the Schrödinger equation for He+ in the quadratic form sense. The rate of convergence of these limits is also discussed.

In this paper, we first show the nondegeneracy and asymptotic behavior of ground states for the nonlinear fourth-order Schrödinger equation with mixed dispersion: δ Δ 2 u − Δu + u = | u | 2 σ u , u ∈ H 2 ( R N ) , where δ > 0 is sufficiently small, 0 < σ < 2 ( N − 2 ) + , 2 ( N − 2 ) + = 2 N − 2 for N ≥ 3 and 2 ( N − 2 ) + = + ∞ for N=2,3. This work extends some results in Bonheure, Casteras, Dos Santos, and Nascimento [Orbitally stable standing waves of a mixed dispersion nonlinear Schrödinger equation. SIAM J Math Anal. 2018;50:5027–5071]. Next, suppose P ( x ) and Q ( x ) are two positive, radial and continuous functions satisfying that as r = | x | → + ∞ , P ( r ) = 1 + a 1 r m 1 + O ( 1 r m 1 + θ 1 ) , Q ( r ) = 1 + a 2 r m 2 + O ( 1 r m 2 + θ 2 ) , where a 1 , a 2 ∈ R , m 1 , m 2 > 1 , θ 1 , θ 2 > 0 . We use the Lyapunov–Schmidt reduction method developed by Wei and Yan [Infinitely many positive solutions for the nonlinear Schrödinger equations in RN. Calc Var. 2010;37:423–439] to construct infinitely many nonradial positive and sign-changing solutions with arbitrary large energy for the following equation: δ Δ 2 u − Δu + P ( x ) u = Q ( x ) | u | 2 σ u , u ∈ H 2 ( R N ) .

We are concerned with the existence of positive solutions to the following boundary value problem in $$(0,\infty ),$$ $$\begin{aligned} \frac{1}{A}\left( A\phi \left( \left| u^{\prime }\right| \right) u^{\prime }\right) ^{\prime }=-a(t)u^{\alpha },t>0,\left( A\phi \left( \left| u^{\prime }\right| \right) u^{\prime }\right) \left( 0\right) =0\text { and}\lim \nolimits _{t\rightarrow +\infty }u(t)=0, \end{aligned}$$ where $$\alpha \ge 0,$$ $$\phi $$ is a nonnegative continuously differentiable function on $$\left[ 0,\infty \right) $$ , A is a continuous function on $$ \left[ 0,\infty \right) $$ , differentiable, positive on $$\left( 0,\infty \right) $$ and a is a nonnegative function satisfying some appropriate assumptions related to Karamata regular variation theory. We give also, estimates on such solutions.

Existence and asymptotic behavior of ground states for linearly coupled systems involving exponential growth

We are interested in the asymptotic behavior of ground states for a class of quasilinear elliptic equations in when the nonlinear term has -critical growth. In the previous result [Adachi et al. Asymptotic property of ground states for a class of quasilinear Schrödinger equation with -critical growth. Calc Var Partial Differential Equations. 2019;58(3). Art. 88, 29 pp.], it was shown that, after a suitable scaling, the ground state converges to the Talenti function. However, the uniqueness of the limit of the full sequence was not obtained, which was essentially owning to the fact that the Talenti function does not belong to . In this paper, by constructing a refined test function and performing a detailed asymptotic analysis, we are able to obtain the uniqueness of asymptotic limit of ground states.