On the planar Schrödinger-Poisson system with zero mass potential

In this paper, we prove that the following planar Schrödinger-Poisson system with zero mass -Δu+φu=f(u), x∈R^2, Δφ= 2πu^2, x∈R^2, admits a nontrivial radially symmetric solution under weaker assumptions on f by using some new analytical approaches.

Solitary waves for a fractional Klein–Gordon–Maxwell equations

We investigate existence of solutions for a fractional Klein–Gordon coupled with Maxwell's equation. On the basis of overcoming the lack of compactness, we obtain that there is a radially symmetric solution for the critical system by means of variational methods.

Singular limits and properties of solutions of some degenerate elliptic and parabolic equations

Let n ⩾ 3, 0 ⩽ m < n − 2/n, ρ1 > 0, $\beta>\beta_{0}^{(m)}=(({m\rho_{1}})/({n-2-nm}))$, αm = ((2β + ρ1)/(1 − m)) and α = 2β+ρ1. For any λ > 0, we prove the uniqueness of radially symmetric solution υ(m) of Δ(υm/m) + αmυ + βx · ∇υ = 0, υ > 0, in ℝn∖{0} which satisfies $\lim\nolimits_{|x|\to 0|}|x|^{\alpha _m/\beta }v^{(m)}(x) = \lambda ^{-((\rho _1)/((1-m)\beta ))}$ and obtain higher order estimates of υ(m) near the blow-up point x = 0. We prove that as m → 0+, υ(m) converges uniformly in C2(K) for any compact subset K of ℝn∖{0} to the solution υ of Δlog υ + α υ + β x·∇ υ = 0, υ > 0, in ℝn\{0}, which satisfies $\lim\nolimits_{ \vert x \vert \to 0} \vert x \vert ^{{\alpha}/{\beta}}v(x)=\lambda^{-{\rho_{1}}/{\beta}}$. We also prove that if the solution u(m) of ut = Δ (um/m), u > 0, in (ℝn∖{0}) × (0, T) which blows up near {0} × (0, T) at the rate $ \vert x \vert ^{-{\alpha_{m}}/{\beta}}$ satisfies some mild growth condition on (ℝn∖{0}) × (0, T), then as m → 0+, u(m) converges uniformly in C2 + θ, 1 + θ/2(K) for some constant θ ∈ (0, 1) and any compact subset K of (ℝn∖{0}) × (0, T) to the solution of ut = Δlog u, u > 0, in (ℝn∖{0}) × (0, T). As a consequence of the proof, we obtain existence of a unique radially symmetric solution υ(0) of Δ log υ + α υ + β x·∇ υ = 0, υ > 0, in ℝn∖{0}, which satisfies $\lim\nolimits_{ \vert x \vert \to 0} \vert x \vert ^{{\alpha}/{\beta}}v(x)=\lambda^{-{\rho_{1}}/{\beta}}$.

UNIQUENESS OF POSITIVE RADIAL-SYMMETRICAL SOLUTION OF DIRICHLET PROBLEM IN ANNULAR DOMAINS FOR ONE CLASS OF NONLINEAR DIFFERENTIAL EQUATIONS OF THE SECOND ORDER

The existence and uniqueness of positive radially symmetric solution of Dirichlet problem in annular domain for one class of nonlinear differential equations of the second order is proved.