Positive Radial Functions Research Articles

New type of solutions for Schrödinger equations with critical growth

We consider the following nonlinear Schrödinger equations with critical growth: −Δu+V(|y|)u=uN+2N−2,u>0inRN, where V(|y|) is a bounded positive radial function in C1, N ≥ 5. By using a finite reduction argument, we show that if r2V(r) has either an isolated local maximum or an isolated local minimum at r0 > 0 with V(r0) > 0, there exists infinitely many non-radial large energy solutions which are invariant under some sub-groups of O(3).

Finite time blow‐up to the Cauchy problem in a higher dimensional two species chemotaxis system

In this paper, we study a blow‐up radial solution to the fully parabolic chemotaxis system with the higher dimensional two species Cauchy problem. In addition, we show that the set of positive radial functions in has a dense subset composed of positive radial initial data causing blow‐up in finite time with respect to topology for .

A Note on Hardy Inequalities on Homogeneous Groups

We provide the necessary and sufficient characterizations on a pair of positive radial functions so that the two-weight Hardy inequalities hold true on homogeneous groups, one of most general subclasses of nilpotent Lie groups. We also present a simple condition on a couple of positive radial weight functions for the Lp −Hardy type inequalities to be valid.

Positive or sign-changing solutions for a critical semilinear nonlocal equation

We consider the following critical semilinear nonlocal equation involving the fractional Laplacian $$ (-\Delta)^su=K(|x|)|u|^{2^*_s-2}u,\ \ \hbox{in}\ \ \mathbb{R}^N, $$ where $K(|x|)$ is a positive radial function, $N>2+2s$, $0<s<1$, and $2^*_s=\frac{2N}{N-2s}$. Under some asymptotic assumptions on $K(x)$ at an extreme point, we show that this problem has infinitely many non-radial positive or sign-changing solutions.

Infinitely many positive and sign-changing solutions for nonlinear fractional scalar field equations

We consider the following nonlinear fractional scalar field equation $$ (-\Delta)^s u + u = K(|x|)u^p, u > 0 \hbox{in} \mathbb{R}^N, $$ where $K(|x|)$ is a positive radial function, $N\ge 2$, $0 < s < 1$, and $1 < p < \frac{N+2s}{N-2s}$. Under various asymptotic assumptions on $K(x)$ at infinity, we show that this problem has infinitely many non-radial positive solutions and sign-changing solutions, whose energy can be made arbitrarily large.

Segregated vector solutions for nonlinear schrödinger systems in ℝ2

We study the following nonlinear Schrödinger system {−Δu+P(|x|)u=μu3+βv2u, x∈ℝ2,−Δv+Q(|x|)v=vu3+βu2v, x∈ℝ2,where P(r) and Q(r) are positive radial functions, μ>0,v>0, and β∈ℝ is a coupling constant. This type of system arises, particularly, in models in Bose-Einstein condensates theory. Applying a finite reduction method, we construct an unbounded sequence of non-radial positive vector solutions of segregated type when β is in some suitable interval, which gives an answer to an interesting problem raised by Peng and Wang in Remark 4.1 (Arch. Ration. Mech. Anal., 208(2013), 305–339).

Nonexistence Results for the Schrödinger-Poisson Equations with Spherical and Cylindrical Potentials in

We study the following Schrödinger-Poisson system: , , , where are positive radial functions, , , and is allowed to take two different forms including and with . Two theorems for nonexistence of nontrivial solutions are established, giving two regions on the plane where the system has no nontrivial solutions.

The Schrödinger–Poisson equation under the effect of a nonlinear local term

In this paper we study the problem { − Δ u + u + λ ϕ u = u p , − Δ ϕ = u 2 , lim | x | → + ∞ ϕ ( x ) = 0 , where u , ϕ : R 3 → R are positive radial functions, λ > 0 and 1 < p < 5 . We give existence and nonexistence results, depending on the parameters p and λ. It turns out that p = 2 is a critical value for the existence of solutions.