Non-trivial solutions for Schrödinger-Poisson systems involving critical nonlocal term and potential vanishing at infinity

The present study is concerned with the following Schrödinger-Poisson system involving critical nonlocal term $$\begin{array}{} \displaystyle \begin{cases} -\triangle u+V(x)u-l(x)\phi|u|^{3}u=\eta K(x)f(u),~~\mbox{in}~~\mathbb{R}^{3}, \notag\\ -\triangle\phi=l(x)|u|^{5},~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\mbox{in}~~\mathbb{R}^{3}, \end{cases} \end{array}$$(1.1) where the potential V(x) and K(x) are positive continuous functions that vanish at infinity, and l(x) is bounded, nonnegative continuous function. Under some simple assumptions on V, K, l and f, we prove that the problem (1.1) has a non-trivial solution.

Sign-changing multi-bump solutions for the Chern-Simons-Schrödinger equations in ℝ2

In this paper we consider the nonlinear Chern-Simons-Schrödinger equations with general nonlinearity $$\begin{array}{} \displaystyle -{\it\Delta} u+\lambda V(|x|)u+\left(\frac{h^2(|x|)}{|x|^2}+\int\limits^{\infty}_{|x|}\frac{h(s)}{s}u^2(s)ds\right)u=f(u),\,\, x\in\mathbb R^2, \end{array}$$ where λ > 0, V is an external potential and $$\begin{array}{} \displaystyle h(s)=\frac{1}{2}\int\limits^s_0ru^2(r)dr=\frac{1}{4\pi}\int\limits_{B_s}u^2(x)dx \end{array}$$ is the so-called Chern-Simons term. Assuming that the external potential V is nonnegative continuous function with a potential well Ω := int V–1(0) consisting of k + 1 disjoint components Ω0, Ω1, Ω2 ⋯, Ωk, and the nonlinearity f has a general subcritical growth condition, we are able to establish the existence of sign-changing multi-bump solutions by using variational methods. Moreover, the concentration behavior of solutions as λ → +∞ are also considered.

Uniqueness and local properties of positive solutions to a semilinear elliptic equation with singular potentials

In this paper, we study the uniqueness and local behavior at the origin of positive solutions to [Formula: see text] where [Formula: see text], [Formula: see text]) is a bounded smooth domain and [Formula: see text], and [Formula: see text] is a nonnegative continuous function over [Formula: see text]. In the two cases: (i) [Formula: see text], [Formula: see text] and [Formula: see text] on [Formula: see text]; (ii) [Formula: see text], [Formula: see text] and [Formula: see text] on [Formula: see text], we establish the uniqueness of positive solutions and the exact blow-up rate of positive solutions at the origin. This paper seems to be the first one to deal with the latter case.

Existence and Global Asymptotic Behavior of Singular Positive Solutions for Radial Laplacian

The aim of this paper is to establish existence and uniqueness of a positive continuous solution to the following singular nonlinear problem. {-t1-ntn-1u′′=a(t)uσ, t∈(0,1), limt→0⁡tn-1u′(t)=0, u(1)=0}, where n≥3,σ<1, and a denotes a nonnegative continuous function that might have the property of being singular at t=0 and /or t=1 and which satisfies certain condition associated to Karamata class. We emphasize that the nonlinearity might also be singular at u=0, while the solution could blow-up at 0. Our method is based on the global estimates of potential functions and the Schauder fixed point theorem.

Solitary Waves for a Class of Quasilinear Schrödinger Equations Involving Vanishing Potentials

In this paper we study the existence of weak positive solutions for the following class of quasilinear Schrödinger equations−Δu + V(x)u − [Δ(uwhere h satisfies some “mountain-pass” type assumptions and V is a nonnegative continuous function. We are interested specially in the case where the potential V is neither bounded away from zero, nor bounded from above. We give a special attention to the case when V may eventually vanish at infinity. Our arguments are based on penalization techniques, variational methods and Moser iteration scheme.

Existence and Global Behavior of Positive Solutions for Some Fourth-Order Boundary Value Problems

We establish the existence and uniqueness of a positive solution to the following fourth-order value problem:u(4)(x)=a(x)uσ(x),x∈(0,1)with the boundary conditionsu(0)=u(1)=u'(0)=u'(1)=0, whereσ∈(-1,1)andais a nonnegative continuous function on (0, 1) that may be singular atx=0orx=1. We also give the global behavior of such a solution.

Existence of Positive Solutions for Some Superlinear Fourth-Order Boundary Value Problems

We are concerned with the following superlinear fourth-order equationu4t+utφt,−ut=0, t∈0, 1;−u0=u1=0, −u′0=a, −u′1=-b, wherea,−bare nonnegative constants such thata+b>0andφt,−sis a nonnegative continuous function that is required to satisfy some appropriate conditions related to a classKsatisfying suitable integrability condition. Our purpose is to prove the existence, uniqueness, and global behavior of a classical positive solution to the above problem by using a method based on estimates on the Green function and perturbation arguments.

Existence and Global Asymptotic Behavior of Positive Solutions for Nonlinear Fractional Dirichlet Problems on the Half-Line

We are interested in the following fractional boundary value problem:Dαu(t)+atuσ=0,t∈(0,∞),limt→0⁡t2-αu(t)=0,limt→∞⁡t1-αu(t)=0, where1<α<2,σ∈(-1,1),Dαis the standard Riemann-Liouville fractional derivative, andais a nonnegative continuous function on(0,∞)satisfying some appropriate assumptions related to Karamata regular variation theory. Using the Schauder fixed point theorem, we prove the existence and the uniqueness of a positive solution. We also give a global behavior of such solution.

Existence and Estimates of Positive Solutions for Some Singular Fractional Boundary Value Problems

We establish the existence and uniqueness of a positive solutionufor the following fractional boundary value problem:Dαu(x)=−a(x)uσ(x),x∈(0,1)with the conditionslimx→0+⁡x2−αu(x)=0,u(1)=0, where1<α≤2,σ∈(−1,1), andais a nonnegative continuous function on(0,1)that may be singular atx=0orx=1. We also give the global behavior of such a solution.

Existence and Exact Asymptotic Behavior of Positive Solutions for a Fractional Boundary Value Problem

We establish the existence and uniqueness of a positive solution for the fractional boundary value problem , with the condition , where , and is a nonnegative continuous function on that may be singular at or .