scholarly journals Unbounded principal eigenfunctions and the logistic equation on RN

2003 ◽  
Vol 67 (3) ◽  
pp. 413-427 ◽  
Author(s):  
Wei Dong ◽  
Yihong Du
Keyword(s):  
Positive Solution ◽  
Positive Solutions ◽  
Eigenvalue Problems ◽  
Growth Conditions ◽  
Logistic Equation ◽  
Exponential Rate ◽  
Unique Positive Solution ◽  
Unbounded Coefficients ◽  
Growth Restrictions

We consider the logistic equation − Δu = a (x) u − b (x) up on all of RN with possibly unbounded coefficients near infinity. We show that under suitable growth conditions of the coefficients, the behaviour of the positive solutions of the logistic equation can be largely determined. We also show that certain linear eigenvalue problems on all of RN have principal eigenfunctions that become unbounded near infinity at an exponential rate. Using these results, we finally show that the logistic equation has a unique positive solution under suitable growth restrictions for its coefficients.

scholarly journals MULTIPLE POSITIVE SOLUTIONS FOR A CRITICAL GROWTH PROBLEM WITH HARDY POTENTIAL

2006 ◽  
Vol 49 (1) ◽  
pp. 53-69 ◽  
Author(s):  
Pigong Han
Keyword(s):  
Dirichlet Problem ◽  
Positive Solution ◽  
Positive Solutions ◽  
Variational Approach ◽  
Critical Growth ◽  
Multiple Positive Solutions ◽  
Hardy Potential ◽  
Unique Positive Solution ◽  
Existence And Nonexistence ◽  
Growth Problem

AbstractIn this paper we study the existence and nonexistence of multiple positive solutions for the Dirichlet problem:$$ -\Delta{u}-\mu\frac{u}{|x|^2}=\lambda(1+u)^p,\quad u\gt0,\quad u\in H^1_0(\varOmega), \tag{*} $$where $0\leq\mu\lt(\frac{1}{2}(N-2))^2$, $\lambda\gt0$, $1\ltp\leq(N+2)/(N-2)$, $N\geq3$. Using the sub–supersolution method and the variational approach, we prove that there exists a positive number $\lambda^*$ such that problem (*) possesses at least two positive solutions if $\lambda\in(0,\lambda^*)$, a unique positive solution if $\lambda=\lambda^*$, and no positive solution if $\lambda\in(\lambda^*,\infty)$.

Existence and uniqueness of positive solutions of nonlinear Schrödinger systems

2015 ◽  
Vol 145 (2) ◽  
pp. 365-390 ◽  
Author(s):  
Haidong Liu ◽  
Zhaoli Liu ◽  
Jinyong Chang
Keyword(s):  
Positive Solution ◽  
Positive Solutions ◽  
Ground State ◽  
Existence And Uniqueness ◽  
Unique Positive Solution ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrödinger Systems ◽  
Schrödinger Systems ◽  
Least Energy ◽  
Schrödinger System

We prove that the Schrödinger systemwhere n = 1, 2, 3, N ≥ 2, λ1 = λ2 = … = λN = 1, βij = βji > 0 for i, j = 1, …, N, has a unique positive solution up to translation if the βij (i ≠ j) are comparatively large with respect to the βjj. The same conclusion holds if n = 1 and if the βij (i ≠ j) are comparatively small with respect to the βjj. Moreover, this solution is a ground state in the sense that it has the least energy among all non-zero solutions provided that the βij (i ≠ j) are comparatively large with respect to the βjj, and it has the least energy among all non-trivial solutions provided that n = 1 and the βij (i ≠ j) are comparatively small with respect to the βjj. In particular, these conclusions hold if βij = (i ≠ j) for some β and either β > max{β11, β22, …, βNN} or n = 1 and 0 < β < min{β11, β22, …, βNN}.

scholarly journals Multiple positive solutions for discrete p-Laplacian equations with potential term

2013 ◽  
Vol 7 (2) ◽  
pp. 327-342 ◽  
Author(s):  
Jong-Ho Kim ◽  
Jea-Hyun Park ◽  
June-Yub Lee
Keyword(s):  
Positive Solutions ◽  
Eigenvalue Problems ◽  
Existence Of Solutions ◽  
Growth Conditions ◽  
Multiple Positive Solutions ◽  
Source Terms ◽  
Dirichlet Eigenvalue ◽  
Nonlinear Source ◽  
Potential Term ◽  
Laplacian Equations

We study the existence of solutions to nonlinear discrete boundary value problems with the discrete p-Laplacian, potential, and nonlinear source terms. Using variational methods, we demonstrate that there exist at least two positive solutions. The existence strongly depends on the smallest positive eigenvalue of Dirichlet eigenvalue problems and the growth conditions of the source terms.

scholarly journals Remarks on the uniqueness problem for the logistic equation on the entire space

2006 ◽  
Vol 73 (1) ◽  
pp. 129-137 ◽  
Author(s):  
Yihong Du ◽  
Lishan Liu
Keyword(s):  
Positive Solution ◽  
Logistic Equation ◽  
New Technique ◽  
Uniqueness Problem ◽  
Entire Space ◽  
Unique Positive Solution

We consider the logistic equation −Δu = a (x) u − b (x) uq on all of RN with a (x)/|x|γ and b (x)/|x|τ bounded away from 0 and infinity for all large |x|, where γ > −2, τ ∈ (−∞, ∞). We show that this problem has a unique positive solution. This considerably improves some earlier results. The main new technique here is a Safonov type iteration argument. The result can also be proved by a technique introduced by Marcus and Veron, and the two different techniques are compared.

Nonlocal problems arising from the birth-jump processes

2018 ◽  
Vol 149 (2) ◽  
pp. 447-469
Author(s):  
M. Delgado ◽  
A. Suárez ◽  
I. B. M. Duarte
Keyword(s):  
Positive Solution ◽  
Elliptic Equations ◽  
Existence And Uniqueness ◽  
Solution Method ◽  
Eigenvalue Problems ◽  
Logistic Equation ◽  
Jump Processes ◽  
Nonlocal Problems

In this paper, we prove the existence and uniqueness of a positive solution for a nonlocal logistic equation arising from the birth-jump processes. For this, we establish a sub-super solution method for nonlocal elliptic equations, we perform a study of the eigenvalue problems associated with these equations and we apply these results to the nonlocal logistic equation.

scholarly journals Multiple Positive Solutions of Nonhomogeneous Elliptic Equations in Unbounded Domains

2007 ◽  
Vol 2007 ◽  
pp. 1-19 ◽  
Author(s):  
Tsing-San Hsu
Keyword(s):  
Positive Solution ◽  
Positive Solutions ◽  
Elliptic Equations ◽  
Exterior Domain ◽  
Unbounded Domains ◽  
Multiple Positive Solutions ◽  
Entire Space ◽  
Unique Positive Solution ◽  
Strip Domain ◽  
Unbounded Cylinder

We will show that under suitable conditions onfandh, there exists a positive numberλ∗such that the nonhomogeneous elliptic equation−Δu+u=λ(f(x,u)+h(x))inΩ,u∈H01(Ω),N≥2, has at least two positive solutions ifλ∈(0,λ∗), a unique positive solution ifλ=λ∗, and no positive solution ifλ>λ∗, whereΩis the entire space or an exterior domain or an unbounded cylinder domain or the complement in a strip domain of a bounded domain. We also obtain some properties of the set of solutions.

On Positive Solutions for Some Nonlinear Semipositone Elliptic Boundary Value

2006 ◽  
Vol 11 (4) ◽  
pp. 323-329 ◽  
Author(s):  
G. A. Afrouzi ◽  
S. H. Rasouli
Keyword(s):  
Boundary Value Problems ◽  
Positive Solution ◽  
Bounded Domain ◽  
Positive Solutions ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Laplacian Operator ◽  
Smooth Bounded Domain ◽  
Elliptic Boundary Value ◽  
Existence Of Positive Solutions

This study concerns the existence of positive solutions to classes of boundary value problems of the form−∆u = g(x,u), x ∈ Ω,u(x) = 0, x ∈ ∂Ω,where ∆ denote the Laplacian operator, Ω is a smooth bounded domain in RN (N ≥ 2) with ∂Ω of class C2, and connected, and g(x, 0) < 0 for some x ∈ Ω (semipositone problems). By using the method of sub-super solutions we prove the existence of positive solution to special types of g(x,u).

scholarly journals A singular eigenvalue problem for the Dirichlet (p, q)-Laplacian

2021 ◽  
Author(s):  
Yunru Bai ◽  
Nikolaos S. Papageorgiou ◽  
Shengda Zeng
Keyword(s):  
Dirichlet Problem ◽  
Eigenvalue Problem ◽  
Positive Solution ◽  
Positive Solutions ◽  
Singular Term ◽  
Type Theorem ◽  
Solution Map ◽  
Continuity Properties ◽  
Variational Tools ◽  
Superlinear Reaction

AbstractWe consider a parametric nonlinear, nonhomogeneous Dirichlet problem driven by the (p, q)-Laplacian with a reaction involving a singular term plus a superlinear reaction which does not satisfy the Ambrosetti–Rabinowitz condition. The main goal of the paper is to look for positive solutions and our approach is based on the use of variational tools combined with suitable truncations and comparison techniques. We prove a bifurcation-type theorem describing in a precise way the dependence of the set of positive solutions on the parameter $$\lambda $$ λ . Moreover, we produce minimal positive solutions and determine the monotonicity and continuity properties of the minimal positive solution map.

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

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Habib Mâagli ◽  
Noureddine Mhadhebi ◽  
Noureddine Zeddini
Keyword(s):  
Asymptotic Behavior ◽  
Continuous Function ◽  
Boundary Value Problem ◽  
Positive Solution ◽  
Positive Solutions ◽  
Existence And Uniqueness ◽  
Boundary Value ◽  
Fractional Boundary Value Problem ◽  
Exact Asymptotic Behavior ◽  
Nonnegative Continuous Function

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 .

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