Existence of positive solutions for a class of semipositone problem in whole ℝN

2019 ◽  
Vol 150 (5) ◽  
pp. 2349-2367
Author(s):  
Claudianor O. Alves ◽  
Angelo R. F. de Holanda ◽  
Jefferson A. dos Santos
Keyword(s):  
Variational Method ◽  
Positive Solutions ◽  
Riesz Potential ◽  
Main Tool ◽  
Continuous Functions ◽  
Existence Of Solution ◽  
Subcritical Growth ◽  
Existence Of Positive Solutions ◽  
Semipositone Problem ◽  
Image Position

In this paper we show the existence of solution for the following class of semipositone problem P$$\left\{\matrix{-\Delta u & = & h(x)(f(u)-a) & \hbox{in} & {\open R}^N, \cr u & \gt & 0 & \hbox{in} & {\open R}^N, \cr}\right.$$ where N ≥ 3, a > 0, h : ℝN → (0, + ∞) and f : [0, + ∞) → [0, + ∞) are continuous functions with f having a subcritical growth. The main tool used is the variational method together with estimates that involve the Riesz potential.

Positive solutions of a nonlinear Sturm–Liouville boundary-value problem

2009 ◽  
Vol 52 (3) ◽  
pp. 561-568
Author(s):  
Patricio Cerda ◽  
Pedro Ubilla
Keyword(s):  
Fixed Point ◽  
Continuous Function ◽  
Positive Solutions ◽  
A Priori ◽  
Liouville Problem ◽  
Continuous Functions ◽  
Positive Continuous Function ◽  
Existence Of Positive Solutions ◽  
Sturm Liouville ◽  
Image Position

AbstractWe establish the existence of positive solutions of the Sturm–Liouville problemwhereWe assume g and $\hat{q}$ to be non-negative, continuous functions, a(s) is a positive continuous function, c≥0, p>1, and the function h is sub-quadratic with respect to u′. We combine a priori estimates with a fixed-point result of Krasnosel'skii to obtain the existence of a positive solution.

Existence and asymptotical behavior of solutions for a quasilinear Choquard equation with singularity

2021 ◽  
Vol 19 (1) ◽  
pp. 259-267
Author(s):  
Liuyang Shao ◽  
Yingmin Wang
Keyword(s):  
Asymptotic Behavior ◽  
Positive Solutions ◽  
Riesz Potential ◽  
Asymptotic Behavior Of Solutions ◽  
Suitable Assumption ◽  
Choquard Equation ◽  
Existence Of Positive Solutions

Abstract In this study, we consider the following quasilinear Choquard equation with singularity − Δ u + V ( x ) u − u Δ u 2 + λ ( I α ∗ ∣ u ∣ p ) ∣ u ∣ p − 2 u = K ( x ) u − γ , x ∈ R N , u > 0 , x ∈ R N , \left\{\begin{array}{ll}-\Delta u+V\left(x)u-u\Delta {u}^{2}+\lambda \left({I}_{\alpha }\ast | u{| }^{p})| u{| }^{p-2}u=K\left(x){u}^{-\gamma },\hspace{1.0em}& x\in {{\mathbb{R}}}^{N},\\ u\gt 0,\hspace{1.0em}& x\in {{\mathbb{R}}}^{N},\end{array}\right. where I α {I}_{\alpha } is a Riesz potential, 0 < α < N 0\lt \alpha \lt N , and N + α N < p < N + α N − 2 \displaystyle \frac{N+\alpha }{N}\lt p\lt \displaystyle \frac{N+\alpha }{N-2} , with λ > 0 \lambda \gt 0 . Under suitable assumption on V V and K K , we research the existence of positive solutions of the equations. Furthermore, we obtain the asymptotic behavior of solutions as λ → 0 \lambda \to 0 .

8.—On Second-order Differential Inequalities

1974 ◽  
Vol 72 (2) ◽  
pp. 109-127 ◽  
Author(s):  
F. V. Atkinson
Keyword(s):  
Differential Equations ◽  
Positive Solutions ◽  
Second Order ◽  
Differential Inequalities ◽  
Existence Of Positive Solutions ◽  
Image Position

SynopsisThis paper is devoted to a study of differential equations and inequalities of the formandThe results are mainly concerned with the existence of positive solutions, their uniqueness in the case of (*), and bounds for these solutions.

Existence of Positive Global Solutions of Mixed Sublinear-Superlinear Problems

1988 ◽  
Vol 40 (5) ◽  
pp. 1222-1242
Author(s):  
W. Allegretto ◽  
Y. X. Huang
Keyword(s):  
Positive Solutions ◽  
Special Interest ◽  
Sufficient Conditions ◽  
Global Solutions ◽  
Linear Type ◽  
Existence Of Positive Solutions ◽  
Image Position ◽  
Quasilinear Problem

Consider the elliptic quasilinear problem:1in Rn, n ≧ 3, whereWe are interested in establishing sufficient conditions on f for the existence of positive solutions u(x) with specified behaviour at ∞. Of special interest to us are criteria which guarantee that u(x) decays at least as fast as |x|−α for some α ≧ 0, given below, in the case f(x, u, ∇u) contains terms of typeThat is: f is of mixed sublinear-super linear type. Our main result is Theorem 3 below which explicitly states sufficient conditions for the existence of such solutions.

scholarly journals Positive Solutions of a Kind of Equations Related to the Laplacian andp-Laplacian

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Fangfang Zhang ◽  
Zhanping Liang
Keyword(s):  
Variational Method ◽  
Bounded Domain ◽  
Positive Solutions ◽  
Eigenvalue Problems ◽  
Sufficient Condition ◽  
Nonlinear Eigenvalue Problems ◽  
Existence Of Positive Solutions ◽  
Nonlinear Eigenvalue

Positive solutions of a kind of equations related to the Laplacian andp-Laplacian on a bounded domain inRNwithN⩾1are studied by using variational method. A sufficient condition of the existence of positive solutions is characterized by the eigenvalues of linear and another nonlinear eigenvalue problems.

scholarly journals Generalized Emden-Fowler equations of subcritical growth

1993 ◽  
Vol 54 (2) ◽  
pp. 254-262 ◽  
Author(s):  
Wolfgang Rother
Keyword(s):  
Eigenvalue Problem ◽  
Elliptic Operator ◽  
Positive Solutions ◽  
Subcritical Growth ◽  
Existence Of Positive Solutions

AbstractThe existence of positive solutions, vanishing at infinity, for the semilinear eigenvalue problem Lu = λ f(x, y) in RN is obtained, where L is a strictly elliptic operator. The function f is assumed to be of subcritical growth with respect to the variable u.

Positive solutions of certain elliptic systems with density-dependent diffusions

1995 ◽  
Vol 125 (5) ◽  
pp. 1031-1050 ◽  
Author(s):  
Inkyung Ahn ◽  
Lige Li
Keyword(s):  
Positive Solutions ◽  
Growth Rates ◽  
Smooth Boundary ◽  
Elliptic Systems ◽  
Biological Interactions ◽  
Bounded Region ◽  
Density Dependent ◽  
Existence Of Positive Solutions ◽  
Image Position ◽  
Increasing Functions

Results are obtained on the existence of positive solutions to the following elliptic system:in a bounded region Ω in Rn with a smooth boundary, where the diffusion terms φ ψ are non-negative functions and the system could be degenerate, β γ are strictly increasing functions, k,σ ≧ 0 are constants. We assume also that the growth rates f, g satisfy certain monotonicities. Applications to biological interactions with density-dependent diffusions are given.

scholarly journals SUCCESSIVE ITERATION AND POSITIVE SOLUTIONS FOR A p-LAPLACIAN MULTIPOINT BOUNDARY VALUE PROBLEM

2008 ◽  
Vol 49 (4) ◽  
pp. 551-560 ◽  
Author(s):  
BO SUN ◽  
XIANGKUI ZHAO ◽  
WEIGAO GE
Keyword(s):  
Positive Solutions ◽  
Monotone Iterative Method ◽  
One Dimensional ◽  
Multipoint Boundary ◽  
Monotone Iterative ◽  
Existence Of Positive Solutions ◽  
The One ◽  
Image Position ◽  
Multipoint Boundary Condition ◽  
Multipoint Boundary Value Problem

AbstractIn this paper, we study the existence of positive solutions for the one-dimensional p-Laplacian differential equation, subject to the multipoint boundary condition by applying a monotone iterative method.

scholarly journals Quasilinear elliptic inequalities with Hardy potential and nonlocal terms

2020 ◽  
pp. 1-19
Author(s):  
Marius Ghergu ◽  
Paschalis Karageorgis ◽  
Gurpreet Singh
Keyword(s):  
Riesz Potential ◽  
Sufficient Conditions ◽  
Hardy Potential ◽  
Existence Of Positive Solutions ◽  
Elliptic Inequality ◽  
Image Position ◽  
Nonlocal Terms ◽  
Necessary And Sufficient ◽  
Quasilinear Elliptic ◽  
Elliptic Inequalities

We study the quasilinear elliptic inequality \[ -\Delta_m u - \frac{\mu}{|x|^m}u^{m-1} \geq (I_\alpha*u^p)u^q \quad\mbox{in }\mathbb{R}^N\setminus \overline B_1, N\geq 1, \] where $p>0$ , $q, \mu \in \mathbb {R}$ , $m>1$ and $I_\alpha$ is the Riesz potential of order $\alpha \in (0,N)$ . We obtain necessary and sufficient conditions for the existence of positive solutions.

scholarly journals Oscillation theorems for semilinear hyperbolic and ultrahyperbolic operators

1978 ◽  
Vol 18 (1) ◽  
pp. 55-64 ◽  
Author(s):  
Mamoru Narita
Keyword(s):  
Boundary Conditions ◽  
Positive Solutions ◽  
Sufficient Conditions ◽  
Differential Inequalities ◽  
Hyperbolic Operator ◽  
Oscillation Property ◽  
All Solutions ◽  
Existence Of Positive Solutions ◽  
Image Position ◽  
Oscillation Theorems

The oscillation property of the semilinear hyperbolic or ultra-hyperbolic operator L defined byis studied. Sufficient conditions are provided for all solutions of uL[u] ≤ 0 satisfying certain boundary conditions to be oscillatory. The basis of our results is the non-existence of positive solutions of the associated differential inequalities.

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