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.

Nonexistence Results of Positive Entire Solutions for Quasilinear Elliptic Inequalities

1997 ◽  
Vol 40 (2) ◽  
pp. 244-253 ◽  
Author(s):  
Yūki Naito ◽  
Hiroyuki Usami
Keyword(s):  
Sufficient Conditions ◽  
Radial Symmetry ◽  
Entire Solutions ◽  
Elliptic Inequality ◽  
Quasilinear Elliptic ◽  
Elliptic Inequalities

AbstractThis paper treats the quasilinear elliptic inequalitywhere N ≥ 2, m > 1, σ >m− 1, and p:ℝN → (0, ∞) is continuous. Sufficient conditions are given for this inequality to have no positive entire solutions. When p has radial symmetry, the existence of positive entire solutions can be characterized by our results and some known results.

scholarly journals The existence of positive solutions for Kirchhoff-type problems via the sub-supersolution method

2018 ◽  
Vol 26 (1) ◽  
pp. 5-41 ◽  
Author(s):  
Baoqiang Yan ◽  
Donal O’Regan ◽  
Ravi P. Agarwal
Keyword(s):  
Positive Solutions ◽  
Bifurcation Theory ◽  
Sufficient Conditions ◽  
Global Bifurcation ◽  
Existence Of A Solution ◽  
Kirchhoff Type ◽  
Existence Of Positive Solutions ◽  
Global Bifurcation Theory ◽  
Necessary And Sufficient ◽  
Kirchhoff Type Problems

Abstract In this paper we discuss the existence of a solution between wellordered subsolution and supersolution of the Kirchhoff equation. Using the sub-supersolution method together with a Rabinowitz-type global bifurcation theory, we establish the existence of positive solutions for Kirchhoff-type problems when the nonlinearity is singular or sign-changing. Moreover, we obtain some necessary and sufficient conditions for the existence of positive solutions for the problem when N = 1.

scholarly journals Finite energy solutions to inhomogeneous nonlinear elliptic equations with sub-natural growth terms

2020 ◽  
Vol 13 (1) ◽  
pp. 53-74 ◽  
Author(s):  
Adisak Seesanea ◽  
Igor E. Verbitsky
Keyword(s):  
Sufficient Conditions ◽  
Nonlinear Elliptic Equations ◽  
Classical Case ◽  
Finite Energy ◽  
Natural Growth ◽  
Nonlinear Elliptic ◽  
Borel Measures ◽  
Inhomogeneous Problems ◽  
Necessary And Sufficient ◽  
Quasilinear Elliptic

AbstractWe obtain necessary and sufficient conditions for the existence of a positive finite energy solution to the inhomogeneous quasilinear elliptic equation-\Delta_{p}u=\sigma u^{q}+\mu\quad\text{on }\mathbb{R}^{n}in the sub-natural growth case {0<q<p-1}, where {\Delta_{p}} ({1<p<\infty}) is the p-Laplacian, and σ, μ are positive Borel measures on {\mathbb{R}^{n}}. Uniqueness of such a solution is established as well. Similar inhomogeneous problems in the sublinear case {0<q<1} are treated for the fractional Laplace operator {(-\Delta)^{\alpha}} in place of {-\Delta_{p}}, on {\mathbb{R}^{n}} for {0<\alpha<\frac{n}{2}}, and on an arbitrary domain {\Omega\subset\mathbb{R}^{n}} with positive Green’s function in the classical case {\alpha=1}.

scholarly journals Topological entropy in totally disconnected locally compact groups

2016 ◽  
Vol 37 (7) ◽  
pp. 2163-2186 ◽  
Author(s):  
ANNA GIORDANO BRUNO ◽  
SIMONE VIRILI
Keyword(s):  
Topological Entropy ◽  
Sufficient Conditions ◽  
Invariant Subgroup ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Totally Disconnected ◽  
Dynamical Interpretation

Let $G$ be a topological group, let $\unicode[STIX]{x1D719}$ be a continuous endomorphism of $G$ and let $H$ be a closed $\unicode[STIX]{x1D719}$-invariant subgroup of $G$. We study whether the topological entropy is an additive invariant, that is, $$\begin{eqnarray}h_{\text{top}}(\unicode[STIX]{x1D719})=h_{\text{top}}(\unicode[STIX]{x1D719}\restriction _{H})+h_{\text{top}}(\bar{\unicode[STIX]{x1D719}}),\end{eqnarray}$$ where $\bar{\unicode[STIX]{x1D719}}:G/H\rightarrow G/H$ is the map induced by $\unicode[STIX]{x1D719}$. We concentrate on the case when $G$ is totally disconnected locally compact and $H$ is either compact or normal. Under these hypotheses, we show that the above additivity property holds true whenever $\unicode[STIX]{x1D719}H=H$ and $\ker (\unicode[STIX]{x1D719})\leq H$. As an application, we give a dynamical interpretation of the scale $s(\unicode[STIX]{x1D719})$ by showing that $\log s(\unicode[STIX]{x1D719})$ is the topological entropy of a suitable map induced by $\unicode[STIX]{x1D719}$. Finally, we give necessary and sufficient conditions for the equality $\log s(\unicode[STIX]{x1D719})=h_{\text{top}}(\unicode[STIX]{x1D719})$ to hold.

Moving Weighted Averages

1993 ◽  
Vol 45 (3) ◽  
pp. 449-469 ◽  
Author(s):  
M. A. Akcoglu ◽  
Y. Déniel
Keyword(s):  
Weight Function ◽  
Sufficient Conditions ◽  
Nonnegative Function ◽  
Finite Measure ◽  
Weight Functions ◽  
Ergodic Averages ◽  
Fixed Weight ◽  
Finite Measure Space ◽  
Image Position ◽  
Necessary And Sufficient

AbstractLet ℝ denote the real line. Let {Tt}tєℝ be a measure preserving ergodic flow on a non atomic finite measure space (X, ℱ, μ). A nonnegative function φ on ℝ is called a weight function if ∫ℝ φ(t)dt = 1. Consider the weighted ergodic averagesof a function f X —> ℝ, where {θk} is a sequence of weight functions. Some sufficient and some necessary and sufficient conditions are given for the a.e. convergence of Akf, in particular for a special case in whichwhere φ is a fixed weight function and {(ak, rk)} is a sequence of pairs of real numbers such that rk > 0 for all k. These conditions are obtained by a combination of the methods of Bellow-Jones-Rosenblatt, developed to deal with moving ergodic averages, and the methods of Broise-Déniel-Derriennic, developed to deal with unbounded weight functions.

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 Sufficient conditions for matchings

1972 ◽  
Vol 18 (2) ◽  
pp. 129-136 ◽  
Author(s):  
Ian Anderson
Keyword(s):  
Simple Proof ◽  
Perfect Matching ◽  
Sufficient Conditions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Number Of Components ◽  
Image Position ◽  
Necessary And Sufficient

A graph G is said to possess a perfect matching if there is a subgraph of G consisting of disjoint edges which together cover all the vertices of G. Clearly G must then have an even number of vertices. A necessary and sufficient condition for G to possess a perfect matching was obtained by Tutte (3). If S is any set of vertices of G, let p(S) denote the number of components of the graph G – S with an odd number of vertices. Then the conditionis both necessary and sufficient for the existence of a perfect matching. A simple proof of this result is given in (1).

Oscillation of Semilinear Elliptic Inequalities by Riccati Transformations

1980 ◽  
Vol 32 (4) ◽  
pp. 908-923 ◽  
Author(s):  
E. S. Noussair ◽  
C. A. Swanson
Keyword(s):  
Exterior Domain ◽  
Nonlinear Term ◽  
Sufficient Conditions ◽  
Type Inequality ◽  
Second Derivative ◽  
Semilinear Elliptic ◽  
Generalized Riccati Transformation ◽  
The Matrix ◽  
Elliptic Inequality ◽  
Elliptic Inequalities

A generalized Riccati transformation will be utilized to derive a Riccati-type inequality (3) associated with a semilinear elliptic inequality yL(y; x) ≦ 0 possessing a positive solution y in an exterior domain in Euclidean n-space. On the basis of (3), general sufficient conditions for the elliptic inequality to be oscillatory are developed in § 3. The matrix of coefficients of the second derivative terms in L(y;x) (i.e. (Aij) in (1)) is not restricted in any way beyond the usual ellipticity hypothesis (iv) below, and thereby one of the difficulties mentioned in [9] and inherent in the method there is resolved. Furthermore, the nonlinear term B﹛x, y) in (1) is not required to be one-signed.

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