Infinitely many positive solutions of a singular dirichlet problem involving the p-Laplacian

2009 ◽  
Vol 14 (11) ◽  
pp. 3784-3791 ◽  
Author(s):  
Libo Wang ◽  
Weigao Ge
Keyword(s):  
Dirichlet Problem ◽  
Positive Solutions ◽  
Singular Dirichlet Problem

scholarly journals Continuity results for parametric nonlinear singular Dirichlet problems

2019 ◽  
Vol 9 (1) ◽  
pp. 372-387 ◽  
Author(s):  
Yunru Bai ◽  
Dumitru Motreanu ◽  
Shengda Zeng
Keyword(s):  
Dirichlet Problem ◽  
Positive Solutions ◽  
Point Of View ◽  
List Type ◽  
Dirichlet Problems ◽  
Set Valued Mapping ◽  
Essential Properties ◽  
Singular Dirichlet Problem

Abstract In this paper we study from a qualitative point of view the nonlinear singular Dirichlet problem depending on a parameter λ > 0 that was considered in [32]. Denoting by Sλ the set of positive solutions of the problem corresponding to the parameter λ, we establish the following essential properties of Sλ: there exists a smallest element $\begin{array}{} u_\lambda^* \end{array}$ in Sλ, and the mapping λ ↦ $\begin{array}{} u_\lambda^* \end{array}$ is (strictly) increasing and left continuous; the set-valued mapping λ ↦ Sλ is sequentially continuous.

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.

Refined Boundary Behavior of the Unique Convex Solution to a Singular Dirichlet Problem for the Monge–Ampère Equation

2018 ◽  
Vol 18 (2) ◽  
pp. 289-302
Author(s):  
Zhijun Zhang
Keyword(s):  
Dirichlet Problem ◽  
Crucial Role ◽  
Blow Up ◽  
Boundary Behavior ◽  
Structure Condition ◽  
Bounded Smooth Domain ◽  
Convex Solution ◽  
Bounded Smooth ◽  
Monge Ampere ◽  
Singular Dirichlet Problem

AbstractThis paper is concerned with the boundary behavior of the unique convex solution to a singular Dirichlet problem for the Monge–Ampère equation\operatorname{det}D^{2}u=b(x)g(-u),\quad u<0,\,x\in\Omega,\qquad u|_{\partial% \Omega}=0,where Ω is a strictly convex and bounded smooth domain in{\mathbb{R}^{N}}, with{N\geq 2},{g\in C^{1}((0,\infty),(0,\infty))}is decreasing in{(0,\infty)}and satisfies{\lim_{s\rightarrow 0^{+}}g(s)=\infty}, and{b\in C^{\infty}(\Omega)}is positive in Ω, but may vanish or blow up on the boundary. We find a new structure condition ongwhich plays a crucial role in the boundary behavior of such solution.

scholarly journals Positive solutions of the discrete Dirichlet problem involving the mean curvature operator

2019 ◽  
Vol 17 (1) ◽  
pp. 1055-1064 ◽  
Author(s):  
Jiaoxiu Ling ◽  
Zhan Zhou
Keyword(s):  
Dirichlet Problem ◽  
Positive Solutions ◽  
Critical Point Theory ◽  
Mean Curvature ◽  
Nonlinear Term ◽  
Sufficient Conditions ◽  
Point Theory ◽  
Curvature Operator ◽  
Mean Curvature Operator ◽  
The Mean

Abstract In this paper, by using critical point theory, we obtain some sufficient conditions on the existence of infinitely many positive solutions of the discrete Dirichlet problem involving the mean curvature operator. We show that the suitable oscillating behavior of the nonlinear term near at the origin and at infinity will lead to the existence of a sequence of pairwise distinct nontrivial positive solutions. We also give two examples to illustrate our main results.

scholarly journals Singular Dirichlet Boundary Value Problem for Second Order Ode

2007 ◽  
Vol 14 (2) ◽  
pp. 325-340
Author(s):  
Irena Rachůnková ◽  
Jakub Stryja
Keyword(s):  
Dirichlet Problem ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Value Problem ◽  
Existence Of A Solution ◽  
Absolutely Continuous ◽  
First Derivative ◽  
Carathéodory Conditions ◽  
Time Singularities ◽  
Existence Principle ◽  
Singular Dirichlet Problem

Abstract This paper investigates the singular Dirichlet problem –𝑢″ = 𝑓(𝑡, 𝑢, 𝑢′), 𝑢(0) = 0, 𝑢(𝑇) = 0, where 𝑓 satisfies the Carathéodory conditions on the set and . The function 𝑓(𝑡, 𝑥, 𝑦) can have time singularities at 𝑡 = 0 and 𝑡 = 𝑇 and space singularities at 𝑥 = 0 and 𝑦 = 0. The existence principle for the above problem is given and its application is presented here. The paper provides conditions which guarantee the existence of a solution which is positive on (0; T) and which has the absolutely continuous first derivative on [0, 𝑇].

Uniqueness of positive solutions of a quasilinear Dirichlet problem with exponential nonlinearity

1998 ◽  
Vol 128 (5) ◽  
pp. 895-906 ◽  
Author(s):  
Adimurthi
Keyword(s):  
Dirichlet Problem ◽  
Positive Solutions ◽  
Exponential Nonlinearity ◽  
Unit Radius ◽  
Image Position

Uniqueness of positive solutions of the following equation:has been obtained, where B ⊂ ℝn is the ball of unit radius and λ > 0. Moreover, if n = 2, then the solutions are nondegenerate.

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