scholarly journals Classification of isolated singularities of nonnegative solutions to fractional semi-linear elliptic equations and the existence results

2018 ◽  
Vol 97 (2) ◽  
pp. 196-221 ◽  
Author(s):  
Huyuan Chen ◽  
Alexander Quaas
Keyword(s):  
Elliptic Equations ◽  
Existence Results ◽  
Isolated Singularities ◽  
Nonnegative Solutions

scholarly journals Measure Data Problems for a Class of Elliptic Equations with Mixed Absorption-Reaction

2021 ◽  
Vol 21 (2) ◽  
pp. 261-280
Author(s):  
Marie-Françoise Bidaut-Véron ◽  
Marta Garcia-Huidobro ◽  
Laurent Véron
Keyword(s):  
Dirichlet Problem ◽  
Compact Subset ◽  
Elliptic Equations ◽  
Radon Measure ◽  
Measure Data ◽  
Nonnegative Solutions

Abstract In the present paper, we study the existence of nonnegative solutions to the Dirichlet problem ℒ p , q M ⁢ u := - Δ ⁢ u + u p - M ⁢ | ∇ ⁡ u | q = μ {{\mathcal{L}}^{{M}}_{p,q}u:=-\Delta u+u^{p}-M|\nabla u|^{q}=\mu} in a domain Ω ⊂ ℝ N {\Omega\subset\mathbb{R}^{N}} where μ is a nonnegative Radon measure, when p > 1 {p>1} , q > 1 {q>1} and M ≥ 0 {M\geq 0} . We also give conditions under which nonnegative solutions of ℒ p , q M ⁢ u = 0 {{\mathcal{L}}^{{M}}_{p,q}u=0} in Ω ∖ K {\Omega\setminus K} , where K is a compact subset of Ω, can be extended as a solution of the same equation in Ω.

scholarly journals Loop Type Subcontinua of Positive Solutions for Indefinite Concave-Convex Problems

2019 ◽  
Vol 19 (2) ◽  
pp. 391-412
Author(s):  
Uriel Kaufmann ◽  
Humberto Ramos Quoirin ◽  
Kenichiro Umezu
Keyword(s):  
Elliptic Equations ◽  
Bifurcation Theory ◽  
A Priori ◽  
Global Bifurcation ◽  
Neumann Boundary ◽  
Zero Solution ◽  
Regularization Scheme ◽  
Convex Problems ◽  
Nonnegative Solutions ◽  
Indefinite Weights

AbstractWe establish the existence of loop type subcontinua of nonnegative solutions for a class of concave-convex type elliptic equations with indefinite weights, under Dirichlet and Neumann boundary conditions. Our approach depends on local and global bifurcation analysis from the zero solution in a nonregular setting, since the nonlinearities considered are not differentiable at zero, so that the standard bifurcation theory does not apply. To overcome this difficulty, we combine a regularization scheme with a priori bounds, and Whyburn’s topological method. Furthermore, via a continuity argument we prove a positivity property for subcontinua of nonnegative solutions. These results are based on a positivity theorem for the associated concave problem proved by us, and extend previous results established in the powerlike case.

Classification of nonnegative solutions to a bi-harmonic equation with Hartree type nonlinearity

2018 ◽  
Vol 149 (04) ◽  
pp. 979-994 ◽  
Author(s):  
Daomin Cao ◽  
Wei Dai
Keyword(s):  
Critical Case ◽  
Classical Solutions ◽  
Sobolev Inequalities ◽  
Extremal Functions ◽  
Method Of Moving Planes ◽  
Nonnegative Solutions ◽  
Moving Planes ◽  
Image Position ◽  
Harmonic Equation

AbstractIn this paper, we are concerned with the following bi-harmonic equation with Hartree type nonlinearity $$\Delta ^2u = \left( {\displaystyle{1 \over { \vert x \vert ^8}}* \vert u \vert ^2} \right)u^\gamma ,\quad x\in {\open R}^d,$$where 0 < γ ⩽ 1 and d ⩾ 9. By applying the method of moving planes, we prove that nonnegative classical solutions u to (𝒫γ) are radially symmetric about some point x0 ∈ ℝd and derive the explicit form for u in the Ḣ2 critical case γ = 1. We also prove the non-existence of nontrivial nonnegative classical solutions in the subcritical cases 0 < γ < 1. As a consequence, we also derive the best constants and extremal functions in the corresponding Hardy-Littlewood-Sobolev inequalities.

On the removability of isolated singular points for elliptic equations involving variable exponent

2016 ◽  
Vol 5 (2) ◽  
Author(s):  
Yongqiang Fu ◽  
Yingying Shan
Keyword(s):  
Singular Point ◽  
Elliptic Equations ◽  
Variable Exponent ◽  
Singular Points ◽  
Sufficient Condition ◽  
Variable Exponents ◽  
Isolated Singularities

AbstractIn this paper, we study the problem of removable isolated singularities for elliptic equations with variable exponents. We give a sufficient condition for removability of the isolated singular point for the equations in

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