Two Classes of Nonlinear Singular Dirichlet Problems with Natural Growth: Existence and Asymptotic Behavior

2020 ◽  
Vol 20 (1) ◽  
pp. 77-93 ◽  
Author(s):  
Zhijun Zhang
Keyword(s):  
Asymptotic Behavior ◽  
Bounded Domain ◽  
Smooth Boundary ◽  
Classical Solutions ◽  
Natural Growth ◽  
Dirichlet Problems ◽  
Nonlinear Transformations

AbstractThis paper is concerned with the existence, uniqueness and asymptotic behavior of classical solutions to two classes of models {-\triangle u\pm\lambda\frac{|\nabla u|^{2}}{u^{\beta}}=b(x)u^{-\alpha}}, {u>0}, {x\in\Omega}, {u|_{\partial\Omega}=0}, where Ω is a bounded domain with smooth boundary in {\mathbb{R}^{N}}, {\lambda>0}, {\beta>0}, {\alpha>-1}, and {b\in C^{\nu}_{\mathrm{loc}}(\Omega)} for some {\nu\in(0,1)}, and b is positive in Ω but may be vanishing or singular on {\partial\Omega}. Our approach is largely based on nonlinear transformations and the construction of suitable sub- and super-solutions.

scholarly journals The exterior Dirichlet problems of Monge–Ampère equations in dimension two

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Limei Dai
Keyword(s):  
Asymptotic Behavior ◽  
Unit Ball ◽  
Bounded Domain ◽  
Viscosity Solutions ◽  
Sufficient Conditions ◽  
Radial Solutions ◽  
Dirichlet Problems ◽  
Necessary And Sufficient ◽  
Monge Ampere ◽  
Prescribed Asymptotic Behavior

AbstractIn this paper, we study the Monge–Ampère equations $\det D^{2}u=f$ det D 2 u = f in dimension two with f being a perturbation of $f_{0}$ f 0 at infinity. First, we obtain the necessary and sufficient conditions for the existence of radial solutions with prescribed asymptotic behavior at infinity to Monge–Ampère equations outside a unit ball. Then, using the Perron method, we get the existence of viscosity solutions with prescribed asymptotic behavior at infinity to Monge–Ampère equations outside a bounded domain.

scholarly journals Boundedness and Asymptotic Behavior to a Chemotaxis System with Indirect Signal Generation and Singular Sensitivity

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Jie Wu ◽  
Li Zhao ◽  
Heping Pan
Keyword(s):  
Asymptotic Behavior ◽  
Boundary Conditions ◽  
Bounded Domain ◽  
Smooth Boundary ◽  
Rates Of Convergence ◽  
Signal Generation ◽  
Heat Semigroup ◽  
Singular Sensitivity ◽  
Chemotaxis System ◽  
Global Boundedness

In this paper, we consider the following indirect signal generation and singular sensitivity n t = Δ n + χ ∇ ⋅ n / φ c ∇ c ,   x ∈ Ω , t > 0 , c t = Δ c − c + w ,   x ∈ Ω , t > 0 , w t = Δ w − w + n ,   x ∈ Ω , t > 0 , in a bounded domain Ω ⊂ R N N = 2 , 3 with smooth boundary ∂ Ω . Under the nonflux boundary conditions for n , c , and w , we first eliminate the singularity of φ c by using the Neumann heat semigroup and then establish the global boundedness and rates of convergence for solution.

scholarly journals Weighted fourth order elliptic problems in the unit ball

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Zongming Guo ◽  
Fangshu Wan
Keyword(s):  
Asymptotic Behavior ◽  
Singular Point ◽  
Unit Ball ◽  
Fourth Order ◽  
Existence And Uniqueness ◽  
Weighted Sobolev Spaces ◽  
Radial Solutions ◽  
Dirichlet Problems ◽  
Uniqueness Results ◽  
Positive Radial Solutions

<p style='text-indent:20px;'>Existence and uniqueness of positive radial solutions of some weighted fourth order elliptic Navier and Dirichlet problems in the unit ball <inline-formula><tex-math id="M1">\begin{document}$ B $\end{document}</tex-math></inline-formula> are studied. The weights can be singular at <inline-formula><tex-math id="M2">\begin{document}$ x = 0 \in B $\end{document}</tex-math></inline-formula>. Existence of positive radial solutions of the problems is obtained via variational methods in the weighted Sobolev spaces. To obtain the uniqueness results, we need to know exactly the asymptotic behavior of the solutions at the singular point <inline-formula><tex-math id="M3">\begin{document}$ x = 0 $\end{document}</tex-math></inline-formula>.</p>

Exact asymptotic behavior near the boundary to the solution for singular nonlinear Dirichlet problems

2009 ◽  
Vol 71 (9) ◽  
pp. 4137-4150 ◽  
Author(s):  
Sonia Ben Othman ◽  
Habib Mâagli ◽  
Syrine Masmoudi ◽  
Malek Zribi
Keyword(s):  
Asymptotic Behavior ◽  
Dirichlet Problems ◽  
Exact Asymptotic Behavior

NONTRIVIAL SOLUTIONS FOR N-LAPLACIAN EQUATIONS WITH SUB-EXPONENTIAL GROWTH

2013 ◽  
Vol 11 (03) ◽  
pp. 1350005 ◽  
Author(s):  
ZHONG TAN ◽  
FEI FANG
Keyword(s):  
Dirichlet Problem ◽  
Bounded Domain ◽  
Exponential Growth ◽  
Morse Theory ◽  
Smooth Boundary ◽  
Orlicz Spaces ◽  
Nontrivial Solutions ◽  
Laplacian Equations

Let Ω be a bounded domain in RNwith smooth boundary ∂Ω. In this paper, the following Dirichlet problem for N-Laplacian equations (N > 1) are considered: [Formula: see text] We assume that the nonlinearity f(x, t) is sub-exponential growth. In fact, we will prove the mapping f(x, ⋅): LA(Ω) ↦ LÃ(Ω) is continuous, where LA(Ω) and LÃ(Ω) are Orlicz spaces. Applying this result, the compactness conditions would be obtained. Hence, we may use Morse theory to obtain existence of nontrivial solutions for problem (N).

scholarly journals INTERIOR ERROR ESTIMATE FOR PERIODIC HOMOGENIZATION

2006 ◽  
Vol 04 (01) ◽  
pp. 61-79 ◽  
Author(s):  
GEORGES GRISO
Keyword(s):  
Error Estimate ◽  
Error Estimates ◽  
Bounded Domain ◽  
Smooth Boundary ◽  
Classical Problem ◽  
Global Error ◽  
Periodic Homogenization ◽  
Open Set ◽  
Interior Error Estimate

In a previous paper about homogenization of the classical problem of diffusion in a bounded domain with sufficiently smooth boundary, we proved that the global error is of order ε1/2. Now, for an open set Ω with sufficiently smooth boundary [Formula: see text] and homogeneous Dirichlet or Neumann limit conditions, we show that in any open set strongly included in Ω the error is of order ε. If the open set Ω ⊂ ℝn is of polygonal (n = 2) or polyhedral (n = 3) boundary, we also give the global and interior error estimates.

scholarly journals On the location of the peaks of least-energy solutions to semilinear Dirichlet problems with critical growth

2002 ◽  
Vol 7 (10) ◽  
pp. 547-561 ◽  
Author(s):  
Marco A. S. Souto
Keyword(s):  
Bounded Domain ◽  
Critical Sobolev Exponent ◽  
Critical Growth ◽  
Dirichlet Problems ◽  
Growth Problem ◽  
Sobolev Exponent ◽  
Least Energy Solutions ◽  
Least Energy

We study the location of the peaks of solution for the critical growth problem−ε 2Δu+u=f(u)+u 2*−1,u>0inΩ,u=0on∂Ω, whereΩis a bounded domain;2*=2N/(N−2),N≥3, is the critical Sobolev exponent andfhas a behavior likeup,1<p<2*−1.

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