scholarly journals Global branches of positive weak solutions of semilinear elliptic problems over nonsmooth domains

1994 ◽  
Vol 124 (2) ◽  
pp. 371-388 ◽  
Author(s):  
Timothy J. Healey ◽  
Hansjörg Kielhöfer ◽  
Charles A. Stuart
Keyword(s):  
Boundary Condition ◽  
Weak Solutions ◽  
Dirichlet Boundary ◽  
Nonlinear Eigenvalue Problem ◽  
Order Elliptic Equation ◽  
Semilinear Elliptic ◽  
Nonsmooth Domains ◽  
Second Order Elliptic Equation ◽  
Semilinear Elliptic Problems ◽  
Parameter Dependent

We consider the nonlinear eigenvalue problem posed by a parameter-dependent semilinear second-order elliptic equation on a bounded domain with the Dirichlet boundary condition. The coefficients of the elliptic operator are bounded measurable functions and the boundary of the domain is only required to be regular in the sense of Wiener. The main results establish the existence of an unbounded branch of positive weak solutions.

BOUNDEDNESS OF THE EXTREMAL SOLUTION FOR SEMILINEAR ELLIPTIC PROBLEMS

2002 ◽  
Vol 04 (03) ◽  
pp. 547-558 ◽  
Author(s):  
DONG YE ◽  
FENG ZHOU
Keyword(s):  
Boundary Condition ◽  
Dirichlet Boundary ◽  
Sufficient Conditions ◽  
Semilinear Elliptic Equation ◽  
Extremal Solution ◽  
Extremal Solutions ◽  
Bounded Smooth Domain ◽  
Semilinear Elliptic ◽  
Semilinear Elliptic Problems ◽  
Bounded Smooth

We investigate here the boundedness of extremal solutions for some semilinear elliptic equation -Δu=λf(u) posed on a bounded smooth domain of ℝN with Dirichlet boundary condition. Some sufficient conditions for f are established to ensure the regularity of extremal solutions when N ≤ 9, which cover all well-known cases.

scholarly journals On the extremal solutions of semilinear elliptic problems

2005 ◽  
Vol 2005 (1) ◽  
pp. 1-9 ◽  
Author(s):  
Lamia Ben Chaabane
Keyword(s):  
Boundary Condition ◽  
Elliptic Equation ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Semilinear Elliptic Equation ◽  
Extremal Solutions ◽  
Bounded Smooth Domain ◽  
Semilinear Elliptic ◽  
Semilinear Elliptic Problems ◽  
Bounded Smooth

We investigate here the properties of extremal solutions for semilinear elliptic equation−Δu=λf(u)posed on a bounded smooth domain ofℝnwith Dirichlet boundary condition and withfexploding at a finite positive valuea.

A Note on Symmetry of Solutions for a Class of Singular Semilinear Elliptic Problems

2016 ◽  
Vol 16 (3) ◽  
Author(s):  
Alessandro Trombetta
Keyword(s):  
Elliptic Equation ◽  
Dirichlet Boundary ◽  
Semilinear Elliptic Equation ◽  
Dirichlet Boundary Conditions ◽  
Moving Plane Method ◽  
Plane Method ◽  
Semilinear Elliptic ◽  
Moving Plane ◽  
Semilinear Elliptic Problems ◽  
Bounded Smooth

AbstractWe prove symmetry and monotonicity properties for positive solutions of the singular semilinear elliptic equationin bounded smooth domains with zero Dirichlet boundary conditions. The well-known moving plane method is applied.

On the Number of Solutions to Semilinear Boundary Value Problems

2004 ◽  
Vol 4 (3) ◽  
Author(s):  
Markus Kunze ◽  
Rafael Ortega
Keyword(s):  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Upper Bound ◽  
Dirichlet Boundary ◽  
Neumann Boundary ◽  
Neumann Boundary Conditions ◽  
Dirichlet Boundary Conditions ◽  
Number Of Solutions ◽  
Semilinear Elliptic ◽  
Semilinear Elliptic Problems

AbstractWe consider semilinear elliptic problems of the form Δu + g(u) = f(x) with Neumann boundary conditions or Δu+λ1u+g(u) = f(x) with Dirichlet boundary conditions, and we derive conditions on g and f under which an upper bound on the number of solutions can be obtained.

scholarly journals Multiplicity of solutions for singular semilinear elliptic equations with critical Hardy-Sobolev exponents

2008 ◽  
Vol 2 (2) ◽  
pp. 158-174 ◽  
Author(s):  
Qianqiao Guo ◽  
Pengcheng Niu ◽  
Jingbo Dou
Keyword(s):  
Boundary Condition ◽  
Variational Methods ◽  
Elliptic Problem ◽  
Dirichlet Boundary Condition ◽  
Elliptic Equations ◽  
Dirichlet Boundary ◽  
Nontrivial Solutions ◽  
Existence And Multiplicity ◽  
Semilinear Elliptic Problem ◽  
Semilinear Elliptic

We consider the semilinear elliptic problem with critical Hardy-Sobolev exponents and Dirichlet boundary condition. By using variational methods we obtain the existence and multiplicity of nontrivial solutions and improve the former results.

Sequences of Weak Solutions for Non-Local Elliptic Problems with Dirichlet Boundary Condition

2014 ◽  
Vol 57 (3) ◽  
pp. 779-809 ◽  
Author(s):  
Giovanni Molica Bisci ◽  
Pasquale F. Pizzimenti
Keyword(s):  
Boundary Condition ◽  
Dirichlet Problem ◽  
Variational Methods ◽  
Weak Solutions ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Infinitely Many Solutions ◽  
Distinct Sequence ◽  
Non Local ◽  
Kirchhoff Type Problems

AbstractIn this paper the existence of infinitely many solutions for a class of Kirchhoff-type problems involving the p-Laplacian, with p > 1, is established. By using variational methods, we determine unbounded real intervals of parameters such that the problems treated admit either an unbounded sequence of weak solutions, provided that the nonlinearity has a suitable behaviour at ∞, or a pairwise distinct sequence of weak solutions that strongly converges to 0 if a similar behaviour occurs at 0. Some comparisons with several results in the literature are pointed out. The last part of the work is devoted to the autonomous elliptic Dirichlet problem.

scholarly journals Global weak solutions for a kinetic-fluid model with local alignment force in a bounded domain

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Fucai Li ◽  
Yue Li
Keyword(s):  
Boundary Condition ◽  
Bounded Domain ◽  
Weak Solutions ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Stokes Equations ◽  
Fluid Model ◽  
Navier Stokes ◽  
Local Alignment ◽  
Fokker Planck

<p style='text-indent:20px;'>We study a kinetic-fluid model in a <inline-formula><tex-math id="M1">\begin{document}$ 3D $\end{document}</tex-math></inline-formula> bounded domain. More precisely, this model is a coupling of the Vlasov-Fokker-Planck equation with the local alignment force and the compressible Navier-Stokes equations with nonhomogeneous Dirichlet boundary condition. We prove the global existence of weak solutions to it for the isentropic fluid (adiabatic coefficient <inline-formula><tex-math id="M2">\begin{document}$ \gamma&gt; 3/2 $\end{document}</tex-math></inline-formula>) and hence extend the existence result of Choi and Jung [Asymptotic analysis for a Vlasov-Fokker-Planck/Navier-Stokes system in a bounded domain, arXiv: 1912.13134v2], where the velocity of the fluid is supplemented with homogeneous Dirichlet boundary condition.</p>

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