Non-existence of solutions for some nonlinear elliptic equations involving measures

2000 ◽  
Vol 130 (1) ◽  
pp. 167-187 ◽  
Author(s):  
L. Orsina ◽  
A. Prignet
Keyword(s):  
Elliptic Equations ◽  
Dirichlet Boundary ◽  
Existence Of Solutions ◽  
Model Problem ◽  
Nonlinear Elliptic Equations ◽  
Dirichlet Boundary Conditions ◽  
Regular Functions ◽  
Nonlinear Elliptic ◽  
Bounded Open Subset ◽  
Image Position

In this paper, we study the non-existence of solutions for the following (model) problem in a bounded open subset Ω of RN: with Dirichlet boundary conditions, where p > 1, q > 1 and μ is a bounded Radon measure. We prove that if λ is a measure which is concentrated on a set of zero r capacity (p < r ≤ N), and if q > r (p − 1)/(r − p), then there is no solution to the above problem, in the sense that if one approximates the measure λ with a sequence of regular functions fn, and if un is the sequence of solutions of the corresponding problems, then un converges to zero.We also study the non-existence of solutions for the bilateral obstacle problem with datum a measure λ concentrated on a set of zero p capacity, with u in for every υ in K, finding again that the only solution obtained by approximation is u = 0.

Padé approximant bounds on the positive solutions of some nonlinear elliptic equations

1978 ◽  
Vol 80 (3-4) ◽  
pp. 183-200
Author(s):  
M. F. Barnsley ◽  
D. Bessis
Keyword(s):  
Laplace Transform ◽  
Elliptic Equations ◽  
Positive Measure ◽  
Dirichlet Boundary ◽  
Homogeneous Equation ◽  
Nonlinear Elliptic Equations ◽  
Dirichlet Boundary Conditions ◽  
Upper And Lower Bounds ◽  
Nonlinear Elliptic ◽  
Positive Functions

SynopsisWe consider the equation Lφ − λpφ + γqφ2 = f on a bounded domain in Rn with homogeneous Neumann-Dirichlet boundary conditions. L is a negative definite uniformly elliptic differential operator, while, p, q and f are positive functions. We show that there exists exactly one positive solution for each λ ∈ R and γ > 0. This solution can be analytically continued throughout Re γ > 0: it is a Laplace transform of a positive measure. The measure is bounded prior to the bifurcation point of the associated “homogeneous” equation and unbounded after. Noting that any Laplace transform of positive measure has associated with it a natural sequence of Tchebycheff systems, it now follows that one can obtain monotonically converging upper and lower bounds which are provided by the generalized Padé approximants generated from the Tchebycheff systems.

Some Elliptic Problems With Degenerate Coercivity

2006 ◽  
Vol 6 (1) ◽  
pp. 1-12 ◽  
Author(s):  
Lucio Boccardo
Keyword(s):  
Open Subset ◽  
Elliptic Equations ◽  
Principal Part ◽  
Existence Of Solutions ◽  
Elliptic Problems ◽  
Nonlinear Elliptic Equations ◽  
Model Case ◽  
Nonlinear Elliptic ◽  
Degenerate Coercivity ◽  
Bounded Open Subset

AbstractIn this paper we are interested in existence of solutions for some nonlinear elliptic equations with principal part having degenerate coercivity. The model case iswith Ω bounded open subset of ℝ

On some nonlinear elliptic equations involving derivatives of the nonlinearity

1985 ◽  
Vol 100 (3-4) ◽  
pp. 281-294 ◽  
Author(s):  
J. Carrillo ◽  
M. Chipot
Keyword(s):  
Boundary Value Problems ◽  
Elliptic Equations ◽  
Existence And Uniqueness ◽  
Elliptic Boundary ◽  
Nonlinear Elliptic Equations ◽  
Elliptic Boundary Value Problems ◽  
Elliptic Boundary Value ◽  
Nonlinear Elliptic ◽  
Image Position ◽  
Derivatives Of

SynopsisWe give some results on existence and uniqueness for the solution of elliptic boundary value problems of typewhen the βi are not necessarily smooth.

scholarly journals A Class of Degenerate Nonlinear Elliptic Equations in Weighted Sobolev Space

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Rasmita Kar
Keyword(s):  
Weak Solution ◽  
Sobolev Space ◽  
Boundary Value Problem ◽  
Elliptic Equations ◽  
Dirichlet Boundary ◽  
Weighted Sobolev Space ◽  
Nonlinear Elliptic Equations ◽  
Dirichlet Boundary Value Problem ◽  
Nonlinear Elliptic ◽  
Bounded Nonlinearity

We prove the existence of a weak solution for the degenerate nonlinear elliptic Dirichlet boundary-value problem Lu-μug1+hu,∇ug2=f in Ω, u=0 on ∂Ω, in a suitable weighted Sobolev space, where Ω⊂ℝn is a bounded domain and h is a continuous bounded nonlinearity.

scholarly journals Existence Results For A Class Of Nonlinear Degenerate Elliptic Equations

2019 ◽  
Vol 5 (2) ◽  
pp. 164-178
Author(s):  
Albo Carlos Cavalheiro
Keyword(s):  
Dirichlet Problem ◽  
Elliptic Equations ◽  
Existence Of Solutions ◽  
Weighted Sobolev Spaces ◽  
Nonlinear Elliptic Equations ◽  
Existence Results ◽  
Degenerate Elliptic Equations ◽  
Nonlinear Elliptic ◽  
Degenerate Elliptic ◽  
Nonlinear Degenerate

AbstractIn this paper we are interested in the existence of solutions for Dirichlet problem associated with the degenerate nonlinear elliptic equations\left\{ {\matrix{ { - {\rm{div}}\left[ {\mathcal{A}\left( {x,\nabla u} \right){\omega _1} + \mathcal{B}\left( {x,u,\nabla u} \right){\omega _2}} \right] = {f_0}\left( x \right) - \sum\limits_{j = 1}^n {{D_j}{f_j}\left( x \right)\,\,{\rm{in}}} \,\,\,\,\,\Omega ,} \hfill \cr {u\left( x \right) = 0\,\,\,\,{\rm{on}}\,\,\,\,\partial \Omega {\rm{,}}} \hfill \cr } } \right.in the setting of the weighted Sobolev spaces.

scholarly journals Regularity of Extremal Solutions to Nonlinear Elliptic Equations with Quadratic Convection and General Reaction

2020 ◽  
Vol 17 (6) ◽  
Author(s):  
Asadollah Aghajani ◽  
Fatemeh Mottaghi ◽  
Vicenţiu D. Rădulescu
Keyword(s):  
Elliptic Equations ◽  
Nonlinear Elliptic Equation ◽  
Dirichlet Boundary ◽  
Nonlinear Elliptic Equations ◽  
Extremal Solution ◽  
General Reaction ◽  
Smooth Bounded Domain ◽  
Nonlinear Elliptic ◽  
Regularity Results ◽  
Increasing Function

AbstractWe consider the nonlinear elliptic equation with quadratic convection $$ -\Delta u + g(u) |\nabla u|^2=\lambda f(u) $$ - Δ u + g ( u ) | ∇ u | 2 = λ f ( u ) in a smooth bounded domain $$ \Omega \subset {\mathbb {R}}^N $$ Ω ⊂ R N ($$ N \ge 3$$ N ≥ 3 ) with zero Dirichlet boundary condition. Here, $$ \lambda $$ λ is a positive parameter, $$ f:[0, \infty ):(0\infty ) $$ f : [ 0 , ∞ ) : ( 0 ∞ ) is a strictly increasing function of class $$C^1$$ C 1 , and g is a continuous positive decreasing function in $$ (0, \infty ) $$ ( 0 , ∞ ) and integrable in a neighborhood of zero. Under natural hypotheses on the nonlinearities f and g, we provide some new regularity results for the extremal solution $$u^*$$ u ∗ . A feature of this paper is that our main contributions require neither the convexity (even at infinity) of the function $$ h(t)=f(t)e^{-\int _0^t g(s)ds}$$ h ( t ) = f ( t ) e - ∫ 0 t g ( s ) d s , nor that the functions $$ gh/h'$$ g h / h ′ or $$ h'' h/h'^2$$ h ′ ′ h / h ′ 2 admit a limit at infinity.

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