Nonlinear Elliptic Equations with Dirichlet Boundary Conditions

2013 ◽  
pp. 303-385
Author(s):  
Dumitru Motreanu ◽  
Viorica Venera Motreanu ◽  
Nikolaos Papageorgiou
Keyword(s):  
Boundary Conditions ◽  
Elliptic Equations ◽  
Dirichlet Boundary ◽  
Nonlinear Elliptic Equations ◽  
Dirichlet Boundary Conditions ◽  
Nonlinear Elliptic

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.

ON A CLASS OF FOURTH ORDER NONLINEAR ELLIPTIC EQUATIONS UNDER NAVIER BOUNDARY CONDITIONS

2010 ◽  
Vol 08 (02) ◽  
pp. 185-197 ◽  
Author(s):  
F. J. S. A. CORRÊA ◽  
J. V. GONCALVES ◽  
ANGELO RONCALLI
Keyword(s):  
Boundary Conditions ◽  
Fixed Points ◽  
Fourth Order ◽  
Elliptic Equations ◽  
Nonlinear Elliptic Equations ◽  
Compact Operators ◽  
Schmidt Method ◽  
Navier Boundary Conditions ◽  
Nonlinear Elliptic ◽  
Existence And Multiplicity

We employ arguments involving continua of fixed points of suitable nonlinear compact operators and the Lyapunov–Schmidt method to prove existence and multiplicity of solutions in a class of fourth order non-homogeneous resonant elliptic problems. Our main result extends even similar ones known for the Laplacian.

Multiple Solutions for Perturbed Elliptic Equations in Unbounded Domains

2003 ◽  
Vol 3 (1) ◽  
pp. 1-23 ◽  
Author(s):  
A. Salvatore
Keyword(s):  
Schrödinger Equation ◽  
Boundary Conditions ◽  
Perturbation Method ◽  
Elliptic Equations ◽  
Multiple Solutions ◽  
Schrodinger Equation ◽  
Dirichlet Boundary ◽  
Unbounded Domains ◽  
Dirichlet Boundary Conditions

AbstractWe look for solutions of a nonlinear perturbed Schrödinger equation with nonhomogeneous Dirichlet boundary conditions. By using a perturbation method introduced by Bolle, we prove the existence of multiple solutions in spite of the lack of the symmetry of the problem.

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 Nonlinear elliptic equations involving the p-Laplacian with mixed Dirichlet-Neumann boundary conditions

2019 ◽  
Vol 39 (2) ◽  
pp. 159-174 ◽  
Author(s):  
Gabriele Bonanno ◽  
Giuseppina D'Aguì ◽  
Angela Sciammetta
Keyword(s):  
Boundary Conditions ◽  
Elliptic Equations ◽  
Mixed Boundary ◽  
Neumann Boundary ◽  
Nonlinear Elliptic Equations ◽  
Mixed Boundary Conditions ◽  
Neumann Boundary Conditions ◽  
Laplacian Operator ◽  
Differential Problem ◽  
Nonlinear Elliptic

In this paper, a nonlinear differential problem involving the \(p\)-Laplacian operator with mixed boundary conditions is investigated. In particular, the existence of three non-zero solutions is established by requiring suitable behavior on the nonlinearity. Concrete examples illustrate the abstract results.

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