On nonlinear perturbations of linear second order elliptic boundary value problems

1978 ◽  
Vol 84 (1) ◽  
pp. 143-157 ◽  
Author(s):  
P. M. Fitzpatrick
Keyword(s):  
Asymptotic Behaviour ◽  
Boundary Value Problems ◽  
Elliptic Operator ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Second Order ◽  
Nonlinear Perturbations ◽  
Elliptic Boundary Value ◽  
Image Position

AbstractLet Ω ⊆ n be open and bounded, with ∂Ω smooth. Sufficient conditions on f are given in order that for p > n, the equationhas a solution for every h ∈ Lp; L is a second order symmetric elliptic operator and B represents either the first, second or third boundary value problems. These conditions are in terms of the asymptotic behaviour of f, in its second variable, in relation to those λ for which there is a nontrivial solution of

scholarly journals Fundamental solutions and surface distributions

1969 ◽  
Vol 16 (3) ◽  
pp. 255-257
Author(s):  
R. A. Adams ◽  
G. F. Roach
Keyword(s):  
Elliptic Equation ◽  
Boundary Value Problems ◽  
Bounded Domain ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Fundamental Solutions ◽  
Second Order ◽  
Elliptic Boundary Value Problems ◽  
Elliptic Boundary Value ◽  
Image Position

When studying the solutions of elliptic boundary value problems in a bounded, smoothly bounded domain D⊂Rn we often encounter the formulawhere u(x)∈C2(D)∩C′(D̄) is a solution of the second order self-adjoint elliptic equationand denotes differentiation along the inward normal to ∂D at x∈∂D.

scholarly journals Second order elliptic boundary value problems in spaces with homogeneous norms

1980 ◽  
Vol 29 (1) ◽  
pp. 1-13 ◽  
Author(s):  
A. J. Pryde
Keyword(s):  
Boundary Value Problems ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Partial Differential Operator ◽  
Second Order ◽  
Elliptic Boundary Value ◽  
First Order ◽  
Constant Coefficients ◽  
Regularity Results ◽  
Homogeneous Norms

AbstractWe consider the interior and Dirichiet problems and problems with first order boundary conditions, for a second order homogeneous elliptic partial differential operator with constant coefficients. Under natural conditions on the operators, these problems give rise to isomorphisms between the appropriate spaces with homogeneous norms. From there we obtain a priori estimates and regularity results for boundary value problems in Sobolev spaces.

General edge asymptotics of solutions of second-order elliptic boundary value problems I

1993 ◽  
Vol 123 (1) ◽  
pp. 109-155 ◽  
Author(s):  
Martin Costabel ◽  
Monique Dauge
Keyword(s):  
Boundary Value Problems ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Weighted Sobolev Spaces ◽  
Second Order ◽  
Elliptic Boundary Value Problems ◽  
General Situation ◽  
Opening Angle ◽  
Elliptic Boundary Value ◽  
Basic Facts

SynopsisThis is the first of two papers in which we study the singularities of solutions of second-order linear elliptic boundary value problems at the edges of piecewise analytic domains in ℝ3. When the opening angle at the edge is variable, there appears the phenomenon of “crossing” of the exponents of singularities. For this case, we introduce the appropriate combinations of the simple tensor product singularities that allow us to give estimates in ordinary and weighted Sobolev spaces for the regular part of the solution and for the coefficients of the singularities. These combinations appear in a natural way as sections of an analytic bundle above the edge. Their behaviour is described with the help of divided differences of powers of the distance to the edge. The class of operators considered includes second-order elliptic operators with analytic complex-valued coefficients with mixed Dirichlet, Neumann or oblique derivative conditions. With our description of the singularities we are able to remove some restrictive hypotheses that were previously made in other works. In this first part, we prove the basic facts in a simplified framework. Nevertheless the tools we use are essentially the same in the general situation.

PERTURBATIONS FROM INDEFINITE SYMMETRIC ELLIPTIC BOUNDARY VALUE PROBLEMS

2017 ◽  
Vol 59 (3) ◽  
pp. 635-648 ◽  
Author(s):  
LIANG ZHANG ◽  
XIANHUA TANG
Keyword(s):  
Boundary Value Problems ◽  
Bounded Domain ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Elliptic Boundary Value Problems ◽  
Smooth Bounded Domain ◽  
Multiplicity Of Solutions ◽  
Elliptic Boundary Value ◽  
Formula Group ◽  
Image Position

AbstractIn this paper, we study the multiplicity of solutions for the following problem: $$\begin{equation*} \begin{cases} -\Delta u-\Delta(|u|^{\alpha})|u|^{\alpha-2}u=g(x,u)+\theta h(x,u), \ \ x\in \Omega,\\ u=0, \ \ x\in \partial\Omega, \end{cases} \end{equation*}$$ where α ≥ 2, Ω is a smooth bounded domain in ${\mathbb{R}}$N, θ is a parameter and g, h ∈ C($\bar{\Omega}$ × ${\mathbb{R}}$). Under the assumptions that g(x, u) is odd and locally superlinear at infinity in u, we prove that for any j ∈ $\mathbb{N}$ there exists ϵj > 0 such that if |θ| ≤ ϵj, the above problem possesses at least j distinct solutions. Our results generalize some known results in the literature and are new even in the symmetric situation.

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