The Dirichlet problem for the biharmonic and polyharmonic equations

2000 ◽  
pp. 227-249
Author(s):  
V. Kozlov ◽  
V. Maz’ya ◽  
J. Rossmann
Keyword(s):  
Dirichlet Problem ◽  
Polyharmonic Equations

Green function representation in the Dirichlet problem for polyharmonic equations in a ball

2008 ◽  
Vol 78 (1) ◽  
pp. 528-530 ◽  
Author(s):  
T. Sh. Kalmenov ◽  
B. D. Koshanov ◽  
M. Yu. Nemchenko
Keyword(s):  
Dirichlet Problem ◽  
Green Function ◽  
Function Representation ◽  
Polyharmonic Equations

Solvability of one nonlocal Dirichlet problem for the Poisson equation

2019 ◽  
Vol 50 (1) ◽  
pp. 67-88
Author(s):  
Batirkhan Turmetov ◽  
Valery Karachik
Keyword(s):  
Dirichlet Problem ◽  
Poisson Equation ◽  
The Poisson Equation

scholarly journals Note on a sign-dependent regularity for the polyharmonic Dirichlet problem

2021 ◽  
Vol 279 ◽  
pp. 1-9
Author(s):  
Inka Schnieders ◽  
Guido Sweers
Keyword(s):  
Dirichlet Problem

scholarly journals A singular eigenvalue problem for the Dirichlet (p, q)-Laplacian

2021 ◽  
Author(s):  
Yunru Bai ◽  
Nikolaos S. Papageorgiou ◽  
Shengda Zeng
Keyword(s):  
Dirichlet Problem ◽  
Eigenvalue Problem ◽  
Positive Solution ◽  
Positive Solutions ◽  
Singular Term ◽  
Type Theorem ◽  
Solution Map ◽  
Continuity Properties ◽  
Variational Tools ◽  
Superlinear Reaction

AbstractWe consider a parametric nonlinear, nonhomogeneous Dirichlet problem driven by the (p, q)-Laplacian with a reaction involving a singular term plus a superlinear reaction which does not satisfy the Ambrosetti–Rabinowitz condition. The main goal of the paper is to look for positive solutions and our approach is based on the use of variational tools combined with suitable truncations and comparison techniques. We prove a bifurcation-type theorem describing in a precise way the dependence of the set of positive solutions on the parameter $$\lambda $$ λ . Moreover, we produce minimal positive solutions and determine the monotonicity and continuity properties of the minimal positive solution map.

scholarly journals Optimal regularity of stable solutions to nonlinear equations involving the p-Laplacian

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Xavier Cabré ◽  
Pietro Miraglio ◽  
Manel Sanchón
Keyword(s):  
Dirichlet Problem ◽  
Optimal Condition ◽  
Bounded Domain ◽  
Laplace Operator ◽  
Nonlinear Equations ◽  
Open Problem ◽  
Optimal Range ◽  
Optimal Regularity ◽  
Smooth Bounded Domain ◽  
Stable Solutions

AbstractWe consider the equation {-\Delta_{p}u=f(u)} in a smooth bounded domain of {\mathbb{R}^{n}}, where {\Delta_{p}} is the p-Laplace operator. Explicit examples of unbounded stable energy solutions are known if {n\geq p+\frac{4p}{p-1}}. Instead, when {n<p+\frac{4p}{p-1}}, stable solutions have been proved to be bounded only in the radial case or under strong assumptions on f. In this article we solve a long-standing open problem: we prove an interior {C^{\alpha}} bound for stable solutions which holds for every nonnegative {f\in C^{1}} whenever {p\geq 2} and the optimal condition {n<p+\frac{4p}{p-1}} holds. When {p\in(1,2)}, we obtain the same result under the nonsharp assumption {n<5p}. These interior estimates lead to the boundedness of stable and extremal solutions to the associated Dirichlet problem when the domain is strictly convex. Our work extends to the p-Laplacian some of the recent results of Figalli, Ros-Oton, Serra, and the first author for the classical Laplacian, which have established the regularity of stable solutions when {p=2} in the optimal range {n<10}.

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