On a maximum principle for a class of fourth-order semilinear elliptic equations

1977 ◽  
Vol 77 (3-4) ◽  
pp. 319-323 ◽  
Author(s):  
Philip W. Schaefer
Keyword(s):  
Elliptic Equations ◽  
Existence Of Solutions ◽  
Linear Partial Differential Equation ◽  
First Eigenvalue ◽  
Dirichlet Eigenvalue ◽  
The First Eigenvalue ◽  
Semilinear Elliptic ◽  
Maximum Value ◽  
Non Linear ◽  
Dirichlet Eigenvalue Problem

SynopsisIt is shown that Ф = | grad u |2–uΔu, where u is a solution of Δ2u+pf(u) = 0 in D, assumes its maximum value on the boundary of D. This principle leads one to a lower bound on the first eigenvalue in the non-linear Dirichlet eigenvalue problem and to the non-existence of solutions to this non-linear partial differential equation subject to certain zero boundaryconditions.

scholarly journals On a class of semilinear elliptic equations with boundary conditions and potentials which change sign

2005 ◽  
Vol 2005 (2) ◽  
pp. 95-104
Author(s):  
M. Ouanan ◽  
A. Touzani
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Elliptic Equations ◽  
First Eigenvalue ◽  
Nontrivial Solutions ◽  
Bounded Smooth Domain ◽  
The First Eigenvalue ◽  
Semilinear Elliptic ◽  
Superlinear Growth ◽  
Bounded Smooth

We study the existence of nontrivial solutions for the problemΔu=u, in a bounded smooth domainΩ⊂ℝℕ, with a semilinear boundary condition given by∂u/∂ν=λu−W(x)g(u), on the boundary of the domain, whereWis a potential changing sign,ghas a superlinear growth condition, and the parameterλ∈]0,λ1];λ1is the first eigenvalue of the Steklov problem. The proofs are based on the variational and min-max methods.

scholarly journals Eigenvalues for a combination between local and nonlocal p-Laplacians

2019 ◽  
Vol 22 (5) ◽  
pp. 1414-1436 ◽  
Author(s):  
Leandro M. Del Pezzo ◽  
Raúl Ferreira ◽  
Julio D. Rossi
Keyword(s):  
Eigenvalue Problem ◽  
Bounded Domain ◽  
Limit Problem ◽  
First Eigenvalue ◽  
Order Zero ◽  
Dirichlet Eigenvalue ◽  
The First Eigenvalue ◽  
Homogeneous Operator ◽  
Dirichlet Eigenvalue Problem

Abstract In this paper we study the Dirichlet eigenvalue problem $$\begin{array}{} \displaystyle -\Delta_p u-\Delta_{J,p}u =\lambda|u|^{p-2}u \text{ in } \Omega,\quad u=0 \, \text{ in } \, \Omega^c=\mathbb{R}^N\setminus\Omega. \end{array}$$ Here Ω is a bounded domain in ℝN, Δpu is the standard local p-Laplacian and ΔJ,pu is a nonlocal p-homogeneous operator of order zero. We show that the first eigenvalue (that is isolated and simple) satisfies $\begin{array}{} \displaystyle \lim_{p\to\infty} \end{array}$(λ1)1/p = Λ where Λ can be characterized in terms of the geometry of Ω. We also find that the eigenfunctions converge, u∞ = $\begin{array}{} \displaystyle \lim_{p\to\infty} \end{array}$ up, and find the limit problem that is satisfied in the limit.

scholarly journals A comparison theorem for the first Dirichlet eigenvalue of a domain in a Kaehler submanifold

1994 ◽  
Vol 56 (2) ◽  
pp. 267-277
Author(s):  
Francisco J. Carreras ◽  
Fernando Giménez ◽  
Vicente Miquel
Keyword(s):  
Outer Radius ◽  
Holomorphic Sectional Curvature ◽  
First Eigenvalue ◽  
Kaehler Manifold ◽  
Dirichlet Eigenvalue ◽  
Totally Geodesic Submanifold ◽  
The First Eigenvalue ◽  
First Dirichlet Eigenvalue ◽  
Dirichlet Eigenvalue Problem ◽  
Sharp Lower Bound

AbstractWe give a sharp lower bound for the first eigenvalue of the Dirichlet eigenvalue problem on a domain of a complex submanifold of a Kaehler manifold with curvature bounded from above. The bound on the first eigenvalue is given as a function of the extrinsic outer radius and the bounds on the curvature, and it is attained only on geodesic spheres of a space of constant holomorphic sectional curvature embedded in the Kaehler manifold as a totally geodesic submanifold.

scholarly journals Super and subsolutions for elliptic equations on all ofℝn

2002 ◽  
Vol 32 (1) ◽  
pp. 41-46
Author(s):  
G. A. Afrouzi ◽  
H. Ghasemzadeh
Keyword(s):  
Population Genetics ◽  
Elliptic Equation ◽  
Elliptic Equations ◽  
Existence Of Solutions ◽  
Asymptotic Properties ◽  
Semilinear Elliptic Equation ◽  
Semilinear Elliptic

By construction sub and supersolutions for the following semilinear elliptic equation−△u(x)=λg(x)f(u(x)),x∈ℝnwhich arises in population genetics, we derive some results about the theory of existence of solutions as well as asymptotic properties of the solutions for everynand for the functiong:ℝn→ℝsuch thatgis smooth and is negative at infinity.

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