A MULTIPLICITY RESULT FOR ELLIPTIC EQUATIONS AT CRITICAL GROWTH IN LOW DIMENSION

2003 ◽  
Vol 05 (02) ◽  
pp. 171-177 ◽  
Author(s):  
GIUSEPPE DEVILLANOVA ◽  
SERGIO SOLIMINI
Keyword(s):  
Elliptic Equations ◽  
Critical Sobolev Exponent ◽  
Critical Growth ◽  
Multiplicity Result ◽  
First Eigenvalue ◽  
The First Eigenvalue ◽  
Low Dimension ◽  
Sobolev Exponent ◽  
Regular Subset

We consider the problem -Δu = |u|2*-2u + λu in Ω, u = 0 on ∂Ω, where Ω is an open regular subset of ℝN (N ≥ 3), [Formula: see text] is the critical Sobolev exponent and λ is a constant in ]0, λ1[ where λ1 is the first eigenvalue of -Δ. In this paper we show that, when N ≥ 4, the problem has at least [Formula: see text] (pairs of) solutions, improving a result obtained in [4] for N ≥ 6.

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 On the location of the peaks of least-energy solutions to semilinear Dirichlet problems with critical growth

2002 ◽  
Vol 7 (10) ◽  
pp. 547-561 ◽  
Author(s):  
Marco A. S. Souto
Keyword(s):  
Bounded Domain ◽  
Critical Sobolev Exponent ◽  
Critical Growth ◽  
Dirichlet Problems ◽  
Growth Problem ◽  
Sobolev Exponent ◽  
Least Energy Solutions ◽  
Least Energy

We study the location of the peaks of solution for the critical growth problem−ε 2Δu+u=f(u)+u 2*−1,u>0inΩ,u=0on∂Ω, whereΩis a bounded domain;2*=2N/(N−2),N≥3, is the critical Sobolev exponent andfhas a behavior likeup,1<p<2*−1.

POSITIVE SOLUTIONS FOR A FOURTH-ORDER QUASILINEAR EQUATION WITH CRITICAL SOBOLEV EXPONENT

2010 ◽  
Vol 12 (01) ◽  
pp. 1-33 ◽  
Author(s):  
EDERSON MOREIRA DOS SANTOS
Keyword(s):  
Hamiltonian System ◽  
Positive Solutions ◽  
Fourth Order ◽  
Quasilinear Equation ◽  
Critical Sobolev Exponent ◽  
Critical Growth ◽  
Positive Parameter ◽  
Sobolev Exponent

We consider a fourth-order quasilinear equation depending on a positive parameter ∊ and with critical growth. Such equation is equivalent to a critical Hamiltonian system and the main goal of this work is to prove the existence of at least two positive solutions when the parameter ∊ is sufficiently small.

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