scholarly journals Positive Solutions of a Kind of Equations Related to the Laplacian andp-Laplacian

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Fangfang Zhang ◽  
Zhanping Liang
Keyword(s):  
Variational Method ◽  
Bounded Domain ◽  
Positive Solutions ◽  
Eigenvalue Problems ◽  
Sufficient Condition ◽  
Nonlinear Eigenvalue Problems ◽  
Existence Of Positive Solutions ◽  
Nonlinear Eigenvalue

Positive solutions of a kind of equations related to the Laplacian andp-Laplacian on a bounded domain inRNwithN⩾1are studied by using variational method. A sufficient condition of the existence of positive solutions is characterized by the eigenvalues of linear and another nonlinear eigenvalue problems.

MULTIPLE POSITIVE SOLUTIONS OF NONLINEAR EIGENVALUE PROBLEMS ASSOCIATED TO A CLASS OF p-LAPLACIAN LIKE OPERATORS

2003 ◽  
Vol 05 (05) ◽  
pp. 737-759 ◽  
Author(s):  
NOBUYOSHI FUKAGAI ◽  
KIMIAKI NARUKAWA
Keyword(s):  
Positive Solutions ◽  
Elliptic Problem ◽  
Eigenvalue Problems ◽  
Multiple Positive Solutions ◽  
Combined Effects ◽  
Nonlinear Eigenvalue Problems ◽  
Quasilinear Elliptic Problem ◽  
Quasilinear Elliptic ◽  
Nonlinear Eigenvalue

This paper deals with positive solutions of a class of nonlinear eigenvalue problems. For a quasilinear elliptic problem (#) - div (ϕ(|∇u|)∇u) = λf(x,u) in Ω, u = 0 on ∂Ω, we assume asymptotic conditions on ϕ and f such as ϕ(t) ~ tp0-2, f(x,t) ~ tq0-1as t → +0 and ϕ(t) ~ tp1-2, f(x,t) ~ tq1-1as t → ∞. The combined effects of sub-nonlinearity (p0> q0) and super-nonlinearity (p1< q1) with the subcritical term f(x,u) imply the existence of at least two positive solutions of (#) for 0 < λ < Λ.

scholarly journals Positive Solution for the Elliptic Problems with Sublinear and Superlinear Nonlinearities

2010 ◽  
Vol 2010 ◽  
pp. 1-10
Author(s):  
Chunmei Yuan ◽  
Shujuan Guo ◽  
Kaiyu Tong
Keyword(s):  
Variational Method ◽  
Positive Solution ◽  
Bounded Domain ◽  
Positive Solutions ◽  
Elliptic Problems ◽  
Real Parameter ◽  
Existence Of Positive Solutions

This paper deals with the existence of positive solutions for the elliptic problems with sublinear and superlinear nonlinearities-Δu=λa(x)up+b(x)uqinΩ,u>0inΩ,u=0on∂Ω, whereλ>0is a real parameter,0<p<1<q.Ωis a bounded domain inRN  (N≥3), anda(x)andb(x)are some given functions. By means of variational method and super-subsolution method, we obtain some results about existence of positive solutions.

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