scholarly journals Sign changing solutions and multiple solutions of an elliptic eigenvalue problem with constraint in H1(RN)

2010 ◽  
Vol 59 (8) ◽  
pp. 3005-3013 ◽  
Author(s):  
Jingkun Liu ◽  
Jianqing Chen
Keyword(s):  
Eigenvalue Problem ◽  
Multiple Solutions ◽  
Elliptic Eigenvalue Problem ◽  
Sign Changing Solutions

MULTIPLE SIGN CHANGING SOLUTIONS OF A QUASILINEAR ELLIPTIC EIGENVALUE PROBLEM INVOLVING THE p-LAPLACIAN

2004 ◽  
Vol 06 (02) ◽  
pp. 245-258 ◽  
Author(s):  
THOMAS BARTSCH ◽  
ZHAOLI LIU
Keyword(s):  
Eigenvalue Problem ◽  
Smooth Domain ◽  
Oscillatory Behaviour ◽  
Bounded Smooth Domain ◽  
Elliptic Eigenvalue Problem ◽  
Positive Multiple ◽  
Bounded Smooth ◽  
Sign Changing Solutions ◽  
Quasilinear Elliptic

We consider the eigenvalue problem [Formula: see text] where Ω⊂ℝN is a bounded smooth domain and Δpu denotes the p-Laplacian, 1<p<+∞; λ>0 is a parameter. The nonlinearity f is required to have an oscillatory behaviour. We prove the existence of multiple positive, multiple negative, and in particular, of multiple sign changing solutions depending on λ.

Multiple solutions for a non-homogeneous elliptic equation at the critical exponent

2004 ◽  
Vol 134 (1) ◽  
pp. 69-87 ◽  
Author(s):  
Mónica Clapp ◽  
Manuel Del Pino ◽  
Monica Musso
Keyword(s):  
Elliptic Equation ◽  
Boundary Conditions ◽  
Critical Exponent ◽  
Bounded Domain ◽  
Multiple Solutions ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Sign Changing Solutions ◽  
Homogeneous Elliptic Equation

We consider the equation−Δu = |u|4/(N−2)u + εf(x) under zero Dirichlet boundary conditions in a bounded domain Ω in RN exhibiting certain symmetries, with f ≥ 0, f ≠ 0. In particular, we find that the number of sign-changing solutions goes to infinity for radially symmetric f, as ε → 0 if Ω is a ball. The same is true for the number of negative solutions if Ω is an annulus and the support of f is compact in Ω.

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