DIFFERENtIAl & DIFFERENCE EqUAtIONS ANd APPlICAtIONS

MULTIPLE POSITIVE SOLUTIONS OF SUPERLINEAR
ELLIPTIC PROBLEMS WITH SIGN-CHANGING WEIGHT
DENIS BONHEURE, JOSÉ MARIA GOMES, AND PATRICK HABETS
We prove the existence of multibump solutions to a superlinear elliptic problem where
a sign-changing weight is affected by a large parameter μ. Our method relies in variational
arguments. A special care is paid to the localization of the deformation along lines
of steepest descent of an energy functional constrained to a C 1,1 -manifold in the space
H 1 0(Ω).
Copyright © 2006 Denis Bonheure et al. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is properly cited.
1. Introduction
We consider positive solutions of the boundary value problem
Δu + ( a + (x) − μa − (x) ) |u| γ u = 0, x ∈ Ω,
u(x) = 0,
x ∈ ∂Ω,
(1.1)
where Ω ⊂ R N is a bounded domain of class 1 , a + and a − are continuous functions
which are positive on nonoverlapping domains, and μ is a large parameter. Positive solutions
u aredefinedtobesuchthatu(x) > 0 for almost every x ∈ Ω.
For the ODE equivalent of (1.1) and for large values of μ, complete results were worked
out in [2, 3] concerning, respectively, the cases of the weight a + (t) being positive in two or
three nonoverlapping intervals. In the present note, we summarize the results obtained in
[1]. By using a variational approach, we extend the results obtained in [2, 3] to the PDE
problem (1.1).
Note first that finding positive solutions of problem (1.1)isequivalenttofindingnontrivial
solutions of
Δu + ( a + (x) − μa − (x) ) u γ+1
+ = 0, x ∈ Ω,
u(x) = 0, x ∈ ∂Ω,
(1.2)
Hindawi Publishing Corporation
Proceedings of the Conference on Differential & Difference Equations and Applications, pp. 221–229