868 research outputs found
Multiple positive solutions for a superlinear problem: a topological approach
We study the multiplicity of positive solutions for a two-point boundary
value problem associated to the nonlinear second order equation .
We allow to change its sign in order to cover the case of
scalar equations with indefinite weight. Roughly speaking, our main assumptions
require that is below as and above
as . In particular, we can deal with the situation
in which has a superlinear growth at zero and at infinity. We propose
a new approach based on the topological degree which provides the multiplicity
of solutions. Applications are given for , where we prove
the existence of positive solutions when has positive
humps and is sufficiently large.Comment: 36 pages, 3 PNG figure
Multiple positive solutions of a Sturm-Liouville boundary value problem with conflicting nonlinearities
We study the second order nonlinear differential equation \begin{equation*}
u"+ \sum_{i=1}^{m} \alpha_{i} a_{i}(x)g_{i}(u) - \sum_{j=0}^{m+1} \beta_{j}
b_{j}(x)k_{j}(u) = 0, \end{equation*} where ,
are non-negative Lebesgue integrable functions defined in
, and the nonlinearities are
continuous, positive and satisfy suitable growth conditions, as to cover the
classical superlinear equation , with . When the positive
parameters are sufficiently large, we prove the existence of at
least positive solutions for the Sturm-Liouville boundary value
problems associated with the equation. The proof is based on the Leray-Schauder
topological degree for locally compact operators on open and possibly unbounded
sets. Finally, we deal with radially symmetric positive solutions for the
Dirichlet problems associated with elliptic PDEs.Comment: 23 pages, 6 PNG figure
Unilateral global bifurcation and nodal solutions for the -Laplacian with sign-changing weight
In this paper, we shall establish a Dancer-type unilateral global bifurcation
result for a class of quasilinear elliptic problems with sign-changing weight.
Under some natural hypotheses on perturbation function, we show that
is a bifurcation point of the above problems and there are
two distinct unbounded continua, and
, consisting of the bifurcation branch
from , where is the
-th positive or negative eigenvalue of the linear problem corresponding to
the above problems, . As the applications of the above
unilateral global bifurcation result, we study the existence of nodal solutions
for a class of quasilinear elliptic problems with sign-changing weight.
Moreover, based on the bifurcation result of Dr\'{a}bek and Huang (1997)
[\ref{DH}], we study the existence of one-sign solutions for a class of high
dimensional quasilinear elliptic problems with sign-changing weight
Existence of positive solutions in the superlinear case via coincidence degree: the Neumann and the periodic boundary value problems
We prove the existence of positive periodic solutions for the second order
nonlinear equation , where has superlinear growth at
zero and at infinity. The weight function is allowed to change its sign.
Necessary and sufficient conditions for the existence of nontrivial solutions
are obtained. The proof is based on Mawhin's coincidence degree and applies
also to Neumann boundary conditions. Applications are given to the search of
positive solutions for a nonlinear PDE in annular domains and for a periodic
problem associated to a non-Hamiltonian equation.Comment: 41 page
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