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Half eigenvalues and the Fucik spectrum of multi-point, boundary value problems
We consider the nonlinear boundary value problem consisting of the equation
\tag{1} -u" = f(u) + h, \quad \text{a.e. on ,} where ,
together with the multi-point, Dirichlet-type boundary conditions \tag{2} u(\pm
1) = \sum^{m^\pm}_{i=1}\alpha^\pm_i u(\eta^\pm_i) where are
integers, ,
, and we suppose that We also suppose that is continuous, and We allow --- such a
nonlinearity is {\em jumping}.
Related to (1) is the equation \tag{3} -u" = \lambda(a u^+ - b u^-), \quad
\text{on ,} where , and for . The problem (2)-(3) is `positively-homogeneous'
and jumping. Regarding as fixed, values of for
which (2)-(3) has a non-trivial solution will be called {\em
half-eigenvalues}, while the corresponding solutions will be called {\em
half-eigenfunctions}.
We show that a sequence of half-eigenvalues exists, the corresponding
half-eigenfunctions having specified nodal properties, and we obtain certain
spectral and degree theoretic properties of the set of half-eigenvalues. These
properties lead to solvability and non-solvability results for the problem
(1)-(2). The set of half-eigenvalues is closely related to the `Fucik spectrum'
of the problem, which we briefly describe. Equivalent solvability and
non-solvability results for (1)-(2) are obtained from either the
half-eigenvalue or the Fucik spectrum approach
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
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