Heriot-Watt Mathematics Report Series
HWM98-25, May 8 1998
The existence of positive solutions for a class of indefinite weight semilinear elliptic boundary value problems
B Ko and K J Brown
Abstract
We discuss the existence of positive classical solutions of the boundary value problem:
\begin{center}
$ -\Delta u=\lambda g(x)f(u) \mbox{ in }\Omega; \hspace{.2in}
\displaystyle (1-\alpha)\frac{\partial u}{\partial n}+\alpha
u=0 \mbox{ on } \partial\Omega $
\end{center}
where $\lambda$ and $\alpha < 1$ are real parameters,$\Omega$ is an open bounded region of $ R}^N, \, N \geq 2$ with smooth boundary and $g:\overline \Omega \to \bf R$ is a smooth function which changes sign on $\Omega$ in
the cases where $f(u) = u(1 - |u|^p)$ and $f(u) = u(1 + |u|^p)$.
\par
A constrained variational principle is used to prove the existence of
positive solutions over a certain range of $\lambda$ which may contain
both positive and negative values of $\lambda$; this variational principle
depends on the spectral properties of the corresponding
linearized problem.
\par
The solutions obtained in the case $f(u) = u(1 - |u|^p)$ do not lie on
a branch of solutions bifurcating from the trivial branch of zero
solutions at a principal eigenvalue; such solutions do not exist in
the case where $g$ does not change sign and so the results show the
crucial role played by the presence of the indefinite weight function
in the existence of positive solutions.
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Full text: http://www.ma.hw.ac.uk/~ken/psfiles/kb_nla_98.ps.Z
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