  We consider the semilinear Dirichlet problem \[ \Delta u+kg(u)=\mu _1 \varphi
_1+\cdots +\mu _n \varphi _n+e(x) \;\; \mbox{for $x \in \Omega$}, \;\; u=0 \;\;
\mbox{on $\partial \Omega$}, \] where $\varphi _k$ is the $k$-th eigenfunction
of the Laplacian on $\Omega$ and $e(x) \perp \varphi _k$, $k=1, \ldots, n$.
Write the solution in the form $u(x)= \Sigma _{i=1}^n \xi _i \varphi _i+U(x)$,
with $ U \perp \varphi _k$, $k=1, \ldots, n$. Starting with $k=0$, when the
problem is linear, we continue the solution in $k$ by keeping $\xi =(\xi _1,
\ldots,\xi _n)$ fixed, but allowing for $\mu =(\mu _1, \ldots,\mu _n)$ to vary.
Studying the map $\xi \rightarrow \mu$ provides us with the existence and
multiplicity results for the above problem. We apply our results to problems at
resonance, at both the principal and higher eigenvalues. Our approach is
suitable for numerical calculations, which we implement, illustrating our
results.

Analysis of PDEs