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Curves of equiharmonic solutions, and problems at resonance

We consider the semilinear Dirichlet problem \[ Δu+kg(u)=μ_1 φ_1+\cdots +μ_n φ_n+e(x) \;\; \mbox{for $x \in Ω$}, \;\; u=0 \;\; \mbox{on $\partial Ω$}, \] where $φ_k$ is the $k$-th eigenfunction of the Laplacian on $Ω$ and $e(x) \perp φ_k$, $k=1, \ldots, n$. Write the solution in the form $u(x)= Σ_{i=1}^n ξ_i φ_i+U(x)$, with $ U \perp φ_k$, $k=1, \ldots, n$. Starting with $k=0$, when the problem is linear, we continue the solution in $k$ by keeping $ξ=(ξ_1, \ldots,ξ_n)$ fixed, but allowing for $μ=(μ_1, \ldots,μ_n)$ to vary. Studying the map $ξ\rightarrow μ$ 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.