We consider a semilinear Dirichlet problem driven by the Laplacian and with a reaction which is of the form \(\hat{\lambda }_1x+f(z,x)\) with \(\hat{\lambda }_1>0\) being the principal eigenvalue of ( \(-\Delta , H_0^1(\Omega )\) ) and f(z, x) is a Caratheodory function which exhibits distinct asymptotic behavior as \(x\rightarrow \pm \infty \) . It is strictly sublinear as \(x\rightarrow -\infty \) and linear as \(x\rightarrow +\infty \) (resonant linear problem with jumping reaction). Using variational tools from the critical point theory, together with truncation and comparison techniques, critical groups and flow invariance arguments, we prove two multiplicity theorems producing seven and eight nontrivial smooth solutions all with sign information. It appears that these are the first such multiplicity results for asymmetric resonant problems.
