Constant sign solutions for double phase problems with superlinear nonlinearity

We study parametric double phase problems involving superlinear nonlinearities with a growth that need not necessarily be polynomial. Based on truncation and comparison methods the existence of two constant sign solutions is shown provided the parameter is larger than the first eigenvalue of the $p$-Laplacian. As a result of independent interest we prove a priori estimates for solutions for a general class of double phase problems with convection term.