Solutions with Sign Information for Noncoercive Double Phase Equations

We consider a nonautonomous ( p , q )-equation with unbalanced growth and a reaction which exhibits the combined effects of a parametric “concave" ( ( p - 1 ) -sublinear) term and of a ( p - 1 ) -linear perturbation, which asymptotically stays above the principal eigenvalue λ ^ 1 a > 0 of the Dirichlet - Δ p a operator. Using variational tools, truncation and comparison techniques and critical groups, we show that for all small values of the parameter the problem has at least three nontrivial bounded solutions with sign information (positive, negative, nodal), which are ordered.