In this paper, we consider a Dirichlet problem driven by the anisotropic (p, q)-Laplacian and superlinear reaction which need not satisfy Ambrosetti–Robinowitz condition. By using variational tools together with truncation comparison techniques critical groups, show existence of at least five nontrivial smooth solutions, all sign information: two positive, negative nodal (sign-changing).

In this paper we prove existence of nonnegative solutions to parabolic Cauchy–Dirichlet problems with (eventually) singular superlinear gradient terms. The model equation is $$\begin{aligned} u_t - \Delta _pu=g(u)|\nabla u|^q+h(u)f(t,x)\qquad \text {in }(0,T)\times \Omega , \end{aligned}$$ where $$\Omega $$ an open bounded subset $${{\,\mathrm{{\mathbb {R}}}\,}}^N$$ $$N>2$$ $$0<T<+\infty $$1<p<...

We consider a nonlinear Dirichlet parametric problem with discontinuous right hand side in which we have a competing effect of sub and superlinear nonlinearities. A bifurcation type result is studied when the parameter tends to zero.