We combine the sub-and supersolution method and perturbation arguments to obtain positive solutions of singular quasi-monotone (p, q)-Laplacian systems.

In this work, generalized Sobolev spaces and Sobolev spaces of infinite order are considered. Existence of solutions for strongly nonlinear equation of infinite order of the form Au + g(x, u) = f is established. Here A is an elliptic operator from a functional space of Sobolev type to its dual and g(x, s) is a lower order term satisfying a sign condition on s.

We prove some properties of the first eigenvalue for the elliptic system −Δpu=λ|u|α|v|βv in Ω, −Δqv=λ|u|α|v|βu in Ω, (u,v)∈W01,p(Ω)×W01,q(Ω). In particular, the first eigenvalue is shown to be simple. Moreover, the stability with respect to (p,q) is established

In this paper we study double phase problems with nonlinear boundary condition and gradient dependence. Under quite general assumptions we prove existence results for such problems where the perturbations satisfy a suitable behavior in the origin and at infinity. Our proofs make use of variational tools, truncation techniques and comparison methods. The obtained solutions depend on the first eigenvalues of the Robin and Steklov eigenvalue problems for the p-Laplacian.

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