In this paper, the existence of smooth positive solutions to a Robin boundary-value problem with non-homogeneous differential operator and reaction given by a nonlinear convection term plus a singular one is established. Proofs chiefly exploit sub-super-solution and truncation techniques, set-valued analysis, recursive methods, nonlinear regularity theory, as well as fixed point arguments. A uniqueness result is also presented.

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The existence of a positive entire weak solution to a singular quasi-linear elliptic system with convection terms is established, chiefly through perturbation techniques, fixed point arguments, and a priori estimates. Some regularity results are then employed to show that the obtained solution is actually strong.

A Robin boundary-value problem with non-homogeneous differential operator, indefinite potential, and reaction defined only near zero is investigated. The existence of one or more nodal solutions is achieved by using truncation, perturbation, and comparison techniques, results from Morse theory, besides variational methods.Let (X, · ) be a real Banach space. Given a set V ⊆ X, write V for the closure of V , ∂V for the boundary of V , and int X (V ) or simply int(V ), when no confusion can arise, for the interior of V . If x ∈ X and δ > 0 then B δ (x) := {z ∈ X : z − x < δ} , B δ := B δ (0) .

<p style='text-indent:20px;'>Existence of a generalized solution to a strongly singular convective elliptic equation in the whole space is established. The differential operator, patterned after the <inline-formula><tex-math id="M2">\begin{document}$ (p,q) $\end{document}</tex-math></inline-formula>-Laplacian, can be non-homogeneous. The result is obtained by solving some regularized problems through fixed point theory, variational methods and compactness results, besides exploiting nonlinear regularity theory and comparison principles.</p>