Abstract We study the quasilinear elliptic problem which is resonant at zero. By using Morse theory, we obtain five nontrivial solutions for equation with coercive nonlinearities.

This talk is a survey on some recent results concerning local regularity for weak solutions to some quasilinear degenerate elliptic equations. This topic is a very classical one in the theory of PDE and we start by recalling contributions given by many authors. Early results go back to the outstanding papers by De Giorgi [12], Stampacchia [29], Ladyzhenskaia and Uraltzeva [18] between the end o...

This paper deals with the quasilinear attraction-repulsion chemotaxis system \begin{align*} \begin{cases} u_t=\nabla\cdot \big((u+1)^{m-1}\nabla u -\chi u(u+1)^{p-2}\nabla v +\xi u(u+1)^{q-2}\nabla w\big) +f(u), \\[1.05mm] 0=\Delta v+\alpha u-\beta v, w+\gamma u-\delta w \end{cases} \end{align*} in a bounded domain $\Omega \subset \mathbb{R}^n$ ($n \in \mathbb{N}$) smooth boundary $\partial\Ome...

In this paper, we establish weak continuity results for quasilinear elliptic and subelliptic operators of divergence form, acting on corresponding classes of subharmonic functions. These results are analogous to our earlier results for fully nonlinear k-Hessian operators. From the weak continuity, we derive various potential theoretic results including capacity estimates, potential estimates an...

We study the problem of existence of positive, spherically symmetric strong solutions of quasilinear elliptic equations involving p-Laplacian in the ball. We allow simultaneous strong dependence of the right-hand side on both the unknown function and its gradient. The elliptic problem is studied by relating it to the corresponding singular ordinary integro-differential equation. Solvability ran...

When an unbounded domain is inside a slab, existence of a positive solution is proved for the Dirichlet problem of a class of semilinear elliptic equations that are similar either to the singular Emden-Fowler equation or a sublinear elliptic equation. The result obtained can be applied to equations with coefficients of the nonlinear term growing exponentially. The proof is based on the super an...