Leonelo Iturriaga

Existence and Multiplicity Results for Degenerate Elliptic Equations with Dependence on the Gradient

We study the existence of positive solutions for a class of degenerate nonlinear elliptic equations with gradient dependence. For this purpose, we combine a blowup argument, the strong maximum principle, and Liouville-type theorems to obtain a priori estimates.

Quasilinear Problems Involving Changing-Sign Nonlinearities without an Ambrosetti–Rabinowitz-Type Condition

In this paper quasilinear elliptic boundary value equations without Ambrosetti and Rabinowitz growth condition are considered. Existence of a nontrivial solution result is established. For this, we show the existence of a Cerami’s sequence by using a variant of the Mountain Pass Theorem due to Schechter. The novelty here is that we may consider nonlinearities which satisfy a local p−superlinear condition and may change sign as well.

Local minimizers in spaces of symmetric functions and applications

Abstract We study H 1 versus C 1 local minimizers for functionals defined on spaces of symmetric functions, namely functions that are invariant by the action of some subgroups of O ( N ) . These functionals, in many cases, are associated with some elliptic partial differential equations that may have supercritical growth. So we also prove some results on classical regularity for symmetric weak solutions for a general class of semilinear elliptic equations with possibly supercritical growth. We then apply these results to prove the existence of a large number of classical positive symmetric solutions to some concave-convex elliptic equations of Henon type.

Existence of Solutions to Quasilinear Elliptic Equations With Singular Weights

Abstract In this paper we show the existence of multiple solutions to a class of quasilinear elliptic equations with singular weights when the continuous nonlinearity satisfies a superlinear condition only at zero. In particular, our approach allows us to consider superlinear, critical and supercritical nonlinearities.

On necessary conditions for the comparison principle and the sub- and supersolution method for the stationary Kirchhoff equation

In this paper, we propose a counterexample to the validity of the comparison principle and of the sub- and supersolution method for nonlocal problems like the stationary Kirchhoff equation. This counterexample shows that in general smooth bounded domains in any dimension, these properties cannot hold true if the nonlinear nonlocal term M(u2) is somewhere increasing with respect to the H01-norm of the solution. Comparing with the existing results, this fills a gap between known conditions on M that guarantee or prevent these properties and leads to a condition that is necessary and sufficient for the validity of the comparison principle. It is worth noting that equations similar to the one considered here have gained interest recently for appearing in models of thermo-convective flows of non-Newtonian fluids or of electrorheological fluids, among others.

Liouville theorems for radial solutions of semilinear elliptic equations

ABSTRACT In this work, we obtain some new Liouvilles theorems for positive, radially symmetric solutions of the equation where f is a continuous function in which is positive in . Our methods adapt to cover more general problems, where the nonlinearity is multiplied by some radially symmetric weights and/or the Laplacian is replaced by the p-Laplacian, 1<p<N. Some results for related elliptic systems are also obtained.