A Regularity Result for the p-Laplacian Near Uniform Ellipticity

We consider weak solutions to a class of Dirichlet boundary value problems involving the $p$-Laplace operator, and prove that the second weak derivatives have summability as high as it is desirable, provided p is sufficiently close to 2. And as a consequence the Holder exponent of the gradients approaches 1. We show that this phenomenon is driven by the classical Calderon-Zygmund constant. We believe that this result is particularly interesting in higher dimensions, and it is related to the optimal regularity of $p$-harmonic mappings. which is still an open question

On Coron's problem for the p-Laplacian

We prove that the critical problem for the $p$-Laplacian operator admits a nontrivial solution in annular shaped domains with sufficiently small inner hole. This extends Coron's problem to a class of quasilinear problems.Comment: 6 page

New multiplicity results for critical p-Laplacian problems

We prove new multiplicity results for the Brézis-Nirenberg problem for the p-Laplacian. Our proofs are based on a new abstract critical point theorem involving the Z2-cohomological index that requires less compactness than the (PS) condition

On a class of nonlinear Schrödinger–Poisson systems involving a nonradial charge density

In the spirit of the classical work of P. H. Rabinowitz on nonlinear Schrödinger equations, we prove existence of mountain-pass solutions and least energy solutions to a class of nonlinear Schrödinger-Poisson systems under different assumptions on a weight function at infinity. Our results cover a range of exponents, p,where the lack of compactness phenomena may be due to the combined effect of the invariance by translations of a `limiting problem' at infinity and of the possible unboundedness of the Palais-Smale sequences. Moreover, we find necessary conditions for concentration at points to occur for solutions to a singular perturbation of the same problem in various functional settings which are suitable for both variational and perturbation methods

New multiplicity results for critical pp-Laplacian problems

We prove new multiplicity results for the Brezis-Nirenberg problem for the $p$-Laplacian. Our proofs are based on a new abstract critical point theorem involving the ${\mathbb Z}_2$-cohomological index that requires less compactness than the (PS) condition.Comment: arXiv admin note: text overlap with arXiv:2102.0913

Groundstate asymptotics for a class of singularly perturbed p-Laplacian problems in RN{{\mathbb {R}}}^N

We study the asymptotic behavior of positive groundstate solutions to the quasilinear elliptic equation in the whole space; we give a characterisation of asymptotic regimes as a function of the parameters and show that the behavior of the groundstates is sensitive to the relation of the growth on the nonlinearities and the critical Sobolev exponent