Pablo L. De Nápoli

Mountain pass solutions to equations of p-Laplacian type

Abstract This work is devoted to study the existence of solutions to equations of p-Laplacian type. We prove the existence of at least one solution, and under further assumptions, the existence of infinitely many solutions. In order to apply mountain pass results, we introduce a notion of uniformly convex functional that generalizes the notion of uniformly convex norm.

Quasilinear elliptic systems of resonant type and nonlinear eigenvalue problems

This work is devoted to the study of a quasilinear elliptic system of resonant type. We prove the existence of infinitely many solutions of a related nonlinear eigenvalue problem. Applying an abstract minimax theorem, we obtain a solution of the quasilinear system −Δpu=Fu(x, u, v), −Δqv=F v(x, u, v), under conditions involving the first and the second eigenvalues.

Multiple solutions for the p-Laplace operator with critical growth

Abstract In this paper we show the existence of at least three nontrivial solutions to the following quasilinear elliptic equation − Δ p u = | u | p ∗ − 2 u + λ f ( x , u ) in a smooth bounded domain Ω of R N with homogeneous Dirichlet boundary conditions on ∂ Ω , where p ∗ = N p / ( N − p ) is the critical Sobolev exponent and Δ p u = div ( | ∇ u | p − 2 ∇ u ) is the p -Laplacian. The proof is based on variational arguments and the classical concentration compactness method.

Elementary Proofs of Embedding Theorems for Potential Spaces of Radial Functions

We present elementary proofs of weighted embedding theorems for radial potential spaces and some generalizations of Ni’s and Strauss’ inequalities in this setting.

Detailed asymptotic of eigenvalues on time scales

Let 𝕋 = {a n } n ∪{0} be a time scale with zero Minkowski (or box) dimension, where {a n } n is a monotonically decreasing sequence converging to zero, and a 1 = 1. In this paper, we find an upper bound for the eigenvalue counting function of the linear problem − u ΔΔ = λu σ, with Dirichlet boundary conditions. We obtain that the nth-eigenvalue is bounded below by . We show that the bound is optimal for the q-difference equations arising in quantum calculus.

Radial Solutions for Hamiltonian Elliptic Systems With Weights

Abstract We prove the existence of infinitely many radial solutions for elliptic systems in ℝn with power weights. A key tool for the proof will be a weighted imbedding theorem for fractional-order Sobolev spaces, that could be of independent interest.

Eigenvalues of the -Laplacian and disconjugacy criteria

We derive oscillation and nonoscillation criteria for the one-dimensional-Laplacian in terms of an eigenvalue inequality for a mixed problem. We generalize the results obtained in the linear case by Nehari and Willett, and the proof is based on a Picone-type identity.

On a generalization of Lazer-Leach conditions for a system of second order ODE's

We study the existence of periodic solutions for a nonlinear second order system of ordinary differential equations. Assuming suitable Lazer-Leach type conditions, we prove the existence of at least one solution applying topological degree methods.

Equations of p-Laplacian Type in Unbounded Domains

Abstract This work is devoted to study the existence of solutions to equations of the p Laplacian type in unbounded domains. We prove the existence of at least one solution, and under further assumptions, the existence of infinitely many solutions. We apply the mountain pass theorem in weighted Sobolev spaces.

Weighted inequalities for the fractional Laplacian and the existence of extremals

In this article we obtain improved versions of Stein-Weiss and Caffarelli-Kohn-Nirenberg inequalities, involving Besov norms of negative smoothness. As an application of the former, we derive the existence of extremals of the Stein-Weiss inequality in certain cases, some of which are not contained in the celebrated theorem of E. Lieb.