Hardy potential versus lower order terms in Dirichlet problems: regularizing effects

The first author is supported by FEDER-MINECO (Spain) grant PGC2018-096422-B-I00 and Junta de Andaluc ' ia (Spain) grant FQM-116.In this paper, dedicated to Ireneo Peral, we study the regularizing e ect of some lower order terms in Dirichlet problems despite the presence of Hardy potentials in the right hand side.FEDER-MINECO (Spain) PGC2018-096422-B-I00Junta de Andalucia European Commission FQM-11

Continuum of solutions for an elliptic problem with critical growth in the gradient

We consider the boundary value problem \begin{equation*} - \Delta u = \lambda c(x)u+ \mu(x) |\nabla u|^2 + h(x), \quad u \in H^1_0(\Omega) \cap L^{\infty}(\Omega) \eqno{(P_{\lambda})} \end{equation*} where $\Omega \subset \R^N, N \geq 3$ is a bounded domain with smooth boundary. It is assumed that $c\gneqq 0$, $c,h$ belong to $L^p(\Omega)$ for some $p > N/2$ and that $\mu \in L^{\infty}(\Omega).$ We explicit a condition which guarantees the existence of a unique solution of $(P_{\lambda})$ when $\lambda <0$ and we show that these solutions belong to a continuum. The behaviour of the continuum depends in an essential way on the existence of a solution of $(P_0)$. It crosses the axis $\lambda =0$ if $(P_0)$ has a solution, otherwise if bifurcates from infinity at the left of the axis $\lambda =0$. Assuming that $(P_0)$ has a solution and strenghtening our assumptions to $\mu(x)\geq \mu_1>0$ and $h\gneqq 0$, we show that the continuum bifurcates from infinity on the right of the axis $\lambda =0$ and this implies, in particular, the existence of two solutions for any $\lambda >0$ sufficiently small.Comment: This second version include added Reference

Diffeomorphism-invariant properties for quasi-linear elliptic operators

For quasi-linear elliptic equations we detect relevant properties which remain invariant under the action of a suitable class of diffeomorphisms. This yields a connection between existence theories for equations with degenerate and non-degenerate coerciveness.Comment: 16 page

A nondifferentiable extension of a theorem of Pucci and Serrin and applications

We study the multiplicity of critical points for functionals which are only differentiable along some directions. We extend to this class of functionals the three critical point theorem of Pucci and Serrin and we apply it to a one-parameter family of functionals $J_\lambda$, $\lambda \in I\subset \mathbb R$. Under suitable assumptions, we locate an open subinterval of values $\lambda$ in $I$ for which $J_\lambda$ possesses at least three critical points. Applications to quasilinear boundary value problems are also given

On two problems studied by A. Ambrosetti

We study the Ambrosetti-Prodi and Ambrosetti-Rabinowitz problems. We prove for the first one the existence of a continuum of solutions with shape of a reflected C (⊃-shape). Next, we show that there is a relationship between these two problems

Relativistic equations with singular potentials

Funding for open access publishing: Universidad de Granada/CBUAThe first part of this paper concern with the study of the Lorentz force equation [GRAPHICS] in the relevant physical configuration where the electric field (E) over right arrow has a singularity in zero. By using Szulkin's critical point theory, we prove the existence of T-periodic solutions provided that T and the electric and magnetic fields interact properly. In the last part, we employ both a variational and a topological argument to prove that the scalar relativistic pendulum-type equation [GRAPHICS] admits at least a periodic solution when h is an element of L-1(0, T) and G is singular at zero.Universidad de Granada/CBUASpanish Government PGC2018-096422-B-I00, PID2021-122122NB-I00Junta de Andalucia FQM-116Ministry of Education, Universities and Research (MIUR) 2017JPCAPN_00

Bifurcation for some quasilinear operators

This paper deals with existence, uniqueness and multiplicity results of positive solutions for the quasilinear elliptic boundary-value problem \begin{array}{c} -\mbox{div}\, (A(x,u)\nabla u) = f(\lambda,x, u), \quad \mbox{ in } \Omega , \\u = 0, \quad \mbox{ on } \partial \Omega , \end{array} where Ω is a bounded open domain in RN with smooth boundary. Under suitable assumptions on the matrix A(x, s), and depending on the behaviour of the function f near u = 0 and near u = +∞, we can use bifurcation theory in order to give a quite complete analysis on the set of positive solutions. We will generalize in different directions some of the results in the papers by Ambrosetti et al., Ambrosetti and Hess, and Artola and Boccardo

Bifurcation for quasilinear elliptic singular BVP

For a continuous function $g\geq 0$ on $(0,+\infty)$ (which may be singular at zero), we confront a quasilinear elliptic differential operator with natural growth in $\nabla u$, $-\Delta u +g(u)|\nabla u|^{2}$, with a power type nonlinearity, $\lambda u^{p}+ f_{0}(x)$. The range of values of the parameter $\lambda$ for which the associated homogeneous Dirichlet boundary value problem admits positive solutions depends on the behavior of $g$ and on the exponent $p$. Using bifurcations techniques we deduce sufficient conditions for the boundedness or unboundedness of the cited range