Related papers: On homoclinic solutions for a second order differe…
We are concerned with the existence of solution of the problem $ -\Delta ^H_pu+|u|^{p-2}u=\lambda|u|^{q-2}u+ |u|^{p^*-2}u\quad \mbox{in}\quad\Omega,$ $u>0\quad \mbox{in}\quad\Omega,$ $a(\nabla u)\cdot \nu =0\quad \mbox{on}\quad\partial…
Let $\Delta_k$ be the Dunkl Laplacian on $\mathbb R^d$ associated with a reflection group $W$ and a multiplicity function $k$. The purpose of this paper is to establish necessary and sufficient condition under which there exists a positive…
We prove that for $p\ge 2$ solutions of equations modeled by the fractional $p$-Laplacian improve their regularity on the scale of fractional Sobolev spaces. Moreover, under certain precise conditions, they are in $W^{1,p}_{loc}$ and their…
We discuss recent advances in the theory of quasilinear equations of the type $ -\Delta_{p} u = \sigma u^{q} \; \; \text{in} \;\; \mathbb{R}^n, $ in the case $0<q< p-1$, where $\sigma$ is a nonnegative measurable function, or measure, for…
We study existence and multiplicity of nontrivial solutions of the following problem $$ \left\{ \begin{array}{rcll} -\Delta_p u+(-\Delta_p)^{s} u & = & \lambda|u|^{q-2}u+|u|^{p^{\ast}-2}u & \mbox{ in }\Omega,\\ u & = & 0 & \mbox{ on }…
The aim of this paper is to deal with the $k$-Hessian counterpart of the Laplace equation involving a nonlinearity studied by Matukuma. Namely, our model is the problem \begin{equation*} (1)\;\;\;\begin{cases} S_k(D^2u)= \lambda…
This paper investigates the Harnack inequality for nonnegative solutions to second-order parabolic equations in double divergence form. We impose conditions where the principal coefficients satisfy the Dini mean oscillation condition in…
A sharp pointwise differential inequality for vectorial second-order partial differential operators, with Uhlenbeck structure, is offered. As a consequence, optimal second-order regularity properties of solutions to nonlinear elliptic…
We study the existence of homoclic solutions for reversible Hamiltonian systems taking the family of differential equations u^4+au^2-u+f(u,b)=0 as a model. Here f is an analytic function and a, b real parameters. These equations are…
We obtain conditions for the differentiability of weak solutions for a second-order uniformly elliptic equation in divergence form with a homogeneous co-normal boundary condition. The modulus of continuity for the coefficients is assumed to…
We consider a nonlinear elliptic equation driven by the Robin $p$-Laplacian plus an indefinite potential. In the reaction we have the competing effects of a strictly $(p-1)$-sublinear parametric term and of a $(p-1)$-linear and nonuniformly…
In this paper we study second order non-linear periodic systems driven by the ordinary vector $p$-Laplacian with a non-smooth, locally Lipschitz potential function. Our approach is variational and it is based on the non-smooth critical…
We consider the problem of Ambrosetti-Prodi type \begin{equation}\label{0}\quad\begin{cases} \Delta u + e^u = s\phi_1 + h(x) &\hbox{in} \Omega, u=0 & \hbox{on} \partial \Omega, \end{cases} \nonumber \end{equation} where $\Omega$ is a…
We discuss the H\"older regularity of solutions to the semilinear equation involving the fractional Laplacian $(-\Delta)^s u=f(u)$ in one dimension. We put in evidence a new regularity phenomenon which is a combined effect of the…
We make the split of the integral fractional Laplacian as $(-\Delta)^s u=(-\Delta)(-\Delta)^{s-1}u$, where $s\in(0,\frac{1}{2})\cup(\frac{1}{2},1)$. Based on this splitting, we respectively discretize the one- and two-dimensional integral…
In this article, we study the following nonlinear doubly nonlocal problem involving the fractional Laplacian in the sense of Hardy-Littlewood-Sobolev inequality \begin{equation*} \left\{\begin{aligned} (-\Delta)^s u & =…
We establish the local H\"older continuity of possibly sign-changing solutions to a class of doubly nonlinear parabolic equations whose prototype is \[ \partial_t\big(|u|^{q-1}u\big)-\Delta_p u=0,\quad 1<p<2,\quad 0<p-1<q. \] The proof…
The purpose of this paper is to investigate a condition \begin{equation*} (C_{p}) \hspace{1cm} \alpha \int_{0}^{u}f(s)ds \leq uf(u)+\beta u^{p}+\gamma,\,\,u>0 \end{equation*} for some $\alpha>2$, $\gamma>0$, and…
Our purpose in this paper is to provide a self contained account of the inhomogeneous Dirichlet problem $\Delta_\infty u=f(x,u)$ where $u$ takes a prescribed continuous data on the boundary of bounded domains. We employ a combination of…
Let $\Omega:=\left( a,b\right) \subset\mathbb{R}$, $m\in L^{1}\left( \Omega\right) $ and $\lambda>0$ be a real parameter. Let $\mathcal{L}$ be the differential operator given by $\mathcal{L}u:=-\phi\left( u^{\prime}\right) ^{\prime}+r\left(…