Related papers: Differential inclusions involving oscillatory term…

We present a methodology for numerically integrating ordinary differential equations containing rapidly oscillatory terms. This challenge is distinct from that for differential equations which have rapidly oscillatory solutions: here the…

Utilizing a new variational principle that allows dealing with problems beyond the usual locally compactness structure, we study problems with a supercritical nonlinearity of the type $ -\Delta u + u= a(x) f(u)$ in $ \Omega$ with…

In this paper, we investigate a class of critical Ambrosetti-Prodi type problems involving the sub-Laplacian on a Carnot group. Specifically, we consider \[ \left\{ \begin{aligned} -\Delta_{\mathbb{G}} u &= \lambda u + u_{+}^{2_{Q}^{*}-1} +…

Let $\Delta_{\Lambda}\le \lambda_{\Lambda}$ be a semi-bounded self-adjoint realization of the Laplace operator with boundary conditions (Dirichlet, Neumann, semi-transparent) assigned on the Lipschitz boundary of a bounded obstacle…

We investigate a perturbed Gelfand problem involving a mixed local-nonlocal $p$-Laplacian operator with singular nonlinearity: \begin{equation*} \begin{aligned} -\Delta_p u + (-\Delta_p)^s u = \lambda \frac{f(u)}{u^{\beta}}\ \text{in} \…

In this paper, we consider the solutions for elliptic equations involving regional fractional Laplacian \begin{equation}\label{0} \arraycolsep=1pt \begin{array}{lll} \displaystyle (-\Delta)^\alpha_\Omega u=f \qquad & {\rm in}\quad…

In this paper, we study the existence and the summability of solutions to a Robin boundary value problem whose prototype is the following: $$ \begin{cases} -\text{div}(b(|u|)\nabla u)=f &\text{in }\Omega,\\[.2cm] \displaystyle\frac{\partial…

In this paper, we are concerned with the boundedness of all the solutions for a kind of second order differential equations with p-Laplacian and an oscillating term $(\phi_p(x'))'+a\phi_p(x^+)-b\phi_p(x^-)=G_x(x,t)+f(t)$, where$x^+=\max…

We consider the quasi-linear eigenvalue problem $-\Delta_p u = \lambda g(u)$ subject to Dirichlet boundary conditions on a bounded open set $\Omega$, where $g$ is a locally Lipschitz continuous functions. Imposing no further conditions on…

We investigate the existence, non-existence, uniqueness, and multiplicity of positive solutions to the following problem: \begin{align}\label{P} \left\{ \begin{array}{l} D_{0+}^\alpha u + h(t)f(u) = 0, \quad 0<t<1, \\[1ex] u(0)=u(1)=0,…

We study the extremal solution for the problem $(-\Delta)^s u=\lambda f(u)$ in $\Omega$, $u\equiv0$ in $\R^n\setminus\Omega$, where $\lambda>0$ is a parameter and $s\in(0,1)$. We extend some well known results for the extremal solution when…

We consider the optimization problem of minimizing $\int_{\Omega}|\nabla u|^{p(x)}+ \lambda \chi_{\{u>0\}} dx$ in the class of functions $W^{1,p(\cdot)}(\Omega)$ with $u-\phi_0\in W_0^{1,p(\cdot)}(\Omega)$, for a given $\phi_0\geq 0$ and…

In this paper we consider the following Dirichlet problem for the $p$-Laplacian in the positive parameters $\lambda$ and $\beta$: [{{array} [c]{rcll}% -\Delta_{p}u & = & \lambda h(x,u)+\beta f(x,u,\nabla u) & \text{in}\Omega u & = & 0 &…

We study the problems of the existence, uniqueness and continuous dependence of Lipschitzian solutions $\varphi$ of equations of the form $$ \varphi(x)=\int_{\Omega}g(\omega)\varphi\big(f(x,\omega)\big)\mu(d\omega)+F(x), $$ where $\mu$ is a…

Let $D$ be a bounded $C^2$-domain. Consider the following Dirichlet initial-boundary problem of nonlocal operators with a drift: $$ \partial_t u={\mathscr L}^{(\alpha)}_\kappa u+b\cdot \nabla u+f\ \mathrm{in}\ \mathbb R_+\times D,\ \…

Let $\Omega $ be a smooth bounded domain in $\R^N, N>1$ and let $n\in \N^*$. We are concerned here with the existence of nonnegative solutions $u\_n$ in $BV(\Omega)$, to the problem $$(P\_n) \begin{cases} -{\rm div} \sigma +2n (\int\_…

This paper is devoted to the study of the existence of positive solutions for a problem related to a higher order fractional differential equation involving a nonlinear term depending on a fractional differential operator,…

We propose a natural gradient term for a class of second-order partial differential equations of the form \begin{equation}\nonumber M(x,Du,D^2u)+g(u)N(x,Du, D^2u)+f(x,u)=0 \;\;\mbox{in}\;\; \Omega, \end{equation} where…

We obtain the inequality $$\int_{\Omega}|\nabla u(x)|^ph(u(x))dx\leq C(n,p)\int_{\Omega} \left( \sqrt{ |\Delta u(x)||{\cal T}_{h,C}(u(x))|}\right)^{p}h(u(x))dx,$$ where $\Omega\subset \mathbf{R}^n$ is a bounded Lipschitz domain, $u\in…

We consider the "Method of particular solutions" for numerically computing eigenvalues and eigenfunctions of the Laplacian $\Delta$ on a smooth, bounded domain Omega in RR^n with either Dirichlet or Neumann boundary conditions. This method…