Related papers: On generalized eigenvalue problems of fractional $…
We study the existence and multiplicity of positive solutions for the following concave-critical problem driven by an operator of mixed order obtained by the sum of the classical $p$-Laplacian and of the fractional $p$-Laplacian,…
Let $\Omega \subset \mathbb{R}^N$, $N \geq 2$, be a smooth bounded domain. For $s \in (1/2,1)$, we consider a problem of the form \[ \left\{\begin{aligned} (-\Delta)^s u & = \mu(x)\, \mathbb{D}_s^{2}(u) + \lambda f(x)\,, & \quad \mbox{in}…
A rigorous methodology for the analysis of initial boundary value problems on the half-line, $0<x<\infty$, $t>0$, for integrable nonlinear evolution PDEs has recently appeared in the literature. As an application of this methodology the…
Suppose $d\ge 2$ and $0<\beta<\alpha<2$. We consider the non-local operator $\mathcal{L}^{b}=\Delta^{\alpha/2}+\mathcal{S}^{b}$, where $$\mathcal{S}^{b}f(x):=\lim_{\varepsilon\to…
In this paper the existence of solutions, $(\lambda,u)$, of the problem $$-\Delta u=\lambda u -a(x)|u|^{p-1}u \quad \hbox{in }\Omega, \qquad u=0 \quad \hbox{on}\;\;\partial\Omega,$$ is explored for $0 < p < 1$. When $p>1$, it is known that…
The first goal of this paper is to establish the existence of a positive solution for the singular boundary value problem (1.1), where $\mathcal{B}$ is a general boundary operator of Dirichlet, Neumann or Robin type, either classical or…
In 2009 Loc and Schmitt established a result on sufficient conditions for multiplicity of solutions of a class of nonlinear eignvalue problems for the p-Laplace operator under Dirichlet boundary conditions, extending an earlier result of…
In this paper, we investigate the $(p_{1}(x), p_{2}(x))$-Laplace operator, the properties of the corresponding integral functional and weak solutions to the related differential equations. We show that the integral functional admits a…
We consider fractional operators of the form $$\mathcal{H}^s=(\partial_t -\mathrm{div}_{x} ( A(x,t)\nabla_{x}))^s,\ (x,t)\in\mathbb R^n\times\mathbb R,$$ where $s\in (0,1)$ and $A=A(x,t)=\{A_{i,j}(x,t)\}_{i,j=1}^{n}$ is an accretive,…
In this work, we consider a mixed local and nonlocal Dirichlet problem with supercritical nonlinearity. We first establish a multiplicity result for the problem \begin{equation} Lu=|u|^{p-2}u+\mu|u|^{q-2}u~~\text{in}~~\Omega,~~~~~…
Consider the eigenvalue problem of a linear second order elliptic operator: \begin{equation} \nonumber -D\Delta \varphi -2\alpha\nabla m(x)\cdot \nabla\varphi+V(x)\varphi=\lambda\varphi\ \ \hbox{ in }\Omega, \end{equation} complemented by…
The main purpose of this article is to reconstruct the nonnegative coefficient $a$ in the double phase problem $\mathrm{div}\,(|\nabla u|^{p-2}\nabla u+a|\nabla u|^{q-2}\nabla u)=0$ in a domain $\Omega$, $u=f$ on $\partial\Omega$, from the…
We show optimal existence, nonexistence and regularity results for nonnegative solutions to Dirichlet problems as $$ \begin{cases} \displaystyle -\Delta_1 u = g(u)|D u|+h(u)f & \text{in}\;\Omega,\\ u=0 & \text{on}\;\partial\Omega,…
In the present work we study existence of sequences of variational eigenvalues to non-local non-standard growth problems ruled by the fractional $g-$Laplacian operator with different boundary conditions (Dirichlet, Neumann and Robin). Due…
This paper is divided in two parts. In the first part, we prove coercivity results and minimization of the Euler energy functional. In the second part, we focus on the existence and multiplicity of a positive solution of fractional…
Consider a classical elliptic pseudodifferential operator $P$ on ${\Bbb R}^n$ of order $2a$ ($0<a<1)$ with even symbol. For example, $P=A(x,D)^a$ where $A(x,D)$ is a second-order strongly elliptic differential operator; the fractional…
In this paper, we consider the dual fractional parabolic problem in the right half space. We prove that the positive solutions are strictly increasing in $x_1$ direction without assuming the solutions be bounded. So far as we know, this is…
The objective of this work is to investigate a nonlocal problem involving singular and critical nonlinearities:\begin{equation*}\left\{\begin{array}{ll} ([u]_{s,p}^p)^{\sigma-1}(-\Delta)^s_p u = \frac{\lambda}{u^{\gamma}}+u^{ p_s^{*}-1…
We consider the eigenvalue problem for the {\it fractional $p-$Laplacian} in an open bounded, possibly disconnected set $\Omega \subset \mathbb{R}^n$, under homogeneous Dirichlet boundary conditions. After discussing some regularity issues…
We establish the existence and uniqueness of the solution to the Dirichlet problem for the variable exponent $p$-Laplacian on a bounded, smooth domain $\Omega \subset {\mathbb R}^n$, where the boundary datum belongs to $W^{1,p}(\Omega)$.…