Related papers: Elliptic Problems Involving Mixed Local-Nonlocal O…

In this paper, we consider an elliptic operator obtained as the superposition of a classical second-order differential operator and a nonlocal operator of fractional type. Though the methods that we develop are quite general, for…

In this article, we consider a combination of local and nonlocal Laplace equation with singular nonlinearities. For such mixed problems, we establish existence of at least one weak solution for a parameter dependent singular nonlinearity…

In this paper, we investigate a mixed elliptic equation involving both local and nonlocal Laplacian operators, with a power-type nonlinearity. Specifically, we consider a Lane-Emden type equation of the form \[-\Delta u + (-\Delta)^s u =…

We show the existence of solution for some classes of nonlocal problems. Our proof combines the presence of sub and supersolution with the pseudomonotone operators theory.

We consider a class of elliptic and parabolic problems, featuring a specific nonlocal operator of fractional-laplacian type, where integration is taken on variable domains. Both elliptic and parabolic problems are proved to be uniquely…

In this work, we study an elliptic problem involving an operator of mixed order with both local and nonlocal aspects, and in either the presence or the absence of a singular nonlinearity. We investigate existence or non-existence…

In this article, we study an elliptic problem of mixed order with both local and nonlocal aspects involving singular nonlinearity in combination with critical Hartree-type nonlinearity. Using variational methods together with the critical…

We analyze the existence and multiplicity of positive solutions to a nonlocal elliptic problem involving the spectral fractional Laplace operator endowed with homogeneous mixed Dirichlet-Neumann boundary conditions and weighted critical…

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 }…

We study the existence and non-existence of positive solutions for the following class of nonlinear elliptic problems in the hyperbolic space $$ -\Delta_{\mathbb{B}^N} u-\lambda u=a(x)u^{p-1} \, + \, \varepsilon u^{2^*-1}…

This work concerns with the existence of solutions for the following class of nonlocal elliptic problems \begin{eqnarray}\label{eq:0.1} &&\left\{\begin{array}{l} (-\Delta)^{s} u+u=|u|^{p-2} u \text { in } \Omega_{r} \\ u \geq 0 \quad \text…

This article is divided into two parts. In the first part, we examine the Brezis-Oswald problem involving a mixed anisotropic and nonlocal $p$-Laplace operator. We establish results on existence, uniqueness, boundedness, and the strong…

We consider the following quasilinear critical problem involving the mixed local-nonlocal operator: \begin{equation}\label{main_prob_abstract_1}\tag{$\mathcal{P}_p$} -\Delta_p u+(-\Delta_p)^s u=|u|^{p^*-2}u+f(x)\text{ in }\mathbb{R}^N,…

In this paper, we consider systems of semilinear elliptic equations \displaystyle -\Delta_{\mathbb{H}^{N}}u=|v|^{p-1}v, \displaystyle -\Delta_{\mathbb{H}^{N}}v=|u|^{q-1}u, in the whole of Hyperbolic space $\mathbb{H}^{N}$. We establish…

In this article, we study an elliptic problem involving an operator of mixed order with both local and nonlocal aspects and in the presence of critical nonlinearity of Hartree type. To this end, we first investigate the corresponding…

We investigate elliptic and parabolic equations involving mixed local and nonlocal operators of the form $(-\Delta)^s-\Delta$, as well as their parabolic counterparts with both the Marchaud fractional time derivative and the classical…

We study some critical elliptic problems involving the difference of two nonlocal operators, or the difference of a local operator and a nonlocal operator. The main result is the existence of two nontrivial weak solutions, one with negative…

In this article, we consider mixed local and nonlocal Sobolev $(q,p)$-inequalities with extremal in the case $0<q<1<p<\infty$. We prove that the extremal of such inequalities is unique up to a multiplicative constant that is associated with…

We establish local boundedness and local H\"older continuity of weak solutions to the following prototype problem: $$ -\operatorname{div}\left(|x|^{-2 \beta}|\nabla u|^{\mathbf{q}-2} \nabla u\right)+(-\Delta)_{p(\cdot, \cdot),…

We consider nonlinear perturbations of the hyperbolic equation in the Hilbert space. Necessary and sufficient conditions for the existence of solutions of boundary-value problem for the corresponding equation and iterative procedures for…