Related papers: The Pohozaev identity for mixed local-nonlocal ope…

In this paper we prove the Pohozaev identity for the semilinear Dirichlet problem $(-\Delta)^s u = f(u)$ in $\Omega$, $u \equiv 0$ in $\mathbb R^n\setminus\Omega$. Here, $s\in(0,1)$, $(-\Delta)^s$ is the fractional Laplacian in $\mathbb…

In this paper we prove a Pohozaev-type identity for both the problem $(-\Delta+m^2)^su=f(u)$ in $\mathbb{R}^N$ and its harmonic extension to $\mathbb{R}^{N+1}_+$ when $0<s<1$. So, our setting includes the pseudo-relativistic operator…

In this paper, we prove a Pohozaev identity for the Spectral Fractional Laplacian (SFL). This identity allows us to establish non-existence results for the semilinear Dirichlet problem $(-\Delta|_{\Omega})^su = f(u)$ in star-shaped domains.…

We establish a new integration by parts formula for the regional fractional laplacian $(-\Delta)^s_\Omega$ in bounded open sets of class $C^2$. As a direct application, we prove that weak solutions to the corresponding Dirichlet problem…

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 expository paper we survey some recent results on Dirichlet problems of the form $Lu=f(x,u)$ in $\Omega$, $u\equiv0$ in $\mathbb R^n\backslash\Omega$. We first discuss in detail the boundary regularity of solutions, stating the main…

We study the Dirichlet problem for the semilinear equations involving the pseudo-relativistic operator on a bounded domain, (\sqrt{-\Delta + m^2} - m)u =|u|^{p-1}u \quad \textrm{in}~\Omega, with the Dirichlet boundary condition $u=0$ on…

We establish Pohozaev identities and integration by parts type formulas for anisotropic integro-differential operators of order $2s$, with $s\in(0,1)$. These identities involve local boundary terms, in which the quantity…

We prove Pohozaev-type identities for smooth solutions of Euler-Lagrange equations of second and fourth order that arise from functional depending on homogeneous H\"{o}rmander vector fields. We then exploit such integral identities to prove…

In this note we present the Pohozaev identity for the fractional Laplacian. As a consequence of this identity, we prove the nonexistence of nontrivial bounded solutions to semilinear problems with supercritical nonlinearities in star-shaped…

In this paper, we study the Pohozaev identity associated with a H$\acute{e}$non-Lane-Emden system involving the fractional Laplacian: \begin{equation} \left\{\begin{array}{ll} (-\triangle)^su=|x|^av^p,&x\in\Omega,…

We prove a fractional Pohozaev type identity in a generalized framework and discuss its applications. Specifically, we shall consider applications to nonexistence of solutions in the case of supercritical semilinear Dirichlet problems and…

We consider the numerical solution of the equation - \Delta u - f(u) = g, for the unknown u satisfying Dirichlet conditions in a bounded domain. The nonlinearity f has bounded, continuous derivative. The algorithm uses the finite element…

We establish an integration by parts formula in bounded domains for the higher order fractional Laplacian $(-\Delta)^s$ with $s>1$. We also obtain the Pohozaev identity for this operator. Both identities involve local boundary terms, and…

By virtue of a suitable approximation argument, we prove a Pohozaev identity for nonlinear nonlocal problems on $\mathbb{R}^N$ involving the fractional $p-$Laplacian operator. Furthermore we provide an application of the identity to show…

We consider positive solutions, possibly unbounded, to the semilinear equation $-\Delta u=f(u)$ on continuous epigraphs bounded from below. Under the homogeneous Dirichlet boundary condition, we prove new monotonicity results for $u$, when…

For $2a$-order strongly elliptic operators $P$ generalizing $(-\Delta )^a$, $0<a<1$, the treatment of the homogeneous Dirichlet problem on a bounded open set $\Omega \subset R^n$ by pseudodifferential methods, has been extended in a recent…

We are concerned with the inverse boundary problem of determining anomalies associated with a semilinear elliptic equation of the form $-\Delta u+a(\mathbf x, u)=0$, where $a(\mathbf x, u)$ is a general nonlinear term that belongs to a…

We study the existence, multiplicity, and certain qualitative properties of solutions to the zero Dirichlet problem for the equation $-\Delta_p u = \lambda |u|^{p-2}u + a(x)|u|^{q-2}u$ in a bounded domain $\Omega \subset \mathbb{R}^N$,…

In this article, we establish Pohozaev-type identities for a class of quasilinear elliptic equations and systems involving both local and nonlocal $p$-Laplace operators. Specifically, we obtain these identities in $\mathbb{R}^n$ for the…