Related papers: Quantitative uniqueness for the power of Laplacian…

The purpose of the present paper is to establish appropriate cut-off resolvent estimates for the Dirichlet Laplacian on exterior domains. The geometrical assumptions on domains are rather general, for example, non-trapping condition is not…

In this article a nonlocal elliptic problem involving $p$-Laplacian on unbounded domain is considered. Using variational methods and under suitable conditions, the existence of a sequence of radially symmetric weak solutions, in two…

In this paper, we study a class of quasilinear elliptic equations involving both local and nonlocal operators with variable exponents. The problem exhibits singular nonlinearities along with a subcritical superlinear growth term and a…

We study a nonlinear elliptic problem defined in a bounded domain involving fractional powers of the Laplacian operator together with a concave-convex term. We characterize completely the range of parameters for which solutions of the…

The aim of this paper is investigating the existence and multiplicity of weak solutions to non--local equations involving the {\em magnetic fractional Laplacian}, when the nonlinearity is subcritical and asymptotically linear at infinity.…

The article is an attempt to investigate the issues of asymptotic analysis for problems involving fractional Laplacian where the domains tend to become unbounded in one-direction. Motivated from the pioneering work on second order elliptic…

We prove an existence result for a $p$-Laplacian problem set in the whole Euclidean space and exhibiting a critical term perturbed by a singular, convective reaction. The approach used combines variational methods, truncation techniques,…

The Calder\'on problem is an inverse problem with applications to electrical impedance tomography and geophysical prospection. We prove uniqueness in the Calder\'on problem in spatial dimension $n \geq 3$ for scalar conductivities in the…

We extend monotonicity-based inversion methods to an inverse coefficient problem for the isotropic nonlocal elliptic equation \[ (-\nabla \cdot \sigma \nabla)^s u = 0 \quad \text{in } \Omega \subset \mathbb{R}^n, \] where $0 < s < 1$, $n…

We consider the fractional Schrodinger equation with a logarithmic nonlinearity, when the power of the Laplacian is between zero and one. We prove global existence results in three different functional spaces: the Sobolev space…

Recent advances in quantitative unique continuation properties for solutions to uniformly elliptic, divergence form equations (with Lipschitz coefficients) has led to a good understanding of the vanishing order and size of singular and zero…

We prove uniqueness results for a Calderon type inverse problem for the Hodge Laplacian acting on graded forms on certain manifolds in three dimensions. In particular, we show that partial measurements of the relative-to-absolute or…

In this article, we study the following fractional $p$-Laplacian equation with critical growth singular nonlinearity \begin{equation*} \quad (-\De_{p})^s u = \la u^{-q} + u^{\alpha}, u>0 \; \text{in}\; \Om,\quad u = 0 \; \mbox{in}\; \mb R^n…

We study via Carleman estimates the sharpest possible exponential decay for {\it waveguide} solutions to the Laplace equation $$(\partial^2_t+\triangle)u=Vu+W\cdot(\partial_t,\nabla)u,$$ and find a necessary quantitative condition on the…

We study exceptional quotient singularities. In particular, we prove an exceptionality criterion in terms of the $\alpha$-invariant of Tian, and utilize it to classify four-dimensional and five-dimensional exceptional quotient…

We study the behaviour of solutions of ordinary differential equations of the second order with singular points, where the coefficients of the second-order derivative vanishes. In particular, we consider solutions entering a singular point…

We investigate the existence of local holomorphic solutions $Y$ of linear partial differential equations in three complex variables whose coefficients are singular along an analytic variety $\Theta$ in $\mathbb{C}^{2}$. The coefficients are…

We study a singularly perturbed problem related to infinity Laplacian operator with prescribed boundary values in a region. We prove that solutions are locally (uniformly) Lipschitz continuous, they grow as a linear function, are strongly…

In these lectures we present some useful techniques to study quantitative properties of solutions of elliptic PDEs. Our aim is to outline a proof of a recent result on propagation of smallness. The ideas are also useful in the study of the…

This paper is devoted to the study, with variational technique, of (p,q)-Laplacian equations in presence of general nonlinearities. Especially we obtain the existence result for the zero mass case, which includes a large class of pure power…