Related papers: Strong unique continuation for general elliptic eq…
We prove the existence of one or more solutions to a singularly perturbed elliptic problema with two potential functions.
We prove the self-improving property of very weak solutions to non-uniformly elliptic problems of double phase type in divergence form under sharp assumptions on the nonlinearity.
We establish the existence of strong solutions to a class of nonlinear strongly coupled and uniform elliptic systems consisting of more than two equations. The existence of of nontrivial and non constant solutions (or pattern formations)…
In this paper we study the local behavior of a solution to second order elliptic operators with sharp singular coefficients in lower order terms. One of the main results is the bound on the vanishing order of the solution, which is a…
We investigate the quantitative unique continuation properties of solutions to second-order elliptic equations with lower-order terms. In particular, we establish quantitative forms of the strong unique continuation property for solutions…
It is known that solutions to second order uniformly elliptic and parabolic equations, either in divergence or nondivergence (general) form, are H\"{o}lder continuous and satisfy the interior Harnack inequality. We show that even in the…
We show that a class of divergence-form elliptic problems with quadratic growth in the gradient and non-coercive zero order terms are solvable, under essentially optimal hypotheses on the coefficients in the equation. In addition, we prove…
In this article we deal with different forms of the unique continuation property for second order elliptic equations with nonlinear potentials of sublinear growth. Under suitable regularity assumptions, we prove the weak and the strong…
In this paper, we give the strong unique continuation property for a general two dimensional anisotropic elliptic system with real coefficients in a Gevrey class under the assumption that the principal symbol of the system has simple…
We prove two results of strong continuity with respect to the initial datum for bounded solutions to the Euler equations in vorticity form. The first result provides sequential continuity and holds for a general bounded solution. The second…
The main result of the present paper is the construction of fundamental solutions for a class of multidimensional elliptic equations with several singular coefficients. These fundamental solutions are directly connected with multiple…
In this article, we study the quantitative uniqueness of solutions to second order elliptic equations with singular lower order terms. We quantify the strong unique continuation property by estimating the maximal vanishing order of…
We show the existence and uniqueness as well as boundedness of weak solutions to linear elliptic equations with $L^2$-drifts of negative divergence and singular zero-order terms which are positive. Our main target is to show the…
We investigate the quantitative unique continuation of solutions to higher order elliptic equations with singular coefficients. Quantitative unique continuation described by the vanishing order is a quantitative form of strong unique…
In two dimensions every weak solution to a nonlinear elliptic system $\rm{div} a(x,u,Du)=0$ has H\"older continuous first derivatives provided that standard continuity, ellipticity and growth assumptions hold with a growth exponent $p \geq…
The main result of the present paper is the construction of fundamental solutions for a class of multidimensional elliptic equations with three singular coefficients, which could be expressed in terms of a confluent hypergeometric function…
In this paper we consider quasilinear elliptic equations with double phase phenomena and a reaction term depending on the gradient. Under quite general assumptions on the convection term we prove the existence of a weak solution by applying…
We establish the validity of a strong unique continuation property for weakly coupled elliptic systems, including competitive ones. Our proof exploits the system structure and uses Carleman estimates. We apply this result to obtain some…
Based on a variant of frequency function, we improve the vanishing order of solutions for Schr\"{o}dinger equations which describes quantitative behavior of strong uniqueness continuation property. For the first time, we investigate the…
We establish a positivity property for a class of semilinear elliptic problems involving indefinite sublinear nonlinearities. Namely, we show that any nontrivial nonnegative solution is positive for a class of problems the strong maximum…