Vitali Liskevich

Second-order semilinear elliptic inequalities in exterior domains

Abstract We study the problem of existence and nonexistence of positive solutions of the semilinear elliptic inequalities in divergence form with measurable coefficients − ∇ ·a· ∇ u+Vu−Wu p ⩾0 in exterior domains in R N , N⩾3 . For W(x)≍|x|−σ (σ∈ R ) at infinity we compute the critical line on the plane (p,σ), which separates the domains of existence and nonexistence, and reveal the class of potentials V that preserves the critical line. Example are provided showing that the class of potentials is maximal possible, in certain sense. The case of (p,σ) on the critical line has also been studied.

Positive solutions to nonlinear p-Laplace equations with Hardy potential in exterior domains

We study the existence and nonexistence of positive (super)solutions to the nonlinear p-Laplace equation −Δpu−μ|x|pup−1=C|x|σuq in exterior domains of RN (N⩾2). Here p∈(1,+∞) and μ⩽CH, where CH is the critical Hardy constant. We provide a sharp characterization of the set of (q,σ)∈R2 such that the equation has no positive (super)solutions. The proofs are based on the explicit construction of appropriate barriers and involve the analysis of asymptotic behavior of super-harmonic functions associated to the p-Laplace operator with Hardy-type potentials, comparison principles and an improved version of Hardys inequality in exterior domains. In the context of the p-Laplacian we establish the existence and asymptotic behavior of the harmonic functions by means of the generalized Prufer transformation.

Positive solutions to superlinear second-order divergence type elliptic equations in cone-like domains

Abstract We study the problem of the existence and nonexistence of positive solutions to the superlinear second-order divergence type elliptic equation with measurable coefficients − ∇ ⋅ a ⋅ ∇ u = u p ( * ) , p > 1 , in an unbounded cone-like domain G ⊂ R N ( N ⩾ 3 ) . We prove that the critical exponent p * ( a , G ) = inf { p > 1 : ( * ) has a positive supersolution at infinity in G } for a nontrivial cone-like domain is always in ( 1 , N N − 2 ) and depends both on the geometry of the domain G and the coefficients a of the equation.

ON L p -SPECTRA AND ESSENTIAL SPECTRA OF SECOND-ORDER ELLIPTIC OPERATORS

The aim of this paper is to investigate spectral properties of second order elliptic operators with measurable coefficients. Namely, we study the problems of L-independence of the spectrum and stability of the essential spectrum. The problem of L-independence of the spectrum for elliptic operators has a long history going back to B. Simon [30] where the question was posed for Schrödinger operators. The main breakthrough was made by R. Hempel and J. Voigt [14] who answered the question in the affirmative for the case that the negative part of the potential is from the Kato class. This result was a starting point for many extensions in different directions [2, 9, 10, 15, 17, 25, 26, 27] (the list is by no means complete). Most of the results deal with cases when the operators are selfadjoint in L and can be defined in all L, 1 6 p <∞. Under these conditions an abstract approach based on a functional calculus was developed by E. B. Davies [9]. In [26] L-independence was established for the Schrödinger operator with form bounded negative part of the potential. In this case the operator exists only in L for p from a certain interval around p = 2. The ideas from [26] were put in a more general context in [25]. Further progress was made by Yu. Semenov [27] who treated selfadjoint elliptic operators with unbounded coefficients, adapting the method from [26]. In the non-symmetric case the

Positive Solutions to Semi-Linear and Quasi-Linear Elliptic Equations on Unbounded Domains

Abstract In this survey we describe recent results on the existence and nonexistence of positive solution to semi-linear and quasi-linear second-order elliptic equations. A typical example is the equation –Δ u = | x | –σ u q in an exterior of the ball in ℝ N or in a cone-like domain in ℝ N . The equations of this type exhibit a phenomenon of presence of critical exponents in the range of the parameter q ∈ ℝ, which separate the zones of the existence from the zones of the nonexistence. The values of the critical exponents depends on the geometry of the domain, the type of the operator in the main part (divergent or nondivergent), the behaviour of the coefficients in lower order terms at infinity. We investigate these dependencies mostly in the cases of the exterior and cone-like domains. The proofs are often based on the explicit construction of appropriate barriers and involve the analysis of asymptotic behavior of super-harmonic functions associated to the corresponding second-order elliptic operator, comparison principles and Hardys inequality in exterior domains. To construct the barriers in the cases of equations with non-smooth coefficients we obtain detailed estimates at infinity of small and large solutions to the corresponding linear equations. Some of the results for the equations with first order term are new and have not been published before. In discussions we list some open problems in this area.

A CRITICAL PHENOMENON FOR SUBLINEAR ELLIPTIC EQUATIONS IN CONE-LIKE DOMAINS

We study positive supersolutions to an elliptic equation

Potential estimates for quasi-linear parabolic equations

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Gradient estimates for degenerate quasi-linear parabolic equations

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SINGULAR SOLUTIONS TO THE HEAT EQUATIONS WITH NONLINEAR ABSORPTION AND HARDY POTENTIALS

-\Delta u=c|x|^{-s}u^p

POSITIVE SOLUTIONS TO SEMILINEAR ELLIPTIC EQUATIONS WITH CRITICAL LOWER ORDER TERMS

,