Elliptic equations and systems with nonstandard growth conditions: Existence, uniqueness and localization properties of solutions

Abstract

We study the Dirichlet problem for the elliptic equations − ∑ i D i ( a i ( x , u ) | D i u | p i ( x ) − 2 D i u ) + c ( x , u ) | u | σ ( x ) − 2 u = f ( x ) in a bounded domain Ω ⊂ R n , and the class of elliptic systems − ∑ j D j ( a i j ( x , ∇ u ) ) = f ( i ) ( x , u ) , i = 1 , … , n , u = ( u ( 1 ) , … , u ( n ) ) , satisfying the growth condition ∀ ( x , s , V ) ∈ Ω × R n 2 ∑ i j a i j ( x , V ) ⋅ V i j ≥ a 0 ∑ i j | V i j | p i j ( x ) , a 0 = const > 0 . The exponents p i j ( x ) , p i ( x ) , σ ( x ) are known functions. These equations are usually referred to as elliptic equations with nonstandard growth conditions. We prove first the theorems of existence of (bounded) weak solutions and establish sufficient conditions of uniqueness of a weak solution. Our main purpose is the study of the localization properties of weak solutions: we show that the weak solution may identically vanish on a set of nonzero measure in Ω (a dead core) and derive estimates on the size and location of these dead cores in terms of the problem data. The study of the localization properties is performed via the method of local energy estimates.