Structural stability for variable exponent elliptic problems, I: The -Laplacian kind problems

Abstract

We study the structural stability (i.e., the continuous dependence on coefficients) of solutions of the elliptic problems under the form b ( u n ) − div a n ( x , ∇ u n ) = f n . The equation is set in a bounded domain Ω of R N and supplied with the homogeneous Dirichlet boundary condition on ∂ Ω . Here b is a non-decreasing function on R , and ( a n ( x , ξ ) ) n is a family of applications which verifies the classical Leray–Lions hypotheses but with a variable summability exponent p n ( x ) , 1 < p − ≤ p n ( ⋅ ) ≤ p + < + ∞ . The need for varying p ( x ) arises, for instance, in the numerical analysis of the p ( x ) -Laplacian problem. Uniqueness and existence for these problems are well understood by now. We apply the stability properties to further generalize the existence results. The continuous dependence result we prove is valid for weak and for renormalized solutions. Notice that, besides being interesting on its own, the renormalized solutions’ framework also permits us to deduce optimal convergence results for the weak solutions. Our technique avoids the use of a fixed duality framework (like the W 0 1 , p ( x ) ( Ω ) – W − 1 , p ′ ( x ) ( Ω ) duality), and thus it is suitable for the study of problems where the summability exponent p also depends on the unknown solution itself, in a local or in a non-local way. The sequel of this paper will be concerned with well-posedness of some p ( u ) -Laplacian kind problems and with existence of solutions to elliptic systems with variable, solution-dependent growth exponent.