Abstract. We consider a Dirichlet problem in divergence form with variable growth, modeled on the p(x)-Laplace equation. We obtain existence and uniqueness of an entropy solution for L 1 data, as well as integrability results for the solution and its gradient. The proofs rely crucially on a priori estimates in Marcinkiewicz spaces with variable exponent, for which we obtain new inclusion results of independent interest.

In this article we consider the p-Laplace equation − p u = λ f (u) on a smooth bounded domain of R N with zero Dirichlet boundary conditions. Under adequate assumptions on f we prove that the extremal solution of this problem is in the energy class W 1, p 0 (Ω) independently of the domain. We also obtain L q and W 1,q estimates for such a solution. Moreover, we prove its boundedness for some range of dimensions depending on the nonlinearity f .

We consider nonnegative solutions of −∆pu = f (x, u), where p > 1 and ∆p is the p-Laplace operator, in a smooth bounded domain of R N with zero Dirichlet boundary conditions. We introduce the notion of semistability for a solution (perhaps unbounded). We prove that certain minimizers, or one-sided minimizers, of the energy are semi-stable, and study the properties of this class of solutions.Under some assumptions on f that make its growth comparable to u m , we prove that every semi-stable solution is bounded if m < mcs. Here, mcs = mcs(N, p) is an explicit exponent which is optimal for the boundedness of semistable solutions. In particular, it is bigger than the critical Sobolev exponent p * − 1.We also study a type of semi-stable solutions called extremal solutions, for which we establish optimal L ∞ estimates. Moreover, we characterize singular extremal solutions by their semi-stability property when the domain is a ball and 1 < p < 2.

In this paper we prove a Sobolev and a Morrey type inequality involving the mean curvature and the tangential gradient with respect to the level sets of the function that appears in the inequalities. Then, as an application, we establish a priori estimates for semi-stable solutions of −∆ p u = g(u) in a smooth bounded domain Ω ⊂ R n . In particular, we obtain new L r and W 1,r bounds for the extremal solution u ⋆ when the domain is strictly convex. More precisely, we prove that u ⋆ ∈ L ∞ (Ω) if n ≤ p + 2 and u ⋆ ∈ L np n−p−2 (Ω) ∩ W 1,p 0 (Ω) if n > p + 2.