In this paper we study sufficient local conditions for the existence of non-trivial solution to a critical equation for the p(x)−Laplacian where the critical term is placed as a source through the boundary of the domain. The proof relies on a suitable generalization of the concentration–compactness principle for the trace embedding for variable exponent Sobolev spaces and the classical mountain...

We study properties of Lipschitz truncations of Sobolev functions with constant and variable exponent. As non-trivial applications we use the Lipschitz truncations to provide a simplified proof of an existence result for incompressible power-law like fluids presented in Frehse, Málek, Steinhauer: SIAM J. Math. Anal., 34, 1064-1083 (2003). We also establish new existence results to a class of in...

We prove the existence of at least three weak solutions for the Dirichlet problem when the nonlinear term f is sublinear and p(x) is greater than n. This Dirichlet problem involves a general elliptic operator in divergence form (in particular, a p(x)-Laplace operator). Our method relies upon a recent critical point theorem obtained by Bonanno and Marano, and is combined with the theory of varia...

We study unbounded weak supersolutions of elliptic partial differential equations with generalized Orlicz (Musielak--Orlicz) growth. show that they satisfy the Harnack inequality optimal exponent provided belong to a suitable Lebesgue or Sobolev space. Furthermore, we establish sharpness our central assumptions.

In this paper, we deal with the existence and nonexistence of nonnegative nontrivial weak solutions for a class of degenerate quasilinear elliptic problems with weights and nonlinearity involving the critical Hardy-Sobolev exponent and a sign-changing function. Some existence results are obtained by splitting the Nerahi manifold and by exploring some properties of the best Hardy-Sobolev constan...

where Ω ⊂ R(N ≥ 4) is an open bounded domain with smooth boundary, β > 0, 0 ∈ Ω, 0 ≤ s < 2, 2∗(s) := 2(N − s) N − 2 is the critical Hardy-Sobolev exponent and, when s = 0, 2∗(0) = 2N N − 2 is the critical Sobolev exponent, 0 ≤ μ < μ := (N − 2) 4 . In [1] A. Ferrero and F. Gazzola investigated the existence of nontrivial solutions for problem (1.1) with β = 1, s = 0. In [2] D. S. Kang and S. J. ...

In this paper we will be concerned with the existence and non-existence of constrained minimizers in Sobolev spaces D(R ), where the constraint involves the critical Sobolev exponent. Minimizing sequences are not, in general, relatively compact for the embedding D(R) →֒ L ∗ (R , Q) when Q is a non-negative, continuous, bounded function. However if Q has certain symmetry properties then all minim...

In this paper, we investigate the existence of a ?weak solutions? for Neumann problems p(x)-Laplacian-like operators, originated from capillary phenomena, following form {?div( |?u|p(x)?2?u + |?u|2p(x)?2?u /?1 |?u|2p(x))= ?f (x, u,?u) in ?,(|?u|p(x)?2?u |?u|2p(x)?2 ?u/ ?|?u|2p(x)) ?u/?? = 0 on ??, setting variable-exponent Sobolev spaces W1,p(x)(?), where ? is smooth bounded domain RN, p(x) C+(...