Let Ω ⊂ ℝn, n ≥ 2, be a bounded domain and let α < n − 1. Motivated by Theorem I.6 and Remark I.18 of [Lions P.-L., The concentration-compactness principle in the calculus of variations. The limit case. I, Rev. Mat. Iberoamericana, 1985, 1(1), 145–201] and by the results of [Černý R., Cianchi A., Hencl S., Concentration-Compactness Principle for Moser-Trudinger inequalities: new results and proofs, Ann. Mat. Pura Appl. (in press), DOI: 10.1007/s10231-011-0220-3], we give a sharp estimate...

Let n ≥ 2 and let Ω ⊂ ℝn be an open set. We prove the boundedness of weak solutions to the problem $$u\in W_0^1{L}^{\Phi }\left(\Omega \right)and-div\left({\Phi }^{\text{'}}\left(\left|\nabla u\right|\right)\frac{\nabla u}{\left|\nabla u\right|}\right)+V\left(x\right){\Phi }^{\text{'}}\left(\left|u\right|\right)\frac{u}{\left|u\right|}=f\left(x,u\right)+\mu h\left(x\right)in\Omega ,$$ where ϕ is a Young function such that the space W 01 L Φ(Ω) is embedded into an exponential or multiple exponential Orlicz space, the nonlinearity f(x, t) has the corresponding critical growth, V(x) is a continuous potential, h ∈ L Φ(Ω) is a non-trivial continuous function and µ ≥ 0 is a small parameter. We consider two classical cases: the case of Ω being an open bounded set and the...

Let $n\ge 2$ and $\Omega \subset {ℝ}^n$ be a bounded set. We give a Moser-type inequality for an embedding of the Orlicz-Sobolev space $W_0{L}^{\Phi }\left(\Omega \right)$, where the Young function $\Phi$ behaves like $t^n{log}^{\alpha }\left(t\right)$, $\alpha <n-1$, for $t$ large, into the Zygmund space $Z_0^{\frac{n-1-\alpha }{n}}\left(\Omega \right)$. We also study the same problem for the embedding of the generalized Lorentz-Sobolev space $W_0^m{L}^{\frac{n}{m},q}{log}^{\alpha }L\left(\Omega \right)$, $m<n$, $q\in (1,\infty ]$, $\alpha <\frac{1}{q^{\text{'}}}$, embedded into the Zygmund space $Z_0^{\frac{1}{q^{\text{'}}}-\alpha }\left(\Omega \right)$.

We give a version of the Moser-Trudinger inequality without boundary condition for Orlicz-Sobolev spaces embedded into exponential and multiple exponential spaces. We also derive the Concentration-Compactness Alternative for this inequality. As an application of our Concentration-Compactness Alternative we prove that a functional with the sub-critical growth attains its maximum.

Consider a class of elliptic equation of the form $$-\Delta u-\frac{\lambda }{{\left|x\right|}^2}u=u^{2^{*}-1}+\mu u^{-q}\text{in}\Omega \setminus \left\{0\right\}$$ with homogeneous Dirichlet boundary conditions, where $0\in \Omega \subset {ℝ}^N$($N\ge 3$), $0<q<1$, $0<\lambda <{(N-2)}^2/4$ and $2^{*}=2N/(N-2)$. We use variational methods to prove that for suitable $\mu$, the problem has at least two positive weak solutions.