Advanced Search

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 of the...

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 case of Ω =...

Let $\mathrm{\Omega }\subset {𝐑}^N$ be an open bounded set with a Lipschitz boundary and let $g:\mathrm{\Omega }\times 𝐑\to 𝐑$ be a Carathéodory function satisfying usual growth assumptions. Then the functional $$\mathrm{\Phi }(u)={\int }_{\mathrm{\Omega }}g(x,u(x))𝑑x$$ is lower semicontinuous with respect to the weak topology on $W^{1,p}(\mathrm{\Omega })$, $1\le p\le \mathrm{\infty }$, if and only if $g$ is convex in the second variable for almost every $x\in \mathrm{\Omega }$.

We prove that the 1-dimensional Hausdorff measure restricted to a simple real analytic curve $\gamma :ℝ\to {ℝ}^N$, $N\ge 2$, is locally 1-monotone.

We prove that the generalized Trudinger inequality for Orlicz-Sobolev spaces embedded into multiple exponential spaces implies a version of an inequality due to Brézis and Wainger.

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.

We give a sufficient condition for a curve $\gamma :ℝ\to {ℝ}^n$ to ensure that the $1$-dimensional Hausdorff measure restricted to $\gamma$ is locally monotone.

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 $Z_0^{\frac{1}{q^{\text{'}}}-\alpha }\left(\Omega \right)$.

Let $\Omega \subset {ℝ}^n$ be a domain and let $\alpha <n-1$. We prove the Concentration-Compactness Principle for the embedding of the space $W_0^1{L}^n{log}^{\alpha }L\left(\Omega \right)$ into an Orlicz space corresponding to a Young function which behaves like $exp\left(t^{n/(n-1-\alpha )}\right)$ for large $t$. We also give the result for the embedding into multiple exponential spaces. Our main result is Theorem where we show that if one passes to unbounded domains, then, after the usual modification of the integrand in the Moser functional, the statement of the Concentration-Compactnes Principle is very...

Let $\Omega \subset {ℝ}^n$, $n\ge 2$, be a bounded connected domain of the class $C^{1,\theta }$ for some $\theta \in (0,1]$. Applying the generalized Moser-Trudinger inequality without boundary condition, the Mountain Pass Theorem and the Ekeland Variational Principle, we prove the existence and multiplicity of nontrivial weak solutions to the problem $$u\in W^1{L}^{\Phi }\left(\Omega \right),-div\left({\Phi }^{\text{'}}\left(\right|\nabla u\left|\right)\frac{\nabla u}{\left|\nabla u\right|}\right)+V\left(x\right){\Phi }^{\text{'}}\left(\right|u\left|\right)\frac{u}{\left|u\right|}=f(x,u)+\mu h\left(x\right)\text{in}\Omega ,\frac{\partial u}{\partial 𝐧}=0\text{on}\partial \Omega ,$$ where $\Phi$ is a Young function such that the space $W^1{L}^{\Phi }\left(\Omega \right)$ is embedded into exponential or multiple exponential Orlicz space, the nonlinearity $f(x,t)$ has the corresponding critical growth, $V\left(x\right)$ is a continuous potential,...

Let $\Omega$ be a bounded open set in ${ℝ}^n$, $n\ge 2$. In a well-known paper , 20, 1077–1092 (1971) Moser found the smallest value of $K$ such that $$sup\left\{{\int }_{\Omega }exp\left({\left(\frac{\left|f\left(x\right)\right|}{K}\right)}^{n/(n-1)}\right):f\in W_0^{1,n}\left(\Omega \right),{\parallel \nabla f\parallel }_{L^n}\le 1\right\}<\infty .$$ We extend this result to the situation in which the underlying space $L^n$ is replaced by the generalized Zygmund space $L^n{log}^{n-1}L{log}^{\alpha }logL$ $(\alpha <n-1)$, the corresponding space of exponential growth then being given by a Young function which behaves like $exp(exp\left(t^{n/(n-1-\alpha )}\right))$ for large $t$. We also discuss the case of an embedding into triple and other multiple exponential cases.

We show that for every $\epsilon >0$, there is a set $A\subset {ℝ}^2$ such that ${\mathscr{H}}^1⌞A$ is a monotone measure, the corresponding tangent measures at the origin are not unique and ${\mathscr{H}}^1⌞A$ has the $1$-dimensional density between $1$ and $3+\epsilon$ everywhere on the support.

We show that for every $\epsilon >0$ there is a set $A\subset {ℝ}^3$ such that ${\mathscr{H}}^1⌞A$ is a monotone measure, the corresponding tangent measures at the origin are non-conical and non-unique and ${\mathscr{H}}^1⌞A$ has the $1$-dimensional density between $1$ and $2+\epsilon$ everywhere in the support.