In this paper, we prove some weighted sharp inequalities of Trudinger–Moser type. The weights considered here have a logarithmic growth. These inequalities are completely new and are established in some new Sobolev spaces where the norm is a mixture of the norm of the gradient in two different Lebesgue spaces. This fact allowed us to prove a very interesting result of sharpness for the case of doubly exponential growth at infinity. Some improvements of these inequalities for the weakly convergent sequences are also proved using a version of the Concentration-Compactness principle of P.L. Lions. Taking profit of these inequalities, we treat in the last part of this work some elliptic quasilinear equation involving the weighted $(N,q)-$Laplacian operator where $1 < q < N$ and a nonlinearities enjoying a new type of exponential growth condition at infinity.

We study the $\Gamma$-convergence of nonconvex vectorial integral functionals whose integrands satisfy possibly degenerate growth and coercivity conditions. The latter involve suitable scale-dependent weight functions. We prove that under appropriate uniform integrability conditions on the weight functions, which shall belong to a Muckenhoupt class, the corresponding functionals $\Gamma$-converge, up to subsequences, to a degenerate integral functional defined on a limit weighted Sobolev space. The general analysis is then applied to the case of random stationary integrands and weights to prove a stochastic homogenization result for the corresponding functionals.

We prove the existence of one positive, one negative and one sign-changing solution of a p-Laplacian equation on ℝN with a p-superlinear subcritical term. Sign-changing solutions of quasilinear elliptic equations set on the whole of ℝN have scarcely been investigated in the literature. Our assumptions here are similar to those previously used by some authors in bounded domains, and our proof uses fairly elementary critical point theory, based on constraint minimization on the nodal Nehari set. The lack of compactness due to the unbounded domain is overcome by working in a suitable weighted Sobolev space.

In this article we develop a posteriori error estimates for second orderlinear elliptic problems with point sources in two- and three-dimensional domains. Weprove a global upper bound and a local lower bound for the error measured in a weightedSobolev space. The weight considered is a (positive) power of the distance to the supportof the Dirac delta source term, and belongs to the Muckenhoupt’s class A2. The theoryhinges on local approximation properties of either Clément or Scott–Zhang interpolationoperators, without need of modifications, and makes use of weighted estimates forfractional integrals and maximal functions. Numerical experiments with an adaptivealgorithm yield optimal meshes and very good effectivity indices.

We study the bifurcation problem

$$ \begin{cases} -\Div(a(x)|Du|^{p-2}Du)+h(x)u^{r-1}= f(\lambda,x,u) & \text{in } \sOm\subset\RR^{N}, \\ a(x)|Du|^{p-2}Du\cdot n+b(x)u^{p-1}=\theta g(x,u) & \text{on }\sGa, \\ u\geq0,\quad u\not\equiv0 & \text{in }\sOm, \end{cases} $$

where $\sOm$ is an unbounded domain with smooth non-compact boundary $\sGa$, $n$ denotes the unit outward normal vector on $\sGa$, and $\lambda>0$, $\theta$ are real parameters. We assume throughout that $p lt r lt p^{*}=pN/ (N-p)$, $1 lt p lt N$, the functions $a$, $b$ and $h$ are positive while $f$, $g$ are subcritical nonlinearities. We show that there exist an open interval $I$ and $\lambda^\star gt 0$ such that the problem has no solution if $\theta\in I$ and $\lambda\in (0,\lambda^\star)$. Furthermore, there exist an open interval $J\subset I$ and $\lambda_0 gt 0$ such that, for any $\theta\in J$, the above problem has at least a solution if $\lambda\geq \lambda_0$, but it has no solution provided that $\lambda\in (0,\lambda_0)$.

AMS 2000 Mathematics subject classification: Primary 35J60; 35P30; 58E05; 58G28

The main result establishes that a weak solution of degenerate elliptic equations can be approximated by a sequence of solutions of non-degenerate elliptic equations. To this end we prove an approximation theorem for $A_p$-weights.

AMS 2000 Mathematics subject classification: Primary 35J50. Secondary 35D05

We propose a variational analysis for a Black and Scholes equation with stochastic volatility. This equation gives the price of a European option as a function of the time, of the price of the underlying asset and of the volatility when the volatility is a function of a mean reverting Orstein-Uhlenbeck process, possibly correlated with the underlying asset. The variational analysis involves weighted Sobolev spaces. It enables to prove qualitative properties of the solution, namely a maximum principle and additional regularity properties. Finally, we make numerical simulations of the solution, by finite element and finite difference methods.