We are concerned with the existence of positive weak solutions, as well as the existence of bound states (i.e. solutions in W1, p (ℝN)), for quasilinear scalar field equations of the form

$$ - \Delta _pu + V(x) \vert u \vert ^{p - 2}u = K(x) \vert u \vert ^{q - 2}u + \vert u \vert ^{p^ * - 2}u,\qquad x \in {\open R}^N,$$

In this paper we are concerned with questions of multiplicity and concentration behaviorof positive solutions of the elliptic problem

$$ (P_{\var})\hspace*{4cm} \left\{ \begin{array}{rcl} \mathcal{L}_{\var}u=f(u) \ \ \mbox{in} \ \ \R^{3},\\[1.5mm] u>0 \ \ \mbox{in} \ \ \R^{3},\\[1.5mm] u \in H^{1}(\R^3), \end{array} \right. $$$\mathrm{\left(}{\mathit{P}}_{\mathit{\epsilon }}\mathrm{\right)}\left\{\begin{array}{c}\\ {\mathrm{ℒ}}_{\mathit{\epsilon }}\mathit{u}\mathrm{=}\mathit{f}\mathrm{\left(}\mathit{u}\mathrm{\right)}\mathrm{in}{\mathrm{IR}}^{\mathrm{3}}\mathit{,}\\ \mathit{u}\mathit{>}\mathrm{0}\mathrm{in}{\mathrm{IR}}^{\mathrm{3}}\mathit{,}\\ \mathit{u}\mathrm{\in }{\mathit{H}}^{\mathrm{1}}\mathrm{\left(}{\mathrm{IR}}^{\mathrm{3}}\mathrm{\right)}\mathit{,}\end{array}\right.$

where ε is a small positiveparameter, f : ℝ → ℝ is a continuous function, $$ \mathcal{L}_{\var} $$${\mathrm{ℒ}}_{\mathit{\epsilon }}$ is a nonlocaloperator defined by

$$ \mathcal{L}_{\var}u=M\left(\dis\frac{1}{\var}\int_{\R^{3}}|\nabla u|^{2}+\frac{1}{\var^{3}}\dis\int_{\R^{3}}V(x)u^{2}\right)\left[-\var^{2}\Delta u + V(x)u \right], $$${\mathrm{ℒ}}_{\mathit{\epsilon }}\mathit{u}\mathrm{=}\mathit{M}\left(\frac{\mathrm{1}}{\mathit{\epsilon }}{\mathrm{\int }}_{{\mathrm{IR}}^{\mathrm{3}}}\mathrm{|}\mathrm{\nabla }\mathit{u}{\mathrm{|}}^{\mathrm{2}}\mathrm{+}\frac{\mathrm{1}}{{\mathit{\epsilon }}^{\mathrm{3}}}{\mathrm{\int }}_{{\mathrm{IR}}^{\mathrm{3}}}\mathit{V}\mathrm{(}\mathit{x}\mathrm{)}{\mathit{u}}^{\mathrm{2}}\right)\mathrm{[}\mathrm{-}{\mathit{\epsilon }}^{\mathrm{2}}\mathrm{\Delta }\mathit{u}\mathrm{+}\mathit{V}\mathrm{(}\mathit{x}\mathrm{)}{\mathit{u}}^{\mathrm{]}}\mathit{,}$

M : IR+ → IR+ andV : IR3 → IR are continuous functions whichverify some hypotheses.

In this paper, we deal with a class of reflected backward stochastic differential equations (RBSDEs) corresponding to the subdifferential operator of a lower semi-continuous convex function, driven by Teugels martingales associated with a Lévy process. We show the existence and uniqueness of the solution for RBSDEs by means of the penalization method. As an application, we give a probabilistic interpretation for the solutions of a class of partial differential-integral inclusions.

In this paper we consider BSDEs with Lipschitzcoefficient reflected on two discontinuous (RCLL) barriers. In thiscase, we prove first the existence and uniqueness of the solution,then we also prove the convergence of the solutions of the penalizedequations to the solution of the RBSDE. Since the method used in thecase of continuous barriers (see Cvitanic and Karatzas, Ann. Probab.24 (1996) 2024–2056 and Lepeltier and San Martín, J. Appl. Probab.41 (2004) 162–175) does notwork, we develop a new method, by considering the solutions of thepenalized equations as the solutions of special RBSDEs and usingsome results of Peng and Xu in Annales of I.H.P.41 (2005) 605–630.

We prove the existence and uniqueness of the solution to certain reflected backward stochastic differential equations (RBSDEs) with one continuous barrier and deterministic terminal time, under monotonicity, and general increasing growth conditions on the associated coefficient. As an application, we obtain, in some constraint cases, the price of an American contingent claim as the unique solution of such an RBSDE.