This paper proves the existence of (at least) one solution of the following variational inequality:
 (*)
 
$$\left\{ {\begin{array}{*{20}c} {u \in W_0^{1,v} (\Omega ) \cap L^\infty (\Omega ), u \geqslant \psi a.e. in \Omega ,} \\ {\langle A(u),\upsilon - u\rangle + \int\limits_\Omega {{\rm H}(x,u,Du)(\upsilon - u) \geqslant 0,} } \\ {\forall \upsilon \in W_0^{1,v} (\Omega ) \cap L^\infty (\Omega ), u \geqslant \psi a.e. in \Omega .} \\ \end{array} } \right.$$

 Here A is an operator of Leray-Lions type acting from W
 0
 1,p
 (Ω) into W−1,p′(Ω) and H grows like ¦Du¦p. The obstacle ψ is a measurable function with values in
 
$$\bar R$$

 , the only hypothesis being
 
$$\{ \upsilon \in W_0^{1,v} (\Omega ) \cap L^\infty (\Omega ):\upsilon \geqslant \psi a.e in \Omega \} \ne 0/$$

 . This allows ψ to be −∞, recovering the case where (*) is an equation. Finally there is no smoothness assumptions on the data: Ω is a bounded open set inR

 N
 , A and H are defined from Caratheodory functions.

https://link.springer.com/content/pdf/10.1007/BF01766148.pdf

Existence of bounded solutions for non linear elliptic unilateral problems

We prove the existence of a solution of the nonlinear elliptic equation: A(u) + g(x, u, Du) = h(x), where A is a Leray-Lions operator from W 0 1 , p ( Ω ) into W−1, p′(Ω) and g is a nonlinear term with “natural” growth with respect to Du [i.e. such that |g(x, u, ξ)| ≦ b(|u|) (|ξ|p + c(x))], satisfying the sign condition g(x, u, ξ)u ≧ 0 but no growth condition with respect to u. Here h belongs to W−1, p′(Ω), thus the solution u of the problem does not in general be more smooth than W 0 1 , p ( Ω ) . The existence of a solution is also proved for the corresponding obstacle problem.

/pdf/on-a-non-linear-partial-differential-equation-having-natural-55sbxeoh7v.pdf

On a non linear partial differential equation having natural growth terms and unbounded solution

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/pdf/resultats-d-existence-pour-certains-problemes-elliptiques-1opi9t2x0i.pdf

Résultats d'existence pour certains problèmes elliptiques quasilinéaires

Introduction The semilinear elliptic equation −∆u = g(x, u) has been the object of several studies in the last twenty years. For instance, let us mention the well-known result proved by Ambrosetti and Rabinowitz (cf. [1]): if g is superlinear and odd with respect to the second variable, then the above equation has a sequence of solutions uh ∈ H 0 (Ω) with ‖uh‖H1 0 →∞. In order to get such a result, a variational technique has been employed. In fact, the above equation is the Euler equation associated with the functional f(u) = 1 2 ∫

/pdf/multiplicity-of-solutions-for-quasilinear-elliptic-equations-1xp8o5v5xz.pdf

Multiplicity of solutions for quasilinear elliptic equations

We consider in this paper the limit behavior of the solutionsuɛ of the problem

$$\begin{gathered} - div(a^\varepsilon Du^\varepsilon ) + \gamma u^\varepsilon = H^\varepsilon (x, u^\varepsilon , Du^\varepsilon ), \hfill \\ u^\varepsilon \in H_0^1 (\Omega ) \cap L^\infty (\Omega ), \hfill \\ \end{gathered}$$

whereHɛ has quadratic growth inDuɛ andaɛ(x) is a family of matrices satisfying the general assumptions of abstract homogenization. We also consider the problem

$$\begin{gathered} - div(a^\varepsilon Du^\varepsilon ) + G^\varepsilon (x, u^\varepsilon , Du^\varepsilon ) = f \in H^{ - 1} (\Omega ), \hfill \\ u^\varepsilon \in H_0^1 (\Omega ), G^\varepsilon (x, u^\varepsilon , Du^\varepsilon ) \in L^1 (\Omega ), u^\varepsilon G^\varepsilon (x, u^\varepsilon , Du^\varepsilon ) \in L^1 (\Omega ) \hfill \\ \end{gathered}$$

whereGɛ has quadratic growth inDuɛ and satisfiesGɛ(x, s, ξ)s ≥ 0. Note that in this last modeluɛ is in general unbounded, which gives extra difficulties for the homogenization process. In both cases we pass to the limit and obtain an homogenized equation having the same structure.

U. Mosco

Papers

A stochastic impulse control problem with quadratic growth Hamiltonian and the corresponding quasi variational inequality