Uniqueness of solutions to some quasilinear elliptic equations whose Hamiltonian has natural growth in the gradient

Abstract

The paper discusses uniqueness of solutions to stationary elliptic problems of the type $$\begin{aligned} A(u)+H(u)=f\in {\mathcal {D}}'(\Omega ), \end{aligned}$$ where $$\Omega \ \in R^{N},\ $$ $$u\in W^{1,p}(\Omega )\ (1\le p\le +\infty ),\ A(u)\ $$ is an elliptic operator, $$H(u)\ $$ is an Hamiltonian that grows with $$\left| {\nabla u}\right| ^{p}$$ and f is given. Methods introduced in Artola (Boll UMI 6(5-B):51–71, 1986), (Proceedings of the International Conference on Generalized Functions, (ICGF 2000). Cambridge Scientific Publishers, Cambridge, 51–92, 2004), (Ricerche di Matematica XLIV, fasc. 2:400–420, 1995) for quasilinear parabolic or elliptic equations, together with properties for some continuity moduli, are used to improve some results from Barles and Murat (Arch Ration Mech Anal 133(1):77–101, 1995) for bounded solutions and from Barles and Porretta (Ann Scuola Norm Sup Pisa Cl Sci 5(1):107–136, 2006), Lions (J Anal Math 45: 234–254, 1985) for unbounded solutions, when 1 $$\le p\le 2.$$ Unilateral problems are considered and the case where f depends on the solution u is also discussed.