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The paper discusses uniqueness of solutions to stationary elliptic problems of the type $$\begin{aligned} A(u)+H(u)=f\in {\mathcal {D}}'(\Omega ), \end{aligned}$$ A ( u ) + H ( u ) = f ∈ D ′ ( Ω ) , where $$\Omega \ \in R^{N},\ $$ Ω ∈ R N , $$u\in W^{1,p}(\Omega )\ (1\le p\le +\infty ),\ A(u)\ $$ u ∈ W 1 , p ( Ω ) ( 1 ≤ p ≤ + ∞ ) , A ( u ) is an elliptic operator, $$H(u)\ $$ H ( u ) is an Hamiltonian that grows with $$\left| {\nabla u}\right| ^{p}$$ ∇ u 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 $$\le p\le 2.$$ ≤ p ≤ 2 . Unilateral problems are considered and the case where f depends on the solution u is also discussed.
Bollettino dell'Unione Matematica Italiana – Springer Journals
Published: Jun 23, 2017
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