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

Uniqueness of solutions to some quasilinear elliptic equations whose Hamiltonian has natural... 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bollettino dell'Unione Matematica Italiana Springer Journals

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

Bollettino dell'Unione Matematica Italiana, Volume 11 (2) – Jun 23, 2017
17 pages

/lp/springer_journal/uniqueness-of-solutions-to-some-quasilinear-elliptic-equations-whose-Mk7QFxUv4E
Publisher
Springer Journals
Subject
Mathematics; Mathematics, general
ISSN
1972-6724
eISSN
2198-2759
DOI
10.1007/s40574-017-0130-4
Publisher site
See Article on Publisher Site

### 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} 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.

### Journal

Bollettino dell'Unione Matematica ItalianaSpringer Journals

Published: Jun 23, 2017

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