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Uniqueness and the maximum principle for quasilinear elliptic equations with quadratic growth conditions. (English) Zbl 0859.35031
This paper is devoted to the study of uniqueness properties for quasilinear uniformly elliptic equations with quadratic growth conditions: \[ -\text{div}(a(x,v,\nabla v))+ b(x,v,\nabla v)=f(x)\quad\text{in }\Omega, \] where \(f\in H^{-1}(\Omega)\) and \(\Omega\) is a bounded domain in \(\mathbb{R}^N\). In order to prove the uniqueness of the solution of the boundary value problems the authors establish the comparison principle. Under some structure condition of the equation involving the first Dirichlet eigenvalue of \(-\Delta\), the authors prove the comparison principle for the solutions \(v\in H^1(\Omega)\cap L^\infty(\Omega)\) satisfying \(a(x,v,\nabla v)\in (L^2(\Omega))^N\) and \(b(x,v,\nabla v)\in L^1(\Omega)\). Therefore, if the natural growth conditions on \(a\) and \(b\), namely, \[ |a(x,v,\xi)|\leq C(|v|)(1+|\xi|)\quad\text{and}\quad |b(x,v,\xi)|\leq C(|v|)(1+|\xi|^2) \] are assumed, the comparison principle holds for the solutions \(v\in H^1(\Omega)\cap L^\infty(\Omega)\). As its applications, there are two typical comparison results: One is the comparison principle for the solutions \(u\in H^1(\Omega)\cap L^\infty(\Omega)\) of the equation \[ -\Delta u+H(x,u,\nabla u)=0\quad\text{in}\quad\Omega \] with \({\partial H\over \partial u}(x,u,p)\geq \alpha_0\geq 0\) and \(|{\partial H\over \partial p}(x,u,p)|\leq C(1+|p|)\), and the other is the comparison principle for the solutions \(u\in H^1(\Omega)\) with \(g(u)|\nabla u|^2\in L^1(\Omega)\) of the equation \[ -\Delta u+g(u)|\nabla u|^2=0\quad\text{in}\quad\Omega \] with \(f\in H^{-1}(\Omega)\), \(g'(\cdot)>0\), and \(g(0)=0\). In both cases, the method of proof consists in making a certain change of function \(u=\varphi(v)\) and in proving that the transformed equation satisfies the structure condition of the main comparison principle.
Existence of such solutions has been obtained in L. Boccardo, the second author and J. P. Puel [Port. Math. 41, 507-534 (1982; Zbl 0544.35040)] and in A. Bensoussan, L. Boccardo and the second author [Ann. Inst. H. Poincaré, Anal. Nonlin. 5, No. 4, 347-364 (1988; Zbl 0696.35042)].

MSC:
35J65 Nonlinear boundary value problems for linear elliptic equations
35B50 Maximum principles in context of PDEs
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
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[1] A. Bensoussan, L. Boccardo & F. Murat: On a nonlinear partial differential equation having natural growth terms and unbounded solution. Ann. Inst. H. Poincaré, Anal, non linéaire, 5 (1988), 347-364. · Zbl 0696.35042
[2] L. Boccardo, F. Murat & J.-P. Puel: Existence de solutions non bornés pour certaines équations quasi-linéaires. Portugaliae Math., 41 (1982), 507-534. · Zbl 0524.35041
[3] L. Boccardo, F. Murat & J.-P. Puel: Existence de solutions faibles pour des équations elliptiques quasi-linéaires à croissance quadratique. Nonlinear Partial Differential Equations and their Applications, Collège de France Seminar, Volume IV, H. Brezis & J.-L. Lions, editors, Research Notes in Mathematics, 84, Pitman, London (1983), 19-73.
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[6] O. A. Lady?enskaja & N. N. Ural’ceva: Équations aux derivées partielles de type elliptique. Dunod, Paris (1968).
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