Entropy solutions of Dirichlet problem for a class of nonlinear elliptic fourth order equations with \(L_1\)-data (Q2735861)

The author deals with the Dirichlet problem for the equation NEWLINE\[NEWLINE\sum_{|\alpha|= 1,2} (-1)^{|\alpha|} D^\alpha A_\alpha (x,\nabla_2 u)= F(x,u) \quad\text{in }\Omega,NEWLINE\]NEWLINE where \(\Omega\) is a bounded open set of \(\mathbb{R}^n\), \(n> 2\); \(F: \Omega\times \mathbb{R}\to \mathbb{R}\) is a Carathéodory function satisfying the conditions: for almost every \(x\in \Omega\) the function \(F(x,\cdot)\) is nonincreasing in \(\mathbb{R}\) and for every \(s\in \mathbb{R}\) the function \(F(\cdot, s)\) belongs to \(L^1(\Omega)\); \(\nabla _2u= \{D^\alpha u:|\alpha|= 1,2\}\); \(A_\alpha: \Omega\times \mathbb{R}^{n,2}\to \mathbb{R}\), \(|\alpha|= 1,2\), are Carathéodory functions satisfying some growth and monotonicity conditions and the following ellipticity condition: for almost every \(x\in \Omega\) and every \(\xi\in \mathbb{R}^{n,2}\), NEWLINE\[NEWLINE\sum_{|\alpha|=1,2} A_\alpha (x,\xi) \xi_\alpha\geq c \Bigl\{ \sum_{|\alpha|=1}|\xi_\alpha|^q+ \sum_{|\alpha|=2}|\xi_\alpha|^p \Bigr\}- g(x)NEWLINE\]NEWLINE with a positive constant \(c\) and a nonnegative function \(g\in L^1(\Omega)\) \((\mathbb{R}^{n,2}\) denotes the space of all sets \(\xi= \{\xi_\alpha: |\alpha|= 1,2\}\) of real numbers). It is supposed that \(1< p< n/2\) and \(2p< q< n\). The author introduces some notions of solutions of the problem under consideration and states results on the existence as well as the uniqueness of an entropy solution and on the existence of so-called \(H\)- and \(W\)-solutions of this problem. In general the author follows the approach of the paper of \textit{Ph. Bénilan, L. Boccardo, Th. Gallouët, R. Gariepy, M. Pierre}, and \textit{J. L. Vazquez} [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 22, 241-273 (1995; Zbl 0866.35037)]. However the realization of this approach is carried out by means of delicate enough considerations connected with some particularities of higher-order equations in comparison with second-order ones.