Quasilinear equations and spaces of Campanato-Morrey type. (English) Zbl 0827.35021

The paper is, in a sense, a continuation of two of the author’s recent papers [J. Differ. Equ. 86, 102-122 (1990; Zbl 0707.35033); J.-M. Rakotoson and W. P. Ziemer, Trans. Am. Math. Soc. 319, 747-764 (1990; Zbl 0708.35023)]. The aim is, for the second-order equation \(Au + F(u, \nabla u) = T\), \(x \in \Omega\), where \(Au = - \sum^N_{i = 1} D_i a_i (x,u, \nabla u)\) with \(a_i\) and \(F\) satisfying certain growth conditions, to study the relation between the type of the weak solution \(u \in W^{1,p}_{\text{loc}} (\Omega)\), \(1 < p < \infty\), and the properties of the right-hand side \(T \in W^{- 1,p}_{\text{loc}} (\Omega)\), \(1/p + 1/q = 1\). According to the results of the first paper mentioned above, the right-hand side \(T\) is considered to belong to \(M^{- 1,q}_{\lambda, \text{loc}} (\Omega)\), \(\lambda > 0\), the so- called Morrey space in \(W^{- 1,q}_{\text{loc}} (\Omega)\), i.e., to satisfy an estimate \(\sup_{\rho > 0} \sup_{x \in \Omega} \rho^{- \lambda/q} |T |_{W^{- 1,q} (\Omega \cap B(x, \rho))} < \infty\), where \(B(x, \rho)\) is the ball with center \(x\) and radius \(\rho\). The main results are as follows:
(1) For every \(T \in M^{- 1,q}_{\lambda, \text{loc}} (\Omega)\), \(N > \lambda > N - p\), there exist functions \(f_i\) from the Morrey space \(L^{q, \lambda}_{\text{loc}} (\Omega)\) such that \(T = - \sum^N_{i = 1} D_i f_i\).
(2) If \(T \in M^{- 1,q}_{\lambda,q} (\Omega)\), \(N - p < \lambda \leq N\), and if \(a_i\), \(F\) satisfy suitable growth conditions, then the weak solution \(u\) is locally bounded. Moreover, if \(A\) is strongly monotone, then \(u \in C^{0, \alpha}_{\text{loc}} (\Omega)\).
(3) If \(Au = - \sum^n_{i,j = 1} D_i (a_{ij} (x,u) D_ju)\) with \(a_{ij} \in C^{0, \beta}_{ \text{loc}} (\Omega \times \mathbb{R})\), \(T \in H^{- 1}_{\text{loc}} (\Omega)\) and if the weak solution \(u\) belongs to \(H^1_{\text{loc}} (\Omega) \cap L^\infty_{\text{loc}} (\Omega)\), then \(u \in C^{1, \alpha}_{\text{loc}} (\Omega)\) if and only if \(T \in M^{- 1,2}_{\lambda, \text{loc}} (\Omega )\) for some \(\lambda \in (N,N + 2]\).
The last result yields the following corollary: Let \(\mu\) be a signed Radon measure whose total variation \(|\mu |\) satisfies the condition that, for every relatively compact subset \(\Omega'\) in \(\Omega\), there are positive numbers \(c\), \(\varepsilon\) such that \(|\mu |(B(x,r)) \leq cr^{N - 1 + \varepsilon}\) for all \(x\), \(r\) for which \(B(x,2r) \subset \Omega'\); then \(\mu \in M^{- 1,2}_{\lambda, \text{loc}} (\Omega)\) for some \(\lambda \in (N,N + 2]\).


35B65 Smoothness and regularity of solutions to PDEs
35J60 Nonlinear elliptic equations
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
35D10 Regularity of generalized solutions of PDE (MSC2000)
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