Existence and uniqueness results for possibly singular nonlinear elliptic equations with measure data (Q723753)

In this paper, the authors consider the elliptic boundary value problem \[ \begin{cases} -\text{div}(a(x,\nabla u))=H(u)\mu \;\;\;&\text{in} \;\Omega,\\ u>0 & \text{in} \;\Omega,\\ u=0 & \text{on} \;\partial \Omega, \end{cases}\tag{P} \] where \(\Omega\) is an open bounded set in \(\mathbb{R}^N\) (\(N\geq 2\)), \(\mu\) is a nonnegative bounded Radon measure, \(a:\Omega\times \mathbb{R}^N\rightarrow \mathbb{R}^N\) is a Carathédory function and \(H:(0,+\infty)\rightarrow (0+\infty)\) is a continuous function, with \(\lim_{s\rightarrow \infty}H(s)<\infty\). The following conditions are assumed on the functions \(a,H\): there exist \(p\in (1+\infty)\), \(\alpha,\beta,C,s_0>0\), and \(\gamma\geq 0\) such that \(\begin{aligned} &(1) \;\;a(x,\xi)\xi\geq a |\xi|^p,\\ &(2) \;\;|a(x,\xi)|\leq \beta |\xi|^{p-1},\\ &(3) \;\;\langle a(x,\xi)-a(x,\xi'),\xi-\xi'\rangle>0, \\ &(4) \;\;H(s)\leq \frac{C}{s^\gamma}, \end{aligned} \) for all \(s\in (0,s_0)\), \(\xi,\xi'\in \mathbb{R}^N\), with \(\xi \neq \xi'\), and for a.e. \(x\in \Omega\). The measure \(\mu\) can be always decomposed as \(\mu=\mu_d+\mu_c\), where \(\mu_d\) is a diffuse measure with respect to the \(p\)-capacity and \(\mu_c\) is a measure concentrated on a set of zero \(p\)-capacity. In this paper, the diffusive measure \(\mu_d\) is assumed non-zero. After introducing the notion of renormalized solution and the notion of distributional solution of problem \((P)\), the authors, under the above assumptions, prove the existence of a distributional solution \(u\) having the following regularity: \(u^{p-1}\in L_{loc}^q\), for each \(q \in [1,\frac{N}{N-p})\), and \(|\nabla u|^{p-1}\in L_{loc}^q\), for each \(q \in [1,\frac{N}{N-1})\). By assuming the additional condition \(\gamma\leq 1\), under which a renormalized solution is also a distributional solution, the authors prove the existence of a renormalized solution \(u\) such that: \((i)\) if \(p>2-\frac{1}{N}\), then \ \(u\in W_0^{1,q}(\Omega)\) for each \(1\leq q<\frac{N(p-1)}{N-1}\), \((ii)\) if \(1<p\leq 2-\frac{1}{N}\) then \(u^{p-1}\in L^q(\Omega)\), for each \(q<\frac{N}{N-p}\), and \(|\nabla u|^{p-1}\in L^q(\Omega)\), for each \(q<\frac{N}{N-1}\). If, in addiction, \(H\) is non-increasing and \(\mu_c\) is non-zero, then the uniqueness of the renormalized solution is established. The method of the proof is as follows: The authors first obtain the above results in the non-singular case (i.e., for \(\gamma=0\)) using an approximation scheme where the measure \(\mu\) is replaced by a sequence \(\mu_n\) of bounded functions satisfying suitable convergence properties. Then, the extension to the singular case \(\gamma>0\) is carried out using again an approximation scheme, where \(H\) is replaced by its truncations at level \(n\) (\(n\in \mathbb{N}\)), and the measure \(\mu_c\) is replaced by a suitable sequence of bounded functions.