Some nonlinear elliptic systems with right-hand side integrable data with respect to the distance to the boundary
Let \(\Omega\) be a bounded smooth open set in \(\mathbb{R}^N\), \(N\geq 2\), let \(\delta\) be the distance from the boundary of \(\Omega\) and let \(m\in (1,\infty)\). The author considers the following boundary value problem for an elliptic nonlinear system \[ -\Delta (|u|^{m-1}u_i)=f_i\quad\text{in }\Omega,\quad u_i=0\quad\text{on }\partial \Omega,\quad i=1,\dots,d,\tag{P} \] where \(f=(f_1,f_2,\dots,f_d)\) is a measurable function which belongs to the weighted space \(L^1(\Omega,\delta,\mathbb{R}^N)\). The author proves that problem \((P)\) admits a unique very weak solution \(u\) such that \(|u|^{m}\in W_0^{1,q}(\Omega,\delta)\cap L^{\bar{q}}(\Omega,\delta)\cap L^{q_0}(\Omega)\), with \[ 1\leq q<\frac{N+1}{N},\quad 1\leq \bar{q}<\frac{N+1}{N-1},\quad 1\leq q_0<\frac{N}{N-1},\tag{1} \] and establishes an upper estimate of the norms of \(|u|^m\) in the spaces \(W_0^{1,q}(\Omega,\delta)\), \(L^{\bar{q}}(\Omega,\delta)\) and \(L^{q_0}(\Omega)\) in terms of the norm of \(f\) in the space \(L^1(\Omega,\delta,\mathbb{R}^N)\). The author also investigates the higher regularity of the solution \(u\) in dependence of the higher summability of the data \(f\). In particular, he proves that: -- if each \(f_i\) is in the Orlicz space generated by the function \(|s|\log(1+|s|)\) with weight function \(\delta\), then the exponents \(q,\bar{q},q_0\) are allowed to take their upper bounds in conditions (1). -- the upper bounds in conditions (1) can be relaxed if \(f\in L^p(\Omega,\delta,\mathbb{R}^N)\), where \(p>1\). In particular, if \(p>\frac{N+1}{2}\), then \(q_0\) is allowed to take the value \(+\infty\). \noindent Finally, the author proves the existence and uniqueness of very weak solutions under the assumption \(f\in L^1(\Omega,\delta^\alpha,\mathbb{R}^n)\), where \(\alpha\in (0,1)\). In this case, the author proves that the solution \(u\) has the following regularity: \(|u|^m\in W_0^{1,q}(\Omega,\delta)\cap L^{\bar{q}}(\Omega,\delta)\cap W_0^{1,\tilde{q}}(\Omega)\), with \[ 1\leq q<\frac{N+\alpha}{N-1+\alpha},\quad 1\leq \bar{q}<\frac{N+\alpha}{N-2+\alpha},\quad 1\leq q_0<\frac{N}{N-1+\alpha},\tag{1'} \] and establishes an upper estimate of the norms of \(|u|^m\) in the spaces \(W_0^{1,q}(\Omega,\delta)\), \(L^{\bar{q}}(\Omega,\delta)\) and \(W_0^{1,\tilde{q}}(\Omega)\) in terms of the norm of \(f\) in the space \(L^1(\Omega,\delta^{\alpha},\mathbb{R}^N)\). \noindent The proof is based on an approximation argument.
scientific article; zbMATH DE number 6376248
scientific article