Nonlinear anisotropic elliptic and parabolic equations in \(\mathbb R^{N}\) with advection and lower order terms and locally integrable data
The paper deals with existence and regularity of distributional solutions for nonlinear anisotropic elliptic equations of the form \[ -\sum_{\ell=1}^N {{\partial}\over{\partial x_\ell}}\left(\beta_\ell(x)\left| {{\partial u}\over{\partial x_\ell}}\right| ^{p_\ell-2} {{\partial u}\over{\partial x_\ell}}\right)-\text{div\,}g(u)+| u| ^{s-1}u=f\quad \text{in}\;\mathbb R^N, \] where \(N\geq2,\) \(\beta_\ell\colon\;\mathbb R^N\to \mathbb R\) are strictly positive and bounded function, \(g=(g_1,\ldots,g_N)\) is a continuous vector field with components growing like \(| u| ^{s-\eta}\) for \(s>1\) and some \(\eta\in(1,s),\) \(f\) is locally integrable function, and suitable restrictions are imposed on the exponents \(p_1,\ldots,p_N,s.\) Similar results are derived for the anisotropic parabolic equations as well. The technique employed relies on a~priori estimates for suitable approximate solutions \(\{u_\varepsilon\}_{0<\varepsilon<1}\) for which the standard variational framework works, and then passing to the limit when \(\varepsilon\to0.\)
scientific article; zbMATH DE number 2162294