A nonsmooth critical point theory approach to some nonlinear elliptic equations in \({\mathbb R}^n\) (Q5933735)

A quasilinear elliptic equation NEWLINE\[NEWLINE - \sum^{n}_{i,j=1} D_{j}\bigl(a_{ij}(x,u)D_{i}u\bigr) + \frac 12\sum^{n}_{i,j=1} {\partial a_{ij}\over\partial u} (x,u)D_{i}uD_{j}u + b(x)u = f(x,u) \tag{1} NEWLINE\]NEWLINE in \(\mathbb R^{n}\) is studied. Suppose that \(b\in L^\infty_{\text{loc}} (\mathbb R^{n})\) and \(b\geq b_{0}>0\) almost everywhere for a constant \(b_0\); define a Hilbert space \(E\) of all functions \(u:\mathbb R^{n} \to \mathbb R\) such that \(\|u\|^2_{E} = \int_{\mathbb R^{n}} (|Du|^2 + b(x)|u|^2) \roman dx <\infty\). Let \(f\) be a subcritical Carathéodory function. Under suitable hypotheses on the functions \(a_{ij}\) and \(f\), existence of a nontrivial nonnegative weak solution \(u\in E\) to (1) is proven by means of nonsmooth critical point theory. Furthermore, right hand sides \(f\) such that \(f(x,\cdot)\) is not continuous are considered if \(a_{ij} = \delta_{ij}\).