Nonlinear elliptic equations in \(\mathbb{R}^ N\) without growth restrictions on the data (Q689088)

The authors examine equations of the form (*) \(-\text{div } A(x,Du)+ g(x,u) =f\), where the principal part of the operator, \(-\text{div } A(x,Du)\) acts like the \(p\)-Laplacian. Provided \(ug(x,u)\) goes to infinity fast enough as \(| u|\to\infty\), they show that equation (*) has a distributional solution in \(\mathbb{R}^ N\) for any locally integrable function \(f\). In addition, the gradient of this solution, which is only assumed to be locally in \(L^{p-1}\), is locally in \(L^ q\) for some \(q>p-1\); the exponent \(q\) is given explicitly in terms of \(p\), \(N\), and the growth of \(g\).