Nonlinear elliptic equations and intrinsic potentials of Wolff type (Q333115)

The authors discuss existence of weak solutions for the problems NEWLINE\[NEWLINE\displaystyle -\Delta_p u = \sigma u^qNEWLINE\]NEWLINE in \(\mathbb{R}^n\), \(\liminf_{x\to\infty} u(x) = 0\), \(u>0\), with \(0<q<p-1\) and \(\sigma\) a locally integrable function, or locally finite Borel measure.NEWLINENEWLINEClosely related are the problems NEWLINE\[NEWLINE\displaystyle -\Delta_p v = b\frac{|\nabla v|^p}{v}+\sigma,NEWLINE\]NEWLINE \(\liminf_{x\to\infty} v(x) = 0,\) \(v>0\), where \(\displaystyle b = \frac{q(p-1)}{p-1-q}\), \ \ \(0<q<p-1\).NEWLINENEWLINEIf for a ball \(B\) one defines \(\varkappa(B) \) as the best constant in NEWLINE\[NEWLINE\left(\int_B |\varphi|^q \, d \sigma\right)^{\frac 1 q} \leq \varkappa(B) \, ||\Delta \varphi||^\frac{1}{p-1}_{L^1(\mathbb{R}^n)},NEWLINE\]NEWLINE for all \(\varphi \in C^2(\mathbb{R}^n)\) vanishing at infinity such that \(-\Delta \varphi \geq 0\), then one can define the nonlinear potential NEWLINE\[NEWLINE {\mathbf K} \sigma (x) = \int_0^{\infty} \left[\frac{ \varkappa(B(x, r))^{\frac{q(p-1)}{p-1-q}}}{r^{n- p}}\right]^{\frac{1}{p-1}}\frac{dr}{r}, \quad x \in \mathbb{R}^n.NEWLINE\]NEWLINE Define also the Wolff potential NEWLINE\[NEWLINE{\mathbf W} \sigma(x) =\int_0^{\infty} \left[\frac{\sigma(B(x,r))}{r^{n-p}}\right]^{\frac{1}{p-1}}\frac{dr}{r}, \quad x \in \mathbb{R}^n.NEWLINE\]NEWLINE A sample result of this paper is that under finiteness conditions on the \(\mathbf{K}\) and \({\mathbf W}\) potentials, there exists a solution \(u\) of the first equation such that for some \(c>0\) (a constant which depends only on \(p\), \(q\), and \(n\)), one has NEWLINE\[NEWLINE c^{-1} \left [{\mathbf K}\sigma + \left({\mathbf W} \sigma\right)^{\frac{p-1}{p-1-q}}\right] \leq u \leq c \left [{\mathbf K} \sigma + \left({\mathbf W} \sigma\right)^{\frac{p-1}{p-1-q}}\right]. NEWLINE\]NEWLINE The results are then extended to operators more general than the \(p\)-Laplace operator.