Pointwise estimates of solutions and existence criteria for sublinear elliptic equations

I. E. Verbitsky

Matematičeskaâ fizika i kompʹûternoe modelirovanie, Tome 20 (2017) no. 3, pp. 18-33

Résumé

We give a survey of recent results on positive solutions to sublinear elliptic equations of the type $-Lu+ V \, u^{q}=f$, where $L$ is an elliptic operator in divergence form, $0$, $f\geq 0$ and $V$ is a function that may change sign, in a domain $\Omega \subseteq \mathbb{R}^{n}$, or in a weighted Riemannian manifold, with a positive Green's function $G$. We discuss the existence, as well as global lower and upper pointwise estimates of classical and weak solutions $u$, and conditions that ensure $u \in L^r(\Omega)$ or $u \in W^{1, p} (\Omega)$. Some of these results are applicable to homogeneous sublinear integral equations $ u = G(u^q d \sigma)$ in $\Omega,$ where $0$, and $\sigma=-V$ is a positive locally finite Borel measure in $\Omega$. Here ${G} (f \, d \sigma)(x) =\int_\Omega G(x, y), \, f(y) \, d \sigma(y)$ is an integral operator with positive (quasi) symmetric kernel $G$ on $\Omega \times \Omega$ which satisfies the weak maximum principle. This includes positive solutions, possibly singular, to sublinear equations involving the fractional Laplacian, $$ (-\Delta)^{\frac{\alpha}{2}} u = \sigma \, u^q, \quad u \ge 0 \quad \text{in} \, \, \Omega, $$ where $0$, $0 \alpha n$ and $u=0$ in $\Omega^c$ and at infinity in domains $\Omega \subseteq \mathbb{R}^{n}$ with positive Green's function $G$.

Détail

Keywords: sublinear elliptic equations, Green’s function, weak maximum principle, fractional Laplacian.