An upper bound for positive solutions of the equation Δ𝑢=𝑢^{𝛼}

Kuznetsov, S.¹

¹ Department of Mathematics, University of Colorado, Boulder, CO 80309-0395

Electronic research announcements of the American Mathematical Society, Tome 10 (2004), pp. 103-112.

Résumé

In 2002 Mselati proved that every positive solution of the equation $\Delta u=u^2$ in a bounded domain of class $C^4$ is the limit of an increasing sequence of moderate solutions. (A solution is called moderate if it is dominated by a harmonic function.) As a part of his proof, he established an upper bound (in terms of the capacity of $K$) for solutions vanishing off a compact subset $K$ of $\partial E$. We use a different kind of capacity (we call it the Poisson capacity) and we establish in terms of this capacity an upper bound for solutions of $\Delta u=u^\alpha$ with $1\alpha \le 2$. This is a part of the program: to classify all positive solutions of this equation.

Détail

DOI : 10.1090/S1079-6762-04-00135-0

Bibliographie

[1] Dynkin, E. B. Diffusions, superdiffusions and partial differential equations 2002

[2] Kuznetsov, S. E. Polar boundary sets for superdiffusions and removable lateral singularities for nonlinear parabolic PDEs Comm. Pure Appl. Math. 1998 303 340