Boundary trace of positive solutions of semilinear elliptic equations in Lipschitz domains: the subcritical case

Marcus, Moshe; Veron, Laurent

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 10 (2011) no. 4, pp. 913-984.

We study the generalized boundary value problem for nonnegative solutions of $-\Delta u+g\left(u\right)=0$ in a bounded Lipschitz domain $\Omega$, when $g$ is continuous and nondecreasing. Using the harmonic measure of $\Omega$, we define a trace in the class of outer regular Borel measures. We amphasize the case where $g\left(u\right)={\left|u\right|}^{q-1}u$, $q>1$. When $\Omega$ is (locally) a cone with vertex $y$, we prove sharp results of removability and characterization of singular behavior. In the general case, assuming that $\Omega$ possesses a tangent cone at every boundary point and $q$ is subcritical, we prove an existence and uniqueness result for positive solutions with arbitrary boundary trace.

Classification : 35K60, 31A20, 31C15, 44A25, 46E35

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