An elliptic semilinear equation with source term and boundary measure data: the supercritical case

Abstract

We give new criteria for the existence of weak solutions to an equation with a super linear source term \begin{align*}-Δu = u^q ~~\text{in}~Ω,~~u=σ~~\text{on }~\partialΩ\end{align*}where $\Omega$ is a either a bounded smooth domain or $\mathbb{R}\_{+}^N$, $q\textgreater{}1$ and $\sigma \in {\mathfrak{M}}^{+}(\partial \Omega )$ is a nonnegative Radon measure on $\partial \Omega$. One of the criteria we obtain is expressed in terms of some Bessel capacities on $\partial \Omega$. We also give a sufficient condition for the existence of weak solutions to equation with source mixed terms. \begin{align*} -Δu = |u|^{q\_1-1}u|\nabla u|^{q\_2} ~~\text{in}~Ω,~~u=σ~~\text{on }~\partialΩ\end{align*} where $q\_1,q\_2\geq 0, q\_1+q\_2\textgreater{}1, q\_2\textless{}2$, $\sigma \in \mathfrak{M}(\partial \Omega )$ is a Radon measure on $\partial \Omega$.

Journal of Functional Analysis 269 (2015) 1995--2017

References in corpus (2)

• Positive solutions of quasilinear elliptic equations with subquadratic growth in the gradient
• Potential estimates and quasilinear parabolic equations with measure data

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• An optimal Hardy-Littlewood-Sobolev inequality on ${\mathbf{R}}^{n-k}\times {\mathbf{R}}^n$ and its consequences
• Isolated singularities for elliptic equations with Hardy operator and source nonlinearity
• Boundary value problem with measures for fractional elliptic equations involving source nonlinearities