Semilinear elliptic problems on the half space with a supercritical nonlinearity

Sho Katayama

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arXiv · 2310.17001

Semilinear elliptic problems on the half space with a supercritical nonlinearity

Sho Katayama

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Abstract

This paper concerns positive solutions to the boundary value problems of the scalar field equation in the half space with a Sobolev supercritical nonlinearity and an inhomogeneous Dirichlet boundary condition, admitting a nontrivial nonnegative Radon measure as the boundary data. Under a suitable integrability assumption on the boundary data and the Joseph--Lundgren subcritical condition on the nonlinear term, we give a complete classification of the existence/nonexistence of a positive solution with respect to the size of the boundary data. Furthermore, we give a result on the existence of multiple positive solutions via bifurcation theory.

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Sho Katayama. 2023-10-25. Semilinear elliptic problems on the half space with a supercritical nonlinearity. https://arxiv.org/abs/2310.17001

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