On elliptic equations with singular potentials and nonlinear boundary conditions

Ferreira, Lucas; Neves, Sérgio

doi:10.1090/qam/1506

Authors: Lucas C. F. Ferreira and Sérgio L. N. Neves
Journal: Quart. Appl. Math. 76 (2018), 699-711
MSC (2010): Primary 35J15, 35J65, 35J91, 35J75, 35A01
DOI: https://doi.org/10.1090/qam/1506
Published electronically: May 29, 2018
MathSciNet review: 3855827
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Abstract: We consider the Laplace equation in the half-space satisfying a nonlinear Neumann condition with boundary potential. This class of problems appears in a number of mathematical and physics contexts and is linked to fractional dissipation problems. Here the boundary potential and nonlinearity are singular and of power-type, respectively. Depending on the degree of singularity of potentials, first we show a nonexistence result of positive solutions in $\mathcal {D}^{1,2}(\mathbb {R}^n_+)$ with a $L^p$-type integrability condition on $\partial \mathbb {R}^n_{+}$. After, considering critical nonlinearities and conditions on the size and sign of potentials, we obtain the existence of positive solutions by means of minimization techniques and perturbation methods.

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