Elliptic problems in the half-space with nonlinear critical boundary conditions

Marcelo Fernandes Furtado () and Karla Carolina Vicente de Sousa
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Marcelo Fernandes Furtado: University of Brasília
Karla Carolina Vicente de Sousa: University of Brasília

Partial Differential Equations and Applications, 2021, vol. 2, issue 6, 1-16

Abstract: Abstract We obtain multiple solutions for the nonlinear boundary value problem $$\begin{aligned} -\Delta u-\dfrac{1}{2}\left( x\cdot \nabla u\right) = f(u), \text{ in } {\mathbb {R}}_{+}^{N}, \qquad \dfrac{\partial u}{\partial \eta }= \beta |u|^{2/(N-2)}u, \text{ on } \partial {\mathbb {R}}_{+}^{N}, \end{aligned}$$ - Δ u - 1 2 x · ∇ u = f ( u ) , in R + N , ∂ u ∂ η = β | u | 2 / ( N - 2 ) u , on ∂ R + N , where $${\mathbb {R}}^N_+ = \{(x',x_N) \in {\mathbb {R}}^N_+ : x' \in {\mathbb {R}}^{N-1},\,x_N>0 \}$$ R + N = { ( x ′ , x N ) ∈ R + N : x ′ ∈ R N - 1 , x N > 0 } , $$\frac{\partial u}{\partial \eta }$$ ∂ u ∂ η is the partial outward normal derivative, $$\beta >0$$ β > 0 is a parameter and f is a superlinear function with subcritical growth.

Keywords: Nonlinear boundary conditions; Critical trace problems; Half-space; Self-similar solutions; Symmetric functionals; Primary 35J60; Secondary 35B33 (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1007/s42985-021-00137-0

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