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Liouville Type Theorems for Two Mixed Boundary Value Problems with General Nonlinearities
Analysis of PDEs · Submitted 20 Oct 2014
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We will prove that these problems possess no positive solution under some assumptions on the nonlinear terms.
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The abstract, sentence by sentence
In this paper, we study the nonexistence of positive solutions for the following two mixed boundary value problems. The first problem is the mixed nonlinear-Neumann boundary value problem $$ \left\{ \begin{array}{ll} \displaystyle -\Delta u=f(u) &{\rm in}\quad \R, \\ \displaystyle \\ \frac{\partial u}{\partial \nu}=g(u) &{\rm on}\quad \Gamma_1,\\ \displaystyle \\ \frac{\partial u}{\partial \nu}=0 &{\rm on}\quad \Gamma_0 \end{array} \right. $$ and the second is the nonlinear-Dirichlet boundary value problem $$ \left\{ \begin{array}{ll} \displaystyle -\Delta u=f(u) &{\rm in}\quad \R, \\ \displaystyle \\ \frac{\partial u}{\partial \nu}=g(u) &{\rm on}\quad \Gamma_1,\\ \displaystyle \\ u=0 &{\rm on}\quad \Gamma_0, \end{array} \right. $$ where $\R=\{x\in \mathbb R^N:x_N>0\}$, $\Gamma_1=\{x\in \mathbb R^N:x_N=0,x_1<0\}$ and $\Gamma_0=\{x\in \mathbb R^N:x_N=0,x_1>0\}$. We will prove that these problems possess no positive solution under some assumptions on the nonlinear terms. The main technique we use is the moving plane method in an integral form.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | In this paper, we study the nonexistence of positive solutions for the following two mixed boundary value problems. | contribution 0.98 | 0.86 | 0.06 | asserted 0.68 |
| s1 | The first problem is the mixed nonlinear-Neumann boundary value problem $$ \left\{ \begin{array}{ll} \displaystyle -\Delta u=f(u) &{\rm in}\quad \R, \\ \displaystyle \\ \frac{\partial u}{\partial \nu}=g(u) &{\rm on}\quad \Gamma_1,\\ \displaystyle \\ \frac{\partial u}{\partial \nu}=0 &{\rm on}\quad \Gamma_0 \end{array} \right. | contribution 0.59 | 0.42 | 0.03 | asserted 0.74 |
| s2 | $$ and the second is the nonlinear-Dirichlet boundary value problem $$ \left\{ \begin{array}{ll} \displaystyle -\Delta u=f(u) &{\rm in}\quad \R, \\ \displaystyle \\ \frac{\partial u}{\partial \nu}=g(u) &{\rm on}\quad \Gamma_1,\\ \displaystyle \\ u=0 &{\rm on}\quad \Gamma_0, \end{array} \right. | contribution 0.55 | 0.38 | 0.03 | asserted 0.67 |
| s3 | $$ where $\R=\{x\in \mathbb R^N:x_N>0\}$, $\Gamma_1=\{x\in \mathbb R^N:x_N=0,x_1<0\}$ and $\Gamma_0=\{x\in \mathbb R^N:x_N=0,x_1>0\}$. | background 0.93 | 0.24 | 0.02 | asserted 0.88 |
| s4 | We will prove that these problems possess no positive solution under some assumptions on the nonlinear terms. | result 0.96 | 0.93 | 0.95 | asserted 0.99 |
| s5 | The main technique we use is the moving plane method in an integral form. | method 1.00 | 0.88 | 0.02 | asserted 1.00 |
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Original abstract
In this paper, we study the nonexistence of positive solutions for the following two mixed boundary value problems. The first problem is the mixed nonlinear-Neumann boundary value problem $$ \left\{ \begin{array}{ll} \displaystyle -\Delta u=f(u) &{\rm in}\quad \R, \\ \displaystyle \\ \frac{\partial u}{\partial \nu}=g(u) &{\rm on}\quad \Gamma_1,\\ \displaystyle \\ \frac{\partial u}{\partial \nu}=0 &{\rm on}\quad \Gamma_0 \end{array} \right. $$ and the second is the nonlinear-Dirichlet boundary value problem $$ \left\{ \begin{array}{ll} \displaystyle -\Delta u=f(u) &{\rm in}\quad \R, \\ \displaystyle \\ \frac{\partial u}{\partial \nu}=g(u) &{\rm on}\quad \Gamma_1,\\ \displaystyle \\ u=0 &{\rm on}\quad \Gamma_0, \end{array} \right. $$ where $\R=\{x\in \mathbb R^N:x_N>0\}$, $\Gamma_1=\{x\in \mathbb R^N:x_N=0,x_1<0\}$ and $\Gamma_0=\{x\in \mathbb R^N:x_N=0,x_1>0\}$. We will prove that these problems possess no positive solution under some assumptions on the nonlinear terms. The main technique we use is the moving plane method in an integral form.
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