Concentrating solutions for an exponential nonlinearity with Robin boundary condition

2015 ◽  
Vol 129 ◽  
pp. 294-319 ◽  
Author(s):  
Wenjing Chen ◽  
Shengbing Deng ◽  
Pablo Figueroa
Keyword(s):  
Boundary Condition ◽  
Robin Boundary Condition ◽  
Exponential Nonlinearity ◽  
Concentrating Solutions ◽  
Robin Boundary

scholarly journals Concentrating solutions for a planar elliptic problem with large nonlinear exponent and Robin boundary condition

2019 ◽  
Vol 8 (1) ◽  
pp. 1252-1285
Author(s):  
Yibin Zhang ◽  
Lei Shi
Keyword(s):  
Boundary Condition ◽  
Boundary Value Problem ◽  
Dirichlet Boundary ◽  
Smooth Boundary ◽  
Robin Boundary Condition ◽  
Unit Vector ◽  
Large Exponent ◽  
Concentrating Solutions ◽  
Robin Boundary ◽  
Robin Boundary Value Problem

Abstract Let Ω ⊂ ℝ2 be a bounded domain with smooth boundary and b(x) > 0 a smooth function defined on ∂Ω. We study the following Robin boundary value problem: $$\begin{array}{} \displaystyle \left\{ \begin{alignedat}{2} &{\it\Delta} u+u^p=0 &\quad& \text{in }{\it\Omega},\\ &u>0 &\quad& \text{in }{\it\Omega},\\ &\frac{\partial u}{\partial\nu} +\lambda b(x)u=0 &\quad& \text{on } \partial{\it\Omega}, \end{alignedat} \right. \end{array}$$ where ν denotes the exterior unit vector normal to ∂Ω, 0 < λ < +∞ and p > 1 is a large exponent. We construct solutions of this problem which exhibit concentration as p → +∞ and simultaneously as λ → +∞ at points that get close to the boundary, and show that in general the set of solutions of this problem exhibits a richer structure than the problem with Dirichlet boundary condition.

scholarly journals Solutions for parametric double phase Robin problems

2021 ◽  
Vol 121 (2) ◽  
pp. 159-170 ◽  
Author(s):  
Nikolaos S. Papageorgiou ◽  
Calogero Vetro ◽  
Francesca Vetro
Keyword(s):  
Boundary Condition ◽  
Existence Theorems ◽  
Robin Boundary Condition ◽  
Phase Problem ◽  
Double Phase ◽  
Robin Boundary ◽  
Double Phase Problem

We consider a parametric double phase problem with Robin boundary condition. We prove two existence theorems. In the first the reaction is ( p − 1 )-superlinear and the solutions produced are asymptotically big as λ → 0 + . In the second the conditions on the reaction are essentially local at zero and the solutions produced are asymptotically small as λ → 0 + .

scholarly journals Multiple solutions to a quasilinear Schrödinger equation with Robin boundary condition

2020 ◽  
Vol 5 (4) ◽  
pp. 3825-3839
Author(s):  
Yin Deng ◽  
◽  
Gao Jia ◽  
Fanglan Li ◽  
◽  
...  
Keyword(s):  
Boundary Condition ◽  
Multiple Solutions ◽  
Robin Boundary Condition ◽  
Dinger Equation ◽  
Robin Boundary
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