Abstract
AbstractLetΩ⊂ ℝ2be a bounded domain with smooth boundary andb(x) > 0 a smooth function defined on∂Ω. We study the following Robin boundary value problem:$$\begin{array}{} \displaystyle \left\{ \begin{alignedat}{2} &{\it\Delta} u+u^p=0 &\quad& \text{in }{\it\Omega},\\ &u>0 &\quad& \text{in }{\it\Omega},\\ &\frac{\partial u}{\partial\nu} +\lambda b(x)u=0 &\quad& \text{on } \partial{\it\Omega}, \end{alignedat} \right. \end{array}$$whereνdenotes the exterior unit vector normal to∂Ω, 0 <λ< +∞ andp> 1 is a large exponent. We construct solutions of this problem which exhibit concentration asp→ +∞ and simultaneously asλ→ +∞ 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.
Highlights
Let Ω ⊂ R be a bounded domain with smooth boundary and b(x) > a smooth function de ned on ∂Ω
The rst result we will establish is that when p and λ satisfy some restrained growth conditions, for any m ≥, problem (1.1) possesses at least two solutions which concentrate at m di erent points with distance to the boundary uniformly approaching zero
When x is a non-degenerate critical point of the mean curvature function κ on ∂Ω, they nd that under a certain restrained condition of p and λ, problem (1.10) has a solution with a concentration point located at distance O( /λ) from x
Summary
Let Ω ⊂ R be a bounded domain with smooth boundary and b(x) > a smooth function de ned on ∂Ω. The rst result we will establish is that when p and λ satisfy some restrained growth conditions, for any m ≥ , problem (1.1) possesses at least two solutions which concentrate at m di erent points with distance to the boundary uniformly approaching zero. There exist p > and λ > such that for any p > p and λ > λ satisfying p log λ = o( ), problem (1.1) has at least two solutions up,λ with a concentration point ξp,λ such that. When x is a non-degenerate critical point of the mean curvature function κ on ∂Ω, they nd that under a certain restrained condition of p and λ, problem (1.10) has a solution with a concentration point located at distance O( /λ) from x.
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