• Positive solutions with single and multi-peak for semilinear elliptic equations with nonlinear boundary condition in the half-space

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    • Keywords


      Elliptic equation; multi-peak solutions; singular perturbation; nonlinear boundary condition

    • Abstract


      We consider the existence of single and multi-peak solutions of thefollowing nonlinear elliptic Neumann problem \begin{align*} &\left\{ \begin{array}{1} -\Delta u + \lambda^{2} u = Q(x)|u|^{p-2}u & {\rm in}\quad {\mathbb R}^N_+,\\ \frac{\partial u}{\partial n} = f(x,u) & {\rm on} \quad \partial {\mathbb R}^N_+, \end{array} \right. \end{align*} where $\lambda$ is a large number, $p \in (2, \frac{2N}{N−2})$ for $N \geq 3, f (x, u)$ is subcritical about $u$ and ${\mathcal Q}$ is positive and has some non-degenerate critical points in ${\mathbb R}^{N}_{+}$. For $\lambda$ large, we can get solutions which have peaks near the non-degenerate critical points of ${\mathcal Q}$.

    • Author Affiliations



      1. School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, People’s Republic of China
    • Dates

  • Proceedings – Mathematical Sciences | News

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