Positive entire solutions of inhomogeneous semilinear elliptic equations with supercritical exponent
Soohyun Bae
We establish that the elliptic equation $\Delta u + K(x)u^p + \mu f(x) = 0 in \mathbb{R}^n$ possesses a continuum of positive entire solutions under a set of assumptions on $K, p, \mu$ and $f$. When $K$ behaves like $1 + d|x|^( - q)$ near $\infty$ for some constants $d$ > 0 and $q$ > 0, separation and uncountable multiplicity of solutions appear for small $\mu$ > 0 provided that $n$ > 10, $p$ is large enough, and $f$ satisfies suitable decay conditions at $\infty$.
keywords: uncountable multiplicity Inhomogeneous semilinear elliptic equations asymptotic behavior positive solutions separation.
Classification of positive solutions of semilinear elliptic equations with Hardy term
Soohyun Bae
We study the elliptic equation $\Delta u+\mu/|x|^2+K(|x|)u^p=0$ in $\mathbb{R}^n \setminus \left \{ 0 \right \}$, where $n\geq1$ and $p>1$. In particular, when $K(|x|)=|x|^l$, a classification of radially symmetric solutions is presented in terms of $\mu$ and $l$. Moreover, we explain the separation structure for the equation, and study the stability of positive radial solutions as steady states.
keywords: Semilinear elliptic equation separation singular solution positive solution stability.
Weighted $L^\infty$ stability of positive steady states of a semilinear heat equation in $\R^n$
Soohyun Bae
This paper deals with the stability of steady states of the semilinear heat equation $u_t=$Δ$u+K(x)u^p+f(x)$ under proper assumptions on $K(x)$ and $f(x)$. We prove the weak asymptotic stability of positive steady states with respect to weighted uniform norms.
keywords: stability Semilinear heat equations weak asymptotic stability. positive steady states
On the elliptic equation Δu+K up = 0 in $\mathbb{R}$n
Soohyun Bae
This paper deals with the elliptic equation Δu+K(|x|) up = 0 in $\mathbb{R}$n\{0} when $r^{-l}K(r)$ for $l>-2$ behaves monotonically near $r=0$ or $\infty$ with $r=|x|$. By the method of phase plane, we present a new proof for the structure of positive radial solutions, and analyze the asymptotic behavior at $\infty$. We also employ the approach to classify singular solutions in terms of the asymptotic behavior at $0$. In particular, when $p=\frac{n+2+2l}{n-2}$, we establish the uniqueness of solutions with asymptotic self-similarity at $0$ and at $\infty$, and the existence of multiple solutions of Delaunay-Fowler type at $0$ and $\infty$.
keywords: singular solution fast decay slow decay Semilinear elliptic equation asymptotically self-similar solution phase-plane analysis Delaunay-Fowler-type solution. positive solution
Standing waves of nonlinear Schrödinger equations with optimal conditions for potential and nonlinearity
Soohyun Bae Jaeyoung Byeon
We consider the singularly perturbed nonlinear elliptic problem \begin{eqnarray*} \varepsilon^2 \Delta v - V(x)v + f(v) =0, v > 0, \lim_{|x|\to \infty} v(x) = 0. \end{eqnarray*} Under almost optimal conditions for the potential $V$ and the nonlinearity $f$, we establish the existence of single-peak solutions whose peak points converge to local minimum points of $V$ as $\varepsilon \to 0$. Moreover, we exhibit a threshold on the condition of $V$ at infinity between existence and nonexistence of solutions.
keywords: optimal conditions. variational method decaying potential standing waves Nonlinear Schrödinger equations

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