Positive entire solutions of inhomogeneous semilinear elliptic equations with supercritical exponent
Soohyun Bae
Conference Publications 2005, 2005(Special): 50-59 doi: 10.3934/proc.2005.2005.50
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
Conference Publications 2013, 2013(special): 31-39 doi: 10.3934/proc.2013.2013.31
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
Discrete & Continuous Dynamical Systems - A 2010, 26(3): 823-837 doi: 10.3934/dcds.2010.26.823
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
Separation structure of radial solutions for semilinear elliptic equations with exponential nonlinearity
Soohyun Bae Yūki Naito
Discrete & Continuous Dynamical Systems - A 2018, 38(9): 4537-4554 doi: 10.3934/dcds.2018198

We consider the semilinear elliptic equation $Δ u + K(|x|)e^u = 0$ in $\mathbf{R}^N$ for $N > 2$, and investigate separation phenomena of radial solutions. In terms of intersection and separation, we classify the solution structures and establish characterizations of the structures. These observations lead to sufficient conditions for partial separation. For $N = 10+4\ell$ with $\ell>-2$, the equation changes its nature drastically according to the sign of the derivative of $r^{-\ell}K(r)$ when $r^{-\ell}K(r)$ is monotonic in $r$ and $r^{-\ell} K(r)\to1$ as $r\to∞$.

keywords: Semilinear elliptic equations exponential nonlinearity separation partial separation critical dimension
On the elliptic equation Δu+K up = 0 in $\mathbb{R}$n
Soohyun Bae
Discrete & Continuous Dynamical Systems - A 2013, 33(2): 555-577 doi: 10.3934/dcds.2013.33.555
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
Communications on Pure & Applied Analysis 2013, 12(2): 831-850 doi: 10.3934/cpaa.2013.12.831
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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