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### Open Access Journals

PROC

Positive entire solutions of inhomogeneous semilinear elliptic equations with supercritical exponent

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 $\infty$.

PROC

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.

DCDS

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.

DCDS

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 $r\to∞$.

DCDS

This paper deals with the elliptic equation
Δu+K(|x|)

*u*^{p}= 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 $\infty$.
CPAA

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.

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