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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

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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