Advances in Differential Equations

A positive solution of a nonlinear elliptic equation in $\Bbb R^N$ with $G$-symmetry

Jun Hirata

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In this paper we consider the following elliptic equation: \begin{equation*} \begin{cases} \; -\Delta u + u = f(x,u) \qquad \text{in}\; \mathbf{R}^N, \\ \quad u \in H^1(\mathbf{R}^N). \end{cases} \end{equation*} Here $f(x,u)$ is invariant under a finite group action $G \subset O(N)$ which acts on $S^{N-1}$ effectively. When $N \geq 3$, we show the existence of a positive solution without global assumptions like: $ u \mapsto \frac{f(x,u)}{u} $ is increasing in $ u > 0$, $f(x,u) \geq f^{\infty}(u)$. We can deal with asymptotically linear equations as well as superlinear equations. Interaction estimates between solution at infinity play important roles in our argument.

Article information

Adv. Differential Equations, Volume 12, Number 2 (2007), 173-199.

First available in Project Euclid: 18 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J60: Nonlinear elliptic equations
Secondary: 35B05: Oscillation, zeros of solutions, mean value theorems, etc. 47J30: Variational methods [See also 58Exx] 58E05: Abstract critical point theory (Morse theory, Ljusternik-Schnirelman (Lyusternik-Shnirel m an) theory, etc.)


Hirata, Jun. A positive solution of a nonlinear elliptic equation in $\Bbb R^N$ with $G$-symmetry. Adv. Differential Equations 12 (2007), no. 2, 173--199.

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