### Existence of entire solutions for semilinear elliptic problems on ${\mathbb R}^{N}$

#### Abstract

In this paper, we consider the existence of positive and negative

entire solutions of semilinear elliptic problem

$$ -\Delta u + u = g(x,u), \quad u \in H^{1}({\mathbb R}^{N})\tag{P}

$$

where $N \geq 2$ and $g:{\mathbb R}^{N} \times {\mathbb R }\to {\mathbb

R}$ is a continuous function with superlinear

growth and $g(x,0) = 0$ on ${\mathbb R}^{N} $.

entire solutions of semilinear elliptic problem

$$ -\Delta u + u = g(x,u), \quad u \in H^{1}({\mathbb R}^{N})\tag{P}

$$

where $N \geq 2$ and $g:{\mathbb R}^{N} \times {\mathbb R }\to {\mathbb

R}$ is a continuous function with superlinear

growth and $g(x,0) = 0$ on ${\mathbb R}^{N} $.

#### Keywords

Nonlinear elliptic problem; positive solution; entire solution

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