### Existence and Multiplicity of Solutions for Nonlinear Elliptic Equations of p-Laplace Type in $R^N$

#### Abstract

In this paper, we discuss the following elliptic equation

\begin{equation*}

-\text{div}(\varphi(x,\nabla u))=\lambda f(x,u) \quad \textmd{in} \quadÂ \Bbb R^{N},

\end{equation*}

where the function $\varphi: \Bbb R^{N} \times \Bbb R^{N} \to \Bbb R^{N}$ is of type $|v|^{p-2}v$ with a real constant $p>1$ and

$f:\Bbb R^{N}\times\Bbb R \to \Bbb R$ satisfies a Carath\'eodory condition.

We show the existence of at least one nontrivial weak solution, and under suitable assumptions,

infinitely many solutions for the problem above by using Mountain pass theorem and Fountain theorem.

Also, we prove that the existence of at least two distinct nontrivial critical points under the appreciate assumptions.

\begin{equation*}

-\text{div}(\varphi(x,\nabla u))=\lambda f(x,u) \quad \textmd{in} \quadÂ \Bbb R^{N},

\end{equation*}

where the function $\varphi: \Bbb R^{N} \times \Bbb R^{N} \to \Bbb R^{N}$ is of type $|v|^{p-2}v$ with a real constant $p>1$ and

$f:\Bbb R^{N}\times\Bbb R \to \Bbb R$ satisfies a Carath\'eodory condition.

We show the existence of at least one nontrivial weak solution, and under suitable assumptions,

infinitely many solutions for the problem above by using Mountain pass theorem and Fountain theorem.

Also, we prove that the existence of at least two distinct nontrivial critical points under the appreciate assumptions.

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