CPAA
Four positive solutions of a quasilinear elliptic equation in $ R^N$
Fang-Fang Liao Chun-Lei Tang
Communications on Pure & Applied Analysis 2013, 12(6): 2577-2600 doi: 10.3934/cpaa.2013.12.2577
This paper deals with the existence of multiple positive solutions of a quasilinear elliptic equation \begin{eqnarray} -\Delta_p u+u^{p-1} = a(x)u^{q-1}+\lambda h(x) u^{r-1}, \text{in} R^N; \\ u\geq 0, \text{ a.e. }x \in R^N;\\ u \in W^{1,p}(R^N), \end{eqnarray} where $1 < p \leq 2$, $N>p$ and $1 < r < p$ $< q < p^* ( = \frac{pN}{N-p})$. A Nehari manifold is defined by a $C^1-$functional $I$ and is decomposed into two parts. Our work is to find four positive solutions of Eq. (1) when parameter $\lambda$ is sufficiently small.
keywords: positive solutions. Nehari manifold $p-$Laplacian Palais-Smale decomposition lemma

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