Advances in Differential Equations

Multiplicity of positive solutions for a class of quasilinear problems

Claudianor O. Alves, Giovany M. Figueiredo, and Uberlandio B. Severo

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In this paper we prove the existence and multiplicity of nontrivial weak solutions for quasilinear elliptic equations of the form $-L_p u +V(x)|u|^{p-2}u= h(u)$ in $\mathbb{R}^N$, where $L_p u\doteq \epsilon^{p}\Delta_p u +\epsilon^{p}\Delta_p (u^2)u$ and $V$ is a positive continuous potential bounded away from zero satisfying some conditions and the nonlinear term $h(u)$ has a subcritical growth type. Here, we use a variational method to get the multiplicity of positive solutions involving the Lusternick-Schnirelman category of the set where $V$ achieves its minimum value.

Article information

Adv. Differential Equations, Volume 14, Number 9/10 (2009), 911-942.

First available in Project Euclid: 18 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35A15: Variational methods 35H30: Quasi-elliptic equations 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10]


Alves, Claudianor O.; Figueiredo, Giovany M.; Severo, Uberlandio B. Multiplicity of positive solutions for a class of quasilinear problems. Adv. Differential Equations 14 (2009), no. 9/10, 911--942.

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