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POSITIVE GROUND STATES FOR A CLASS OF SUPERLINEAR $(p,q)$-LAPLACIAN COUPLED SYSTEMS INVOLVING SCHRÖDINGER EQUATIONS

Published online by Cambridge University Press:  29 July 2019

J. C. DE ALBUQUERQUE
Affiliation:
Department of Mathematics,Federal University of Pernambuco, 50670-901 Recife – PE, Brazil email [email protected], [email protected]
JOÃO MARCOS DO Ó
Affiliation:
Department of Mathematics,University of Brasília, 7, 0910-900 Brasília – DF, Brazil email [email protected]
EDCARLOS D. SILVA*
Affiliation:
Institute of Mathematics and Statistics,Federal University of Goiás, 74001-970, Goiás – GO, Brazil email [email protected]

Abstract

We study the existence of positive ground state solutions for the following class of $(p,q)$-Laplacian coupled systems

$$\begin{eqnarray}\left\{\begin{array}{@{}lr@{}}-\unicode[STIX]{x1D6E5}_{p}u+a(x)|u|^{p-2}u=f(u)+\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D706}(x)|u|^{\unicode[STIX]{x1D6FC}-2}u|v|^{\unicode[STIX]{x1D6FD}}, & x\in \mathbb{R}^{N},\\ -\unicode[STIX]{x1D6E5}_{q}v+b(x)|v|^{q-2}v=g(v)+\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D706}(x)|v|^{\unicode[STIX]{x1D6FD}-2}v|u|^{\unicode[STIX]{x1D6FC}}, & x\in \mathbb{R}^{N},\end{array}\right.\end{eqnarray}$$
where $1<p\leq q<N$. Here the coefficient $\unicode[STIX]{x1D706}(x)$ of the coupling term is related to the potentials by the condition $|\unicode[STIX]{x1D706}(x)|\leq \unicode[STIX]{x1D6FF}a(x)^{\unicode[STIX]{x1D6FC}/p}b(x)^{\unicode[STIX]{x1D6FD}/q}$, where $\unicode[STIX]{x1D6FF}\in (0,1)$ and $\unicode[STIX]{x1D6FC}/p+\unicode[STIX]{x1D6FD}/q=1$. Using a variational approach based on minimization over the Nehari manifold, we establish the existence of positive ground state solutions for a large class of nonlinear terms and potentials.

Type
Research Article
Copyright
© 2019 Australian Mathematical Publishing Association Inc.

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Footnotes

Research supported in part by INCTmat/MCT/Brazil, CNPq and CAPES/Brazil. The third author was also partially supported by Fapeg/CNpq grants 03/2015-PPP.

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