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POSITIVE GROUND STATES FOR A CLASS OF SUPERLINEAR
$(p,q)$-LAPLACIAN COUPLED SYSTEMS INVOLVING SCHRÖDINGER EQUATIONS
- Part of
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- Journal:
- Journal of the Australian Mathematical Society / Volume 109 / Issue 2 / October 2020
- Published online by Cambridge University Press:
- 29 July 2019, pp. 193-216
- Print publication:
- October 2020
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We study the existence of positive ground state solutions for the following class of
$(p,q)$-Laplacian coupled systems where
$$\begin{eqnarray}\left\{\begin{array}{@{}lr@{}}-\unicode[STIX]{x1D6E5}_{p}u+a(x)|u|^{p-2}u=f(u)+\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D706}(x)|u|^{\unicode[STIX]{x1D6FC}-2}u|v|^{\unicode[STIX]{x1D6FD}}, & x\in \mathbb{R}^{N},\\ -\unicode[STIX]{x1D6E5}_{q}v+b(x)|v|^{q-2}v=g(v)+\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D706}(x)|v|^{\unicode[STIX]{x1D6FD}-2}v|u|^{\unicode[STIX]{x1D6FC}}, & x\in \mathbb{R}^{N},\end{array}\right.\end{eqnarray}$$
$1<p\leq q<N$. 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.
MULTIPLE SOLUTIONS FOR A CLASS OF QUASILINEAR ELLIPTIC PROBLEMS
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- Journal:
- Proceedings of the Edinburgh Mathematical Society / Volume 46 / Issue 1 / February 2003
- Published online by Cambridge University Press:
- 27 January 2003, pp. 159-168
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