• Journal of applied mathematics & informatics
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• v.28 no.3_4
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• pp.625-637
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• 2010

In this paper, we are concerned with the quasilinear elliptic systems with boundary blow-up conditions in a smooth bounded domain. Using the method of lower and upper solutions, we prove the sufficient conditions for the existence of the positive solution. Our main results are new and extend the results in [Mingxin Wang, Lei Wei, Existence and boundary blow-up rates of solutions for boundary blow-up elliptic systems, Nonlinear Analysis(In Press)].

• Bulletin of the Korean Mathematical Society
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• v.54 no.1
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• pp.125-144
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• 2017

Three nontrivial nonnegative solutions for some critical quasilinear elliptic systems with lower-order negative perturbations are obtained by using the Ekeland's variational principle and the mountain pass theorem.

• Journal of applied mathematics & informatics
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• v.27 no.5_6
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• pp.1119-1132
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• 2009

In this paper, we consider the quasilinear elliptic system $\\div(|{\nabla}u|^{p-2}{\nabla}u)=u(a_1u^{m1}+b_1(x)u^m+{\delta}_1v^n),\;\\div(|{\nabla}_v|^{q-2}{\nabla}v)=v(a_2v^{r1}+b_2(x)v^r+{\delta}_2u^s)$, in $\Omega$ where m > $m_1$ > p-2, r > $r_1$ > q-, p, q $\geq$ 2, and ${\Omega}{\subset}R^N$ is a smooth bounded domain. By constructing certain super and subsolutions, we show the existence of positive blow-up solutions and give a global estimate.

• Journal of the Korean Mathematical Society
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• v.55 no.6
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• pp.1423-1433
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• 2018

This paper is concerned about several quasilinear elliptic systems with m-Laplacians. According to the Liouville theorems of those systems on ${\mathbb{R}}^n$, we obtain the singularity estimates of the positive $C^1$-weak solutions on bounded or unbounded domain (but it is not ${\mathbb{R}}^n$ and their decay rates on the exterior domain when ${\mid}x{\mid}{\rightarrow}{\infty}$. The doubling lemma which is developed by Polacik-Quittner-Souplet plays a key role in this paper. In addition, the corresponding results of several special examples are presented.

• Journal of applied mathematics & informatics
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• v.28 no.1_2
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• pp.163-173
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• 2010

We consider the system $$\{{{-{\Delta}_pu\;=\;{\lambda}f(\upsilon),\;\;\;x\;{\in}\;{\Omega}, \atop -{\Delta}_q{\upsilon}\;=\;{\mu}g(u),\;\;\;x\;{\in}\;{\Omega},} \atop u\;=\;\upsilon\;=\;0,\;\;\;x\;{\in}\;{\partial\Omega},}$$ where ${\Delta}_pu\;=\;div(|{\nabla}_u|^{p-2}{\nabla}_u)$, ${\Delta}_{q{\upsilon}}\;=\;div(|{\nabla}_{\upsilon}|^{q-2}{\nabla}_{\upsilon})$, p, $q\;{\geq}\;2$, $\Omega$ is a ball in $\mathbf{R}^N$ with a smooth boundary $\partial\Omega$, $N\;{\geq}\;1$, $\lambda$, $\mu$ are positive parameters, and f, g are smooth functions that are negative at the origin and f(x) ~ $x^m$ g(x) ~ $x^n$ for x large for some m, $n\;{\geq}\;0$ with mn < (p - 1)(q - 1). We establish the existence and uniqueness of positive radial solutions when the parameters $\lambda$ and $\mu$ are large.

• Bulletin of the Korean Mathematical Society
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• v.50 no.1
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• pp.57-71
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• 2013

In this paper, we establish the existence of at least three solutions to a Navier boundary problem involving the ($p_1$, ${\cdots}$, $p_n$)-biharmonic systems. We use a variational approach based on a three critical points theorem due to Ricceri [B. Ricceri, A three critical points theorem revisited, Nonlinear Anal. 70 (2009), 3084-3089].