• 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].

• Bulletin of the Korean Mathematical Society
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• v.50 no.6
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• pp.1817-1826
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• 2013

The existence of at least three weak solutions is established for a class of quasilinear elliptic equations involving the p(x)-biharmonic operators with Navier boundary value conditions. The technical approach is mainly based on a three critical points theorem due to Ricceri [11].

• Journal of applied mathematics & informatics
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• v.32 no.5_6
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• pp.721-736
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• 2014

We consider the equation ${\Delta}_mu=p(x)u^{\alpha}+q(x)u^{\beta}$ on $R^N(N{\geq}2)$, where p, q are nonnegative continuous functions and 0 < ${\alpha}{\leq}{\beta}$. Under several hypotheses on p(x) and q(x), we obtain existence and nonexistence of blow-up solutions both for the superlinear and sublinear cases. Existence and nonexistence of entire bounded solutions are established as well.

• Journal of applied mathematics & informatics
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• v.28 no.1_2
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• pp.217-231
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• 2010

We consider a class of elliptic problems of a logistic type $$-div(|{\nabla}_u|^{m-2}{\nabla}_u)\;=\;w(x)u^q\;-\;(a(x))^{\frac{m}{2}}\;f(u)$$ in a bounded domain of $\mathbf{R}^N$ with boundary $\partial\Omega$ of class $C^2$, $u|_{\partial\Omega}\;=\;+{\infty}$, $\omega\;\in\;L^{\infty}(\Omega)$, 0 < q < 1 and $a\;{\in}\;C^{\alpha}(\bar{\Omega})$, $\mathbf{R}^+$ is non-negative for some $\alpha\;\in$ (0,1), where $\mathbf{R}^+\;=\;[0,\;\infty)$. Under suitable growth assumptions on a, b and f, we show the exact blow-up rate and uniqueness of the large solutions. Our proof is based on the method of sub-supersolution.

• Communications of the Korean Mathematical Society
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• v.10 no.4
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• pp.925-929
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• 1995

Consider the problem $-div($\mid$\bigtriangledown_u$\mid$^{p-2}\bigtriangledown_u) = $\mid$u$\mid$^{p^*-2}u + \lambda$\mid$u$\mid$^{q-2}u$ in B, u = 0 on $\partial B$; where $B \subset R^n$ is a ball, $\lambda < 0, 1 < p < n$ and $p^* = \frac{np}{n-p}$ is the critical Sobolev exponent. For given $\lambda > 0$, we show that there exists $k = k(\lambda) \in N$ such that any radial solutions to this problem have at most k noda curves when $p \leq q \leq p^* - 1$.

• Bulletin of the Korean Mathematical Society
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• v.47 no.1
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• pp.29-37
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• 2010

We establish the existence of weak solutions in an infinite subsonic channel in the self-similar plane to the two-dimensional Burgers system. We consider a boundary value problem in a fixed domain such that a part of the domain is degenerate, and the system becomes a second order elliptic equation in the channel. The problem is motivated by the study of the weak shock reflection problem and 2-D Riemann problems. The two-dimensional Burgers system is obtained through an asymptotic reduction of the 2-D full Euler equations to study weak shock reflection by a ramp.

• Journal of applied mathematics & informatics
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• v.32 no.1_2
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• pp.137-152
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• 2014

In this paper, we prove, in the spirit of [3, 12, 20, 22, 23], the existence of infinitely many small solutions to the following quasilinear elliptic equation $-{\Delta}_{p(x)}u+{\mid}u{\mid}^{p(x)-2}u={\mid}u{\mid}^{q(x)-2}u+{\lambda}f(x,u)$ in a smooth bounded domain ${\Omega}$ of ${\mathbb{R}}^N$. We also assume that $\{q(x)=p^*(x)\}{\neq}{\emptyset}$, where $p^*(x)$ = Np(x)/(N - p(x)) is the critical Sobolev exponent for variable exponents. The proof is based on a new version of the symmetric mountainpass lemma due to Kajikiya [22], and property of these solutions are also obtained.

• Journal of applied mathematics & informatics
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• v.29 no.3_4
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• pp.921-936
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• 2011

In this paper, our main purpose is to establish the existence of weak solutions of a weak solutions of a class of p-q-Laplacian system involving concave-convex nonlinearities: $$\{\array{-{\Delta}_pu-{\Delta}_qu={\lambda}V(x)|u|^{r-2}u+\frac{2{\alpha}}{\alpha+\beta}|u|^{\alpha-2}u|v|^{\beta},\;x{\in}{\Omega}\\-{\Delta}p^v-{\Delta}q^v={\theta}V(x)|v|^{r-2}v+\frac{2\beta}{\alpha+\beta}|u|^{\alpha}|v|^{\beta-2}v,\;x{\in}{\Omega}\\u=v=0,\;x{\in}{\partial}{\Omega}}$$ where ${\Omega}$ is a bounded domain in $R^N$, ${\lambda}$, ${\theta}$ > 0, and 1 < ${\alpha}$, ${\beta}$, ${\alpha}+{\beta}=p^*=\frac{N_p}{N_{-p}}$ is the critical Sobolev exponent, ${\Delta}_su=div(|{\nabla}u|^{s-2}{\nabla}u)$ is the s-Laplacian of u. when 1 < r < q < p < N, we prove that there exist infinitely many weak solutions. We also obtain some results for the case 1 < q < p < r < $p^*$. The existence results of solutions are obtained by variational methods.