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A new condition for the concavity method of blow-up solutions to p-Laplacian parabolic equations
Chung, Soon-Yeong,Choi, Min-Jun Elsevier 2018 Journal of differential equations Vol.265 No.12
<P><B>Abstract</B></P> <P>In this paper, we consider an initial-boundary value problem of the p-Laplacian parabolic equation [FORMULA OMISSION] where p ≥ 2 and Ω is a bounded domain of <SUP> R N </SUP> ( N ≥ 1 ) with smooth boundary ∂Ω. The main contribution of this work is to introduce a new condition [FORMULA OMISSION] for some α , β , γ > 0 with 0 < β ≤ ( α − p ) <SUB> λ 1 , p </SUB> p , where <SUB> λ 1 , p </SUB> is the first eigenvalue of p-Laplacian <SUB> Δ p </SUB> , and we use the concavity method to obtain the blow-up solutions to the above equations. In fact, it will be seen that the condition ( <SUB> C p </SUB> ) improves the conditions ever known so far.</P>
Junfang Zhao,Weigao Ge 한국전산응용수학회 2009 Journal of applied mathematics & informatics Vol.27 No.5
In this paper, we are concerned with the following four point boundary value problem with one-dimensional p-Laplacian, (∅p(x'(t)))' + h(t)f(t, x(t), |x'(t)|) = 0, 0 < t < 1, x'(0) −δx(η) = 0, x'(1) + δx(η) = 0, where ∅p(s) = |s|p−2s, p > 1, δ > 0, 1 > η > ξ > 0, ξ +η= 1. By using a fixed point theorem in a cone, we obtain the existence of at least three symmetric positive solutions. The interestingpoint is that the boundary condition is a new Sturm-Liouville-like boundary condition, which has rarely been treated up to now. In this paper, we are concerned with the following four point boundary value problem with one-dimensional p-Laplacian, (∅p(x'(t)))' + h(t)f(t, x(t), |x'(t)|) = 0, 0 < t < 1, x'(0) −δx(η) = 0, x'(1) + δx(η) = 0, where ∅p(s) = |s|p−2s, p > 1, δ > 0, 1 > η > ξ > 0, ξ +η= 1. By using a fixed point theorem in a cone, we obtain the existence of at least three symmetric positive solutions. The interestingpoint is that the boundary condition is a new Sturm-Liouville-like boundary condition, which has rarely been treated up to now.
MULTI-POINT BOUNDARY VALUE PROBLEMS FOR ONE-DIMENSIONAL p-LAPLACIAN AT RESONANCE
Wang, Youyu,Zhang, Guosheng,Ge, Weigao 한국전산응용수학회 2006 Journal of applied mathematics & informatics Vol.22 No.1
In this paper, we consider the multi-point boundary value problems for one-dimensional p-Laplacian at resonance: $({\phi}_p(x'(t)))'=f(t,x(t),x'(t))$, subject to the boundary value conditions: ${\phi}_p(x'(0))={\sum}^{n-2}_{i=1}{\alpha}_i{\phi}_p(x'({\epsilon}i)),\;{\phi}_p(x'(1))={\sum}^{m-2}_{i=1}{\beta}_j{\phi}_p(x'({\eta}_j))$ where ${\phi}_p(s)=/s/^{p-2}s,p>1,\;{\alpha}_i(1,{\le}i{\le}n-2){\in}R,{\beta}_j(1{\le}j{\le}m-2){\in}R,0<{\epsilon}_1<{\epsilon}_2<...<{\epsilon}_{n-2}1,\;0<{\eta}1<{\eta}2<...<{\eta}_{m-2}<1$, By applying the extension of Mawhin's continuation theorem, we prove the existence of at least one solution. Our result is new.
Youfeng Zhang,Zhiyu Zhang,Fengqin Zhang 한국전산응용수학회 2009 Journal of applied mathematics & informatics Vol.27 No.5
In this paper, we consider themultipoint boundary value prob- lem for the one-dimensional p-Laplacian (∅p(u'))'(t) + q(t)f(t,u(t), u'(t)) = 0, t ∈ (0, 1), subject to the boundary conditions: u(0) =<수식> , u(1) =<수식> where ∅p(s) = |s|n−2s, p > 1, ξi ∈ (0, 1) with 0 < ξ1 < ξ2 < · · · < ξn−2 < 1 and αi, βi ∈ [0, 1), 0 < <수식> ,<수식> βi < 1. Using a fixed point theoremdue to Bai and Ge, we study the existence of at least three positive solutions to the above boundary value problem. The important point is that the nonlinear term f explicitly involves a first-order derivative. In this paper, we consider themultipoint boundary value prob- lem for the one-dimensional p-Laplacian (∅p(u'))'(t) + q(t)f(t,u(t), u'(t)) = 0, t ∈ (0, 1), subject to the boundary conditions: u(0) =<수식> , u(1) =<수식> where ∅p(s) = |s|n−2s, p > 1, ξi ∈ (0, 1) with 0 < ξ1 < ξ2 < · · · < ξn−2 < 1 and αi, βi ∈ [0, 1), 0 < <수식> ,<수식> βi < 1. Using a fixed point theoremdue to Bai and Ge, we study the existence of at least three positive solutions to the above boundary value problem. The important point is that the nonlinear term f explicitly involves a first-order derivative.
Yin, Honghui,Yang, Zuodong The Korean Society for Computational and Applied M 2011 Journal of applied mathematics & informatics Vol.29 No.3
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.
ON DISCONTINUOUS ELLIPTIC PROBLEMS INVOLVING THE FRACTIONAL p-LAPLACIAN IN ℝ<sup>N</sup>
Kim, In Hyoun,Kim, Yun-Ho,Park, Kisoeb Korean Mathematical Society 2018 대한수학회보 Vol.55 No.6
We are concerned with the following fractional p-Laplacian inclusion: $$(-{\Delta})^s_pu+V(x){\mid}u{\mid}^{p-2}u{\in}{\lambda}[{\underline{f}}(x,u(x)),\;{\bar{f}}(s,u(x))]$$ in ${\mathbb{R}}^N$, where $(-{\Delta})^s_p$ is the fractional p-Laplacian operator, 0 < s < 1 < p < $+{\infty}$, sp < N, and $f:{\mathbb{R}}^N{\times}{\mathbb{R}}{\rightarrow}{\mathbb{R}}$ is measurable with respect to each variable separately. We show that our problem with the discontinuous nonlinearity f admits at least one or two nontrivial weak solutions. In order to do this, the main tool is the Berkovits-Tienari degree theory for weakly upper semicontinuous set-valued operators. In addition, our main assertions continue to hold when $(-{\Delta})^s_pu$ is replaced by any non-local integro-differential operator.
SOLVABILITY FOR SOME DIRICHLET PROBLEM WITH P-LAPACIAN
김용인 한국수학교육학회 2010 純粹 및 應用數學 Vol.17 No.3
We investigate the existence of the following Dirichlet boundary value problem (ju0jp¡2u0)0 + (p ¡ 1)[®ju+jp¡2u+ ¡ ¯ju¡jp¡2u¡] = (p ¡ 1)h(t);u(0) = u(T) = 0;where p > 1; ® > 0; ¯ > 0 and ®¡1p + ¯¡1p = 2; T = ¼p=®1p ; ¼p = 2¼p sin(¼=p) and h 2 L1(0; T). The results of this paper generalize some early results obtained in [8] and [9]. Moreover, the method used in this paper is elementary and new.
STABILITY RESULTS OF POSITIVE WEAK SOLUTION FOR SINGULAR p-LAPLACIAN NONLINEAR SYSTEM
KHAFAGY, SALAH,SERAG, HASSAN The Korean Society for Computational and Applied M 2018 Journal of applied mathematics & informatics Vol.36 No.3
In this paper, we investigate the stability of positive weak solution for the singular p-Laplacian nonlinear system $-div[{\mid}x{\mid}^{-ap}{\mid}{\nabla}u{\mid}^{p-2}{\nabla}u]+m(x){\mid}u{\mid}^{p-2}u={\lambda}{\mid}x{\mid}^{-(a+1)p+c}b(x)f(u)$ in ${\Omega}$, Bu = 0 on ${\partial}{\Omega}$, where ${\Omega}{\subset}R^n$ is a bounded domain with smooth boundary $Bu={\delta}h(x)u+(1-{\delta})\frac{{\partial}u}{{\partial}n}$ where ${\delta}{\in}[0,1]$, $h:{\partial}{\Omega}{\rightarrow}R^+$ with h = 1 when ${\delta}=1$, $0{\in}{\Omega}$, 1 < p < n, 0 ${\leq}$ a < ${\frac{n-p}{p}}$, m(x) is a weight function, the continuous function $b(x):{\Omega}{\rightarrow}R$ satisfies either b(x) > 0 or b(x) < 0 for all $x{\in}{\Omega}$, ${\lambda}$ is a positive parameter and $f:[0,{\infty}){\rightarrow}R$ is a continuous function. We provide a simple proof to establish that every positive solution is unstable under certain conditions.
STABILITY RESULTS OF POSITIVE WEAK SOLUTION FOR SINGULAR p-LAPLACIAN NONLINEAR SYSTEM
Salah Khafagy,Hassan Serag 한국전산응용수학회 2018 Journal of applied mathematics & informatics Vol.36 No.3
In this paper, we investigate the stability of positive weak solution for the singular $p$-Laplacian\textbf{\ }nonlinear system $\ -div[|x|^{-ap}|\nabla u|^{p-2}\nabla u]+m(x)|u|^{p-2}u=\lambda |x|^{-(a+1)p+c}b(x)f(u)$ \ in$\,\ \Omega ,$ $\ Bu=0$ \ on $\ \partial \Omega ,$ where $\Omega \subset R^{n}$ is a bounded domain with smooth boundary $% Bu=\delta h(x)u+(1-\delta )\frac{\partial u}{\partial n}$ where $\delta \in \lbrack 0,1],$ $h:\partial \Omega \rightarrow R^{+}$ with $h=1$ when $\delta =1,$ $0\in \Omega ,$ $1<p<n,$ $0\leq a<\frac{n-p}{p}$, $m(x)$ is a weight function, the continuous function $b(x):\Omega \rightarrow R$ satisfies either $b(x)>0$ or $b(x)<0$ for all $x\in \Omega $, $\lambda $ is a positive parameter and $f:[0,\infty )\rightarrow R$ is a continuous function. We provide a simple proof to establish that every positive solution is unstable under certain conditions.
정순영,박재현 대한수학회 2015 대한수학회보 Vol.52 No.5
In this paper, we prove the existence of at least three non- trivial solutions to nonlinear discrete boundary value problems { -△p,wu(x) + V(x)|u(x)|q-2u(x) = f(x; u(x)), x ∈ S, u(x) = 0, x ∈ ∂S, involving the discrete p-Laplacian on simple, finite and connected graphs S(S ∪ ∂S,E) with weight ω, where 1 < q < p < ∞. The approach is based on a suitable combine of variational and truncations methods.