• 제목/요약/키워드: Semilinear Elliptic problem

검색결과 20건 처리시간 0.031초

Trends in Researches for Fourth Order Elliptic Equations with Dirichlet Boundary Condition

  • Park, Q-Heung;Yinghua Jin
    • 한국수학사학회지
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    • 제16권4호
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    • pp.107-115
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    • 2003
  • The nonlinear fourth order elliptic equations with jumping nonlinearity was modeled by McKenna. We investigate the trends for the researches of the existence of solutions of a fourth order semilinear elliptic boundary value problem with Dirichlet boundary Condition, ${\Delta}^2u{+}c{\Delta}u=b_1[(u+1)^{-}1]{+}b_2u^+$ in ${\Omega}$, where ${\Omega}$ is a bounded open set in $R^N$ with smooth boundary ${\partial}{\Omega}$.

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CRITICAL POINTS AND MULTIPLE SOLUTIONS OF A NONLINEAR ELLIPTIC BOUNDARY VALUE PROBLEM

  • Choi, Kyeongpyo
    • Korean Journal of Mathematics
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    • 제14권2호
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    • pp.259-271
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    • 2006
  • We consider a semilinear elliptic boundary value problem with Dirichlet boundary condition $Au+bu^+-au^-=t_{1{\phi}1}+t_{2{\phi}2}$ in ${\Omega}$ and ${\phi}_n$ is the eigenfuction corresponding to ${\lambda}_n(n=1,2,{\cdots})$. We have a concern with the multiplicity of solutions of the equation when ${\lambda}_1$ < a < ${\lambda}_2$ < b < ${\lambda}_3$.

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NONEXISTENCE OF NODAL SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATION WITH SOBOLEV-HARDY TERM

  • Choi, Hyeon-Ock;Pahk, Dae-Hyeon
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • 제12권4호
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    • pp.261-269
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    • 2008
  • Let $B_1$ be a unit ball in $R^n(n{\geq}3)$, and $2^*=2n/(n-2)$ be the critical Sobolev exponent for the embedding $H_0^1(B_1){\hookrightarrow}L^{2^*}(B_1)$. By using a variant of Pohoz$\check{a}$aev's identity, we prove the nonexistence of nodal solutions for the Dirichlet problem $-{\Delta}u-{\mu}\frac{u}{{\mid}x{\mid}^2}={\lambda}u+{\mid}u{\mid}^{2^*-2}u$ in $B_1$, u=0 on ${\partial}B_1$ for suitable positive numbers ${\mu}$ and ${\nu}$.

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EXISTENCE OF GROP INVARIANT SOULTIONS OF A SEMILINEAR ELLIPTIC EQUATION

  • Kajinkiya, Ryuji
    • 대한수학회지
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    • 제37권5호
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    • pp.763-777
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    • 2000
  • We investigate the existence of group invariant solutions of the Emden-Fowler equation, - u=$\mid$x$\mid$$\sigma$$\mid$u$\mid$p-1u in B, u=0 on B and u(gx)=u(x) in B for g G. Here B is the unit ball in n 2, 1$\sigma$ 0 and G is a closed subgrop of the orthogonal group. A soultion of the problem is called a G in variant solution. We prove that there exists a G invariant non-radial solution if and only if G is not transitive on the unit sphere.

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NODAL SOLUTIONS FOR AN ELLIPTIC EQUATION IN AN ANNULUS WITHOUT THE SIGNUM CONDITION

  • Chen, Tianlan;Lu, Yanqiong;Ma, Ruyun
    • 대한수학회보
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    • 제57권2호
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    • pp.331-343
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    • 2020
  • This paper is concerned with the global behavior of components of radial nodal solutions of semilinear elliptic problems -Δv = λh(x, v) in Ω, v = 0 on ∂Ω, where Ω = {x ∈ RN : r1 < |x| < r2} with 0 < r1 < r2, N ≥ 2. The nonlinear term is continuous and satisfies h(x, 0) = h(x, s1(x)) = h(x, s2(x)) = 0 for suitable positive, concave function s1 and negative, convex function s2, as well as sh(x, s) > 0 for s ∈ ℝ \ {0, s1(x), s2(x)}. Moreover, we give the intervals for the parameter λ which ensure the existence and multiplicity of radial nodal solutions for the above problem. For this, we use global bifurcation techniques to prove our main results.

MULTIPLE EXISTENCE OF SOLUTIONS FOR A NONHOMOGENEOUS ELLPITIC PROBLEMS ON RN

  • Hirano, Norimichi;Kim, Wan Se
    • East Asian mathematical journal
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    • 제34권5호
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    • pp.703-713
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    • 2018
  • Let $N{\geq}3$, $2^*=2N/(N-2)$ and $p{\in}(2,2^*)$. Our purpose in this paper is to consider multiple existence of solutions of problem $$-{\Delta}u-{\frac{\mu}{{\mid}x{\mid}^2}}+{\alpha}u={\mid}u{\mid}^{p-2}u+{\lambda}f\;u{\in}H^1({\mathbb{R}}^n)$$, where a, ${\lambda}$ > 0, ${\mu}{\in}(0,(N-2)^2/4)f{\in}H^{-1}({\mathbb{R}}^N)$, $f{\geq}0$ and $f{\neq}0$.

EXISTENCE AND MULTIPLICITY RESULTS FOR SOME FOURTH ORDER SEMILINEAR ELLIPTIC PROBLEMS

  • Jin, Yinghua;Wang, Xuechun
    • Korean Journal of Mathematics
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    • 제17권4호
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    • pp.473-480
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    • 2009
  • We prove the existence and multiplicity of nontrivial solutions for a fourth order problem ${\Delta}^2u+c{\Delta}u={\alpha}u-{\beta}(u+1)^-$ in ${\Omega}$, ${\Delta}u=0$ and $u=0$ on ${\partial}{\Omega}$, where ${\lambda}_1{\leq}c{\leq}{\lambda}_2$ (where $({\lambda}_i)_{i{\geq}1}$ is the sequence of the eigenvalues of $-{\Delta}$ in$H_0^1({\Omega})$) and ${\Omega}$ is a bounded open set in $R^N$ with smooth boundary ${\partial}{\Omega}$. The results are proved by applying minimax arguments and linking theory.

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EXISTENCE THEOREMS FOR CRITICAL DEGENERATE EQUATIONS INVOLVING THE GRUSHIN OPERATORS

  • Huong Thi Thu Nguyen;Tri Minh Nguyen
    • 대한수학회논문집
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    • 제38권1호
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    • pp.137-151
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    • 2023
  • In this paper we prove the existence of nontrivial weak solutions to the boundary value problem -G1u = u3 + f(x, y, u) in Ω, u ≥ 0 in Ω, u = 0 on ∂Ω, where Ω is a bounded domain with smooth boundary in ℝ3, G1 is a Grushin type operator, and f(x, y, u) is a lower order perturbation of u3 with f(x, y, 0) = 0. The nonlinearity involved is of critical exponent, which differs from the existing results in [11, 12].