• Title/Summary/Keyword: semilinear elliptic problems

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EXISTENCE OF THE THIRD POSITIVE RADIAL SOLUTION OF A SEMILINEAR ELLIPTIC PROBLEM ON AN UNBOUNDED DOMAIN

  • Ko, Bong-Soo;Lee, Yong-Hoon
    • Journal of the Korean Mathematical Society
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    • v.39 no.3
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    • pp.439-460
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    • 2002
  • We prove the multiplicity of ordered positive radial solutions for a semilinear elliptic problem defined on an exterior domain. The key argument is to prove the existence of the third solution in presence of two known solutions. For this, we obtain some partial results related to three solutions theorem for certain singular boundary value problems. Proof are mainly based on the upper and lower solutions method and degree theory.

RADIAL SYMMETRY OF POSITIVE SOLUTIONS FOR SEMILINEAR ELLIPTIC EQUATIONS IN $R^n$

  • Naito, Yuki
    • Journal of the Korean Mathematical Society
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    • v.37 no.5
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    • pp.751-761
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    • 2000
  • Symmetry properties of positive solutions for semilinear elliptic problems in n are considered. We give a symmetry result for the problem in the feneral case, and then derive various results for certain classes of demilinear elliptic equations. We employ the moving plane method based on the maximum principle on unbounded domains to obtain the result on symmetry.

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

  • Chen, Tianlan;Lu, Yanqiong;Ma, Ruyun
    • Bulletin of the Korean Mathematical Society
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    • v.57 no.2
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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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    • v.34 no.5
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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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    • v.17 no.4
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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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