On positive solutions of quasilinear elliptic systems

Yuanji Cheng

Displaying similar documents to “On positive solutions of quasilinear elliptic systems”

On a semilinear elliptic eigenvalue problem

Mario Michele Coclite (1997)

Annales Polonici Mathematici

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We obtain a description of the spectrum and estimates for generalized positive solutions of -Δu = λ(f(x) + h(u)) in Ω, ${u |}_{\partial Ω} = 0$ , where f(x) and h(u) satisfy minimal regularity assumptions.

Uniqueness of solutions for some degenerate nonlinear elliptic equations

Albo Carlos Cavalheiro (2014)

Applicationes Mathematicae

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We investigate the existence and uniqueness of solutions to the Dirichlet problem for a degenerate nonlinear elliptic equation $- \sum_{i, j = 1}^{n} D_{j} (a_{i j} (x) D_{i} u (x)) + b (x) u (x) + d i v (Φ (u (x))) = g (x) - \sum_{j = 1}^{n} f_{j} (x)$ on Ω in the setting of the space H₀(Ω).

A Note on the Uniqueness and Structure of Solutions to the Dirichlet Problem for Some Elliptic Systems

Chern, Jang-Long, Yotsutani, Shoji, Kawano, Nichiro

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In this note, we consider some elliptic systems on a smooth domain of $R^{n}$ . By using the maximum principle, we can get a more general and complete results of the identical property of positive solution pair, and thus classify the structure of all positive solutions depending on the nonlinarities easily.

The multiplicity of solutions and geometry of a nonlinear elliptic equation

Q. Choi, Sungki Chun, Tacksun Jung (1996)

Studia Mathematica

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Let Ω be a bounded domain in $ℝ^{n}$ with smooth boundary ∂Ω and let L denote a second order linear elliptic differential operator and a mapping from $L^{2} (Ω)$ into itself with compact inverse, with eigenvalues $- λ_{i}$ , each repeated according to its multiplicity, 0 < λ1 < λ2 < λ3 ≤ ... ≤ λi ≤ ... → ∞. We consider a semilinear elliptic Dirichlet problem $L u + b u^{+} - a u^{-} = f (x)$ in Ω, u=0 on ∂ Ω. We assume that $a < λ_{1}$ , $λ_{2} < b < λ_{3}$ and f is generated by $ϕ_{1}$ and $ϕ_{2}$ . We show a relation between the multiplicity of solutions and source terms...

A nonlocal elliptic equation in a bounded domain

Piotr Fijałkowski, Bogdan Przeradzki, Robert Stańczy (2004)

Banach Center Publications

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The existence of a positive solution to the Dirichlet boundary value problem for the second order elliptic equation in divergence form $- \sum_{i, j = 1}^{n} D_{i} (a_{i j} D_{j} u) = f (u, \int_{Ω} g (u^{p}))$ , in a bounded domain Ω in ℝⁿ with some growth assumptions on the nonlinear terms f and g is proved. The method based on the Krasnosel’skiĭ Fixed Point Theorem enables us to find many solutions as well.

Eigenvalue problems of quasilinear elliptic systems on R.

Gong Bao Li (1987)

Revista Matemática Iberoamericana

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In this paper we get the existence results of the nontrivial weak solution (λ,u) of the following eigenvalue problem of quasilinear elliptic systems -D_α (a_αβ(x,u) D_βuⁱ) + 1/2 D_uⁱ a_αβ(x,u)D_αu^j D_βu^j + h(x) uⁱ = ...

A weighted symmetrization for nonlinear elliptic and parabolic equations in inhomogeneous media

Guillermo Reyes, Juan Luis Vázquez (2006)

Journal of the European Mathematical Society

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In the theory of elliptic equations, the technique of Schwarz symmetrization is one of the tools used to obtain a priori bounds for classical and weak solutions in terms of general information on the data. A basic result says that, in the absence of lower-order terms, the symmetric rearrangement of the solution $u$ of an elliptic equation, that we write $u^{*}$ , can be compared pointwise with the solution of the symmetrized problem. The main question we address here is the modification of the...

Generating singularities of solutions of quasilinear elliptic equations using Wolff’s potential

Darko Žubrinić (2003)

Czechoslovak Mathematical Journal

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We consider a quasilinear elliptic problem whose left-hand side is a Leray-Lions operator of $p$ -Laplacian type. If $p < γ < N$ and the right-hand side is a Radon measure with singularity of order $γ$ at $x_{0} \in Ω$ , then any supersolution in $W_{l o c}^{1, p} (Ω)$ has singularity of order at least $\frac{(γ - p)}{(p - 1)}$ at $x_{0}$ . In the proof we exploit a pointwise estimate of $𝒜$ -superharmonic solutions, due to Kilpeläinen and Malý, which involves Wolff’s potential of Radon’s measure.

On a semilinear elliptic equation in $ℍ^{n}$

Gianni Mancini, Kunnath Sandeep (2008)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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We prove existence/nonexistence and uniqueness of positive entire solutions for some semilinear elliptic equations on the Hyperbolic space.

On the principal eigenvalue of elliptic operators in $ℝ^{N}$ and applications

Henry Berestycki, Luca Rossi (2006)

Journal of the European Mathematical Society

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Two generalizations of the notion of principal eigenvalue for elliptic operators in $ℝ^{N}$ are examined in this paper. We prove several results comparing these two eigenvalues in various settings: general operators in dimension one; self-adjoint operators; and “limit periodic” operators. These results apply to questions of existence and uniqueness for some semilinear problems in the whole space. We also indicate several outstanding open problems and formulate some conjectures.