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On Nonlinear Eigen-problems of Quasi-linear Elliptic Operators

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@Article{JPDE-4-56,
author = {Ma Li},
title = {On Nonlinear Eigen-problems of Quasi-linear Elliptic Operators},
journal = {Journal of Partial Differential Equations},
year = {1991},
volume = {4},
number = {3},
pages = {56--72},
abstract = { In this paper, we study the following Eigen-problem {-\frac{∂}{∂x_i}(a_{ij}(x, u)\frac{∂u}{∂x_j}) + \frac{1}{2}a_{iju}(x,u)\frac{∂u}{∂x_i}\frac{∂u}{∂x_j} + h(x)u = μμ\frac{n+2}{n-2} \quad in Ω \qquad (0.1) u = 0 \quad on ∂Ω u > 0 \quad in Ω ⊂ R^n under some assumptions. First. we minimize I(u) = \frac{1}{2}∫_Ωa_{ij}(x, u)\frac{∂u}{∂x_i}\frac{∂u}{∂x_j} + h(x)u² over E_α = {u ∈ H¹_0(Ω); ∫_Ωu^α = 1} ( 2 < α < N = \frac{2n}{n-2}) to give a H¹_0-solution U_α of the perturbation problems of (0.1). Since I is not differentiable in H¹_0(Ω), the key point is the estimate of U_α. Then, we derive local uniform bounds of (U_α) and give a 'bad' solution of (0.1). Last, we remove the singular points of the 'bad' solution to obtain a solution of (0.1), our result is a extension of that of Brezis & Nirenberg.},
issn = {2079-732X},
doi = {https://doi.org/},
url = {http://global-sci.org/intro/article_detail/jpde/5775.html}
}

TY - JOUR
T1 - On Nonlinear Eigen-problems of Quasi-linear Elliptic Operators
AU - Ma Li
JO - Journal of Partial Differential Equations
VL - 3
SP - 56
EP - 72
PY - 1991
DA - 1991/04
SN - 4
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/jpde/5775.html
KW - Non-differentiable
KW - critical
KW - regularity
AB - In this paper, we study the following Eigen-problem {-\frac{∂}{∂x_i}(a_{ij}(x, u)\frac{∂u}{∂x_j}) + \frac{1}{2}a_{iju}(x,u)\frac{∂u}{∂x_i}\frac{∂u}{∂x_j} + h(x)u = μμ\frac{n+2}{n-2} \quad in Ω \qquad (0.1) u = 0 \quad on ∂Ω u > 0 \quad in Ω ⊂ R^n under some assumptions. First. we minimize I(u) = \frac{1}{2}∫_Ωa_{ij}(x, u)\frac{∂u}{∂x_i}\frac{∂u}{∂x_j} + h(x)u² over E_α = {u ∈ H¹_0(Ω); ∫_Ωu^α = 1} ( 2 < α < N = \frac{2n}{n-2}) to give a H¹_0-solution U_α of the perturbation problems of (0.1). Since I is not differentiable in H¹_0(Ω), the key point is the estimate of U_α. Then, we derive local uniform bounds of (U_α) and give a 'bad' solution of (0.1). Last, we remove the singular points of the 'bad' solution to obtain a solution of (0.1), our result is a extension of that of Brezis & Nirenberg.

Ma Li. (1970). On Nonlinear Eigen-problems of Quasi-linear Elliptic Operators.

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*Journal of Partial Differential Equations*.*4*(3). 56-72. doi: