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Volume 18, Issue 3
The Existences of Positive Solutions for p-mean Curvature Operator with Supercritical Potential

Hongzhuo Fu & Yaotian Shen

J. Part. Diff. Eq., 18 (2005), pp. 193-205.

Published online: 2005-08

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  • Abstract

This paper is concerned with the existences of positive solutions of the following Dirichlet problem for p-mean curvature operator with supercritical potential: {-div((1 + |∇u|²) ^{\frac{p-2}{2}} ∇u) = λu^{r-1} + μ\fran{u^{q-1}}{|x|^s}, u > 0 ∈ Ω, u = 0\qquad\qquad x ∈ ∂Ω where u Ω W^{1, p}_0 (Ω), Ω is a bounded domain in R^N(N > p > 1) with smooth boundary ∂Ω and 0 ∈ Ω, 0 < q < p, 0 ≤ s < \frac{N}{p} (p - q) + q, p ≤ r < p∗, p∗ = \frac{Np}{N-p}, μ > 0. It reaches the conclusion where this problem has two positive solutions in the different cases . It discusses the existences of positive solutions of the Dirichlet problem for the p-mean curvature operator with supercritical potential firstly. Meanwhile, it extends some results of the p-Laplace operator to that of p-mean curvature operator for p ≥ 2 .

  • Keywords

Mean curvature operator Mountain Pass Principle (PS) condition Ekeland's variational principle

  • AMS Subject Headings

35J65

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JPDE-18-193, author = {Hongzhuo Fu and Yaotian Shen }, title = {The Existences of Positive Solutions for p-mean Curvature Operator with Supercritical Potential}, journal = {Journal of Partial Differential Equations}, year = {2005}, volume = {18}, number = {3}, pages = {193--205}, abstract = {

This paper is concerned with the existences of positive solutions of the following Dirichlet problem for p-mean curvature operator with supercritical potential: {-div((1 + |∇u|²) ^{\frac{p-2}{2}} ∇u) = λu^{r-1} + μ\fran{u^{q-1}}{|x|^s}, u > 0 ∈ Ω, u = 0\qquad\qquad x ∈ ∂Ω where u Ω W^{1, p}_0 (Ω), Ω is a bounded domain in R^N(N > p > 1) with smooth boundary ∂Ω and 0 ∈ Ω, 0 < q < p, 0 ≤ s < \frac{N}{p} (p - q) + q, p ≤ r < p∗, p∗ = \frac{Np}{N-p}, μ > 0. It reaches the conclusion where this problem has two positive solutions in the different cases . It discusses the existences of positive solutions of the Dirichlet problem for the p-mean curvature operator with supercritical potential firstly. Meanwhile, it extends some results of the p-Laplace operator to that of p-mean curvature operator for p ≥ 2 .

}, issn = {2079-732X}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jpde/5355.html} }
TY - JOUR T1 - The Existences of Positive Solutions for p-mean Curvature Operator with Supercritical Potential AU - Hongzhuo Fu & Yaotian Shen JO - Journal of Partial Differential Equations VL - 3 SP - 193 EP - 205 PY - 2005 DA - 2005/08 SN - 18 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jpde/5355.html KW - Mean curvature operator KW - Mountain Pass Principle KW - (PS) condition KW - Ekeland's variational principle AB -

This paper is concerned with the existences of positive solutions of the following Dirichlet problem for p-mean curvature operator with supercritical potential: {-div((1 + |∇u|²) ^{\frac{p-2}{2}} ∇u) = λu^{r-1} + μ\fran{u^{q-1}}{|x|^s}, u > 0 ∈ Ω, u = 0\qquad\qquad x ∈ ∂Ω where u Ω W^{1, p}_0 (Ω), Ω is a bounded domain in R^N(N > p > 1) with smooth boundary ∂Ω and 0 ∈ Ω, 0 < q < p, 0 ≤ s < \frac{N}{p} (p - q) + q, p ≤ r < p∗, p∗ = \frac{Np}{N-p}, μ > 0. It reaches the conclusion where this problem has two positive solutions in the different cases . It discusses the existences of positive solutions of the Dirichlet problem for the p-mean curvature operator with supercritical potential firstly. Meanwhile, it extends some results of the p-Laplace operator to that of p-mean curvature operator for p ≥ 2 .

Hongzhuo Fu and Yaotian Shen . (2005). The Existences of Positive Solutions for p-mean Curvature Operator with Supercritical Potential. Journal of Partial Differential Equations. 18 (3). 193-205. doi:
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