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Existence of a positive solution for quasilinear elliptic equations with nonlinearity including the gradient

Mieko Tanaka

Author Affiliations

Department of Mathematics, Tokyo University of Science, Kagurazaka 1-3, Shinjyuku-ku, Tokyo, 162-8601, Japan

Boundary Value Problems 2013, 2013:173  doi:10.1186/1687-2770-2013-173

Published: 24 July 2013


We provide the existence of a positive solution for the quasilinear elliptic equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M1">View MathML</a>

in Ω under the Dirichlet boundary condition. As a special case (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M2">View MathML</a>), our equation coincides with the usual p-Laplace equation. The solution is established as the limit of a sequence of positive solutions of approximate equations. The positivity of our solution follows from the behavior of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M3">View MathML</a> as t is small. In this paper, we do not impose the sign condition to the nonlinear term f.

MSC: 35J92, 35P30.

nonhomogeneous elliptic operator; positive solution; the first eigenvalue with weight; approximation