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Existence of Solutions for the p-Laplacian Equations with a Hardy-Sobolev Operator

Author: YanXianWei
Tutor: ZhaoYuanZhang
School: Ocean University of China
Course: Basic mathematics
Keywords: Mountain Pass Lemma Principle of Symmetric Criticality Linking Theorem Nontrivial Solution
CLC: O175.25
Type: Master's thesis
Year: 2011
Downloads: 4
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Abstract


In this paper,we consider the existence of nontrivial solutions for the following p-Laplacian equation with Hardy-Sobolev operators (?)where (?)is unbounded cylinders in (?). Using the well-known Mountain Pass Lemma,we prove the following Theorem 2.1.Theorem (?)satisfying (1). f :Ω×R→R is continuous and satisfying sub-critical growth conditions, i.e.,there exists a constant c andα∈( p , p*) such that(?),whereλ1 is the principal eigenvalue for the p-Laplace operator. Then the problem ( P ) have a nontrivial solution when 2≤p < N. Using the Linking Theorem,we obtain the following Theorem 3.1. Theorem 3.1. If(?) f ( x , u )satisfying the conditions (1)-(4) in Theorem 2.1,and (?)Then problem ( P ) have a nontrivial solution under (?) are the eigenvalues for the p-Laplacian.Theorem 3.2. Under the conditions of Theorem 3.1 and 1τ<τ,then the problem (P) have a mountain pass type nontrivial solution.

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CLC: > Mathematical sciences and chemical > Mathematics > Mathematical Analysis > Differential equations, integral equations > Partial Differential Equations > Elliptic equations
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