Existence Of Solutions For The P-Laplacian Equations With A Hardy-Sobolev Operator

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×Râ†'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 1Ï„<Ï„,then the problem (P) have a mountain pass type nontrivial solution.