Existence And Multiplicity Of Solutions For Elliptic Equations With Hardy Terms And Critical Weighted Hardy-Sobolev Exponents

Firstly, the author investigates a class of semilinear elliptic equationsExistence and multiplicity of solutions are studied by the variational methods and some analysis techniques. The above equations have Hardy terms and critical weighted Hardy-Sobolev exponents. And the first term is singular. WhereΩis an open bounded domain in R^N(N≥3) with smooth boundary (?)Ωand 0∈Ω, 0≤a <(?), 0≤μ<((?)-a)²,a≤b* is the Sobolev critical exponent. Assume that f satisfies:f∈C((?)×R⁺,R), there exists a constantρ> 2 such that(?)=σ>0,(?)=0,x∈(?),and0<ρ^F(x,t)≤f(x,t)t,x∈(?), t∈R⁺\ {0}, where F(x, t) is a primitive function of f(x, t). A detailed analysis on the (PS) sequence of the variational functional corresponding to the equations is given and a local compactness result is obtained. Applying this compactness result, the Mountain Pass Lemma and the Strong Maximum Principle, the author proves the existence and multiplicity of solutions of the above equations.Secondly, we generalize the equation to p- Laplacian: where 1 < p < N, q = q(a, b)(?)(?)is the critical weighted Hardy-Sobolevexponent. Then we prove the existence of positive solutions and nontrivial solutions under suitable conditions.The main results are the following theorems.Theorem 1 Suppose that N≥3(1+ a). 0≤a < (?), 0≤μ< ((?)-a)², a≤b< a + 1, and there exists a constantρ> 2 such thatAssume thatwhereβ(?)(?)andγ(?)(?)-a+β. Then problem (1) has at least a positive solution.Corollary 1 Suppose that N≤4(1 + a), 0≤a < (?), 0≤μ≤((?)-a)²-(1 + a)², a≤b < a + 1. Assume that there exists a constant p > 2 such that (f₁) and (f₂) hold. Then problem (1) has at least a positive solution.Theorem 2 Suppose that N≥3(1 + a), 0≤a < (?),0≤μ< ((?)- a)², a≤b < a + 1, and there exists a constantρ> 2 such thatAssume that (3) holds. Then problem (1) has at least two distinct nontrivial solutions.Corollary 2 Suppose that N≥4(1 + a), 0≤a <(?), 0≤μ≤((?)-a)²-(1 + a)², a≤b< a + 1. Assume that there exists a constantρ> 2 such that (f₃) and (f₄) hold. Then problem (1) has at least two distinct nontrivial solutions.Theorem 3 Suppose that N≥3(1 + a), 0≤a< (?), 0≤μ<((?)-a)^p, a≤b < a+1,ωis some nonempty open subset ofΩ, ((?),η) (?) (0, +∞) is some non-empty open interval and f(x,t) satisfies Then there existsλ₀ > 0 such that problem (2) possesses a positive solution for everyλ>λ₀Theorem 4 Suppose that N≥3(1 + a), 0≤a < (?), 0≤μ<((?)-a)^p, a≤b< a+1,ωis some nonempty open subset ofΩ, (ξ,,η) (?)(0, +∞) is some non-empty open interval and f(x,t) satisfiesThen there exists (?) > 0 such that problem (2) has at least two distinct nontrivial solutions for everyλ≥(?).