Abstract
We consider the semilinear elliptic problem with critical Hardy-Sobolev exponents and Dirichlet boundary condition. By using variational methods we obtain the existence and multiplicity of nontrivial solutions and improve the former results.
Highlights
AND MAIN RESULTSIn this paper we consider the following wide class of semilinear elliptic problems, (1.1) −∆u − μ u |x|2 = g(x, u) + β|u|2∗ (s)−2 |x|s u u=0 in Ω, on ∂Ω, where Ω ⊂ RN (N ≥ 4) is an open bounded domain with smooth boundary, β > 0 ∈ Ω,0 ≤ s < 2, 2∗(s) :=2(N − s) N −2 is the criticalHardy-Sobolev exponent and, when s =
We prove the statement u = 0 if c ∈
We prove the statement when (C2), (C5) and (H) hold
Summary
In this paper we consider the following wide class of semilinear elliptic problems,. |u|2∗ (s)−2 |x|s u u=0 in Ω, on ∂Ω, where Ω ⊂ RN (N ≥ 4) is an open bounded domain with smooth boundary, β >. Dealt with (1.1) with β = 1 and g(x, t) = λ|t|q−2t and obtained the existence of one positive solution for suitable q and λ. They proved in [3] that (1.1) has one nontrivial solution for g(x, t) = λt (λ > 0) and in [9] that (1.1) has one pair of sign-changing solutions for g(x, t) = λt (λ > 0) with some additional assumptions. Let Ω ⊂ RN be an open bounded domain with smooth boundary and assume one of the following three cases holds:. In the sequel we always denote a positive constant by C. With Ω |∇um ε |2 dx ≤ RN |∇u∗ε|2 dx (2.2) follows
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