Abstract
In this paper, we are concerned with a nonlinear p-Laplace equation with critical Sobolev-Hardy exponents and Robin boundary conditions. Through a compactness analysis of the functional corresponding to the problem, we obtain the existence of positive solutions for this problem under different assumptions.
Highlights
We are concerned with the following class of boundary value problems: ⎧ ⎨– pu μ |u|p– |x|p u + λ|u|p– u =|u|p∗ (s)– u |x|s η|u|q– u, ⎩|∇ u|p–∂u ∂ν α(x)|u|p– u, in, on ∂
1 Introduction We are concerned with the following class of boundary value problems:
A global compact result for a semilinear elliptic problem with critical Sobolev nonlinearities on bounded domains was obtained by Struwe [ ]
Summary
We are concerned with the following class of boundary value problems:. , η ≥ and λ ∈ R are parameters, α(x) ∈ C(∂ ), α(x) ≥ . is a bounded domain with a smooth C boundary, ν denotes the unit outward normal to ∂. For second-order semilinear elliptic differential equations on bounded domains, Brezis and Lieb [ ] obtained an existence result of solutions for a class of elliptic equations with critical Sobolev nonlinearities by verifying a sub-level which satisfies the Palais-Smale conditions. A global compact result for a semilinear elliptic problem with critical Sobolev nonlinearities on bounded domains was obtained by Struwe [ ]. The boundary conditions make great influence on our noncompact analysis Does it change the form of our limiting equations, but it adds more limiting equations which induce new blow-up bubble such as Dμ We shall represent any diverging Palais-Smale sequence as the sum of critical points of a family of limiting functionals, which are invariant under scaling. The first result of this paper is the following global compactness theorem.
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