Abstract
The paper considers the existence of multiple solutions of the singular nonlocal elliptic problem −M(∫Ω | x|−ap | ∇u|p)div(|x|−ap | ∇u|p−2∇u) = λh(x) | u|r−2 u, x ∈ Ω, M(∫Ω | x|−ap | ∇u|p) | x|−ap | ∇u|p−2 (∂u/∂ν) = g(x) | u|q−2 u, on ∂Ω, where 1 < (N + 1)/2 < p < N, a < (N − p)/p. By the variational method on the Nehari manifold, we prove that the problem has at least two positive solutions when some conditions are satisfied.
Highlights
Introduction and Main ResultIn this paper, we consider the existence of multiple solutions for the singular elliptic problem:− M (∫ |x|−ap|∇u|pdx) div (|x|−ap|∇u|p−2∇u) Ω= λh (x) |u|r−2u, x ∈ Ω, (1) M (∫ |x|−ap|∇u|p) |x|−ap |∇u|p−2 ∂u ∂]= g (x) |u|q−2u, on ∂Ω, where 1 < (N + 1)/2 < p < N, a < (N − p)/p, λ > 0, Ω is an exterior domain of RN: that is, and Ω = RN \ D, where D is a bounded domain in RN with the smooth boundary ∂D( = ∂Ω), and 0 ∈ Ω. g(x) and h(x) are continuous functions, M(s) = αs + β with the parameters α, β > 0
We consider the existence of multiple solutions for the singular elliptic problem:
Motivated by [4, 5] and our previous work [14], we consider the existence of multiple solutions for problem (1) on the Nehari manifold by variational methods
Summary
We consider the existence of multiple solutions for the singular elliptic problem:. Problem like (1) is usually called nonlocal problem because of the presence of the integral over the entire domain, and this implies that (1) is no longer a pointwise identity. Such kind of problem can be traced back to the work of Kirchhoff. Motivated by [4, 5] and our previous work [14], we consider the existence of multiple solutions for problem (1) on the Nehari manifold by variational methods.
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