Abstract
In this paper, we study the Neumann boundary value problem to a quasilinear elliptic equation with the critical Sobolev exponent and critical Hardy–Sobolev exponent, and prove the existence of nontrivial nonnegative solution by means of variational method.
Highlights
Introduction and main resultsIn this paper, we discuss the following quasilinear elliptic problem with critical Sobolev exponent and critical Hardy–Sobolev exponent: ⎧ ⎨– p u + λ|u|p–2 u = Q(x)|u|p∗ P(x)|u|p∗ (t)–2 |x|t u ⎩|∇ u|p–2
Motivated by the results of the above-mentioned papers, in this paper we aim to show the existence of nontrivial nonnegative solution to problem (1.1) by the variational method
We prove that the energy functional Jλ(u) satisfies the geometry of mountain pass lemma
Summary
Problem (1.1) has at least one nontrivial nonnegative solution for each λ > 0 and N > 2p – 1. Theorem 1.3 Suppose that λ ≤ 0, and the coefficients Q(x), P(x) satisfy the following conditions:. There exists a constant μ∗ > 0 such that Problem (1.1) has at least one nontrivial nonnegative solution for all 0 ≤ μ < μ∗, where μ = –λ, and μ∗ will be given in Sect. The proof of Theorem 1.3 is given in Sect. Definition 2.1 A function u ∈ W 1,p(Ω) is said to be a weak solution of Problem (1.1) if it satisfies. The corresponding nonnegative solutions of Problem (1.1) are equivalent to the critical points of the energy functional. In order to obtain the existence of solution to Problem (1.1), we need the following lemmas.
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