Positive Solutions for Elliptic Problems with the Nonlinearity Containing Singularity and Hardy-Sobolev Exponents

Yong-Yi Lan,Bi-Yun Tang,Xian Hu

doi:10.1155/2020/6727414

Yong-Yi Lan, Bi-Yun Tang + Show 1 more

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https://doi.org/10.1155/2020/6727414

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Abstract

In this paper, we study multiplicity of positive solutions for a class of semilinear elliptic equations with the nonlinearity containing singularity and Hardy-Sobolev exponents. Using variational methods, we establish the existence and multiplicity of positive solutions for the problem.

Highlights

Introduction and Main ResultsConsider the following semilinear elliptic equations withDirichlet boundary value conditions: BBBB@−Δu − u > 0, μ u jxj2 =juj2∗ ðsÞ−2 jxjs u + λu−γ, in Ω, in Ω, ð1Þ u = 0, in ∂Ω, where Ω is a smooth bounded domain in RN ðN ≥ 3Þ, 0 < s < 2, 2∗ðsÞ = 2ðN − sÞ/N − 2 is the Hardy-Sobolev critical exponent, 2∗ = 2∗ð0Þ = 2N/ðN − 2Þ is the Sobolev critical exponent, μ < μ = Δ ðN − 2Þ2/4, and γ ∈ ð0, 1Þ.The energy functional associated with problem (1) is defined by IλðuÞ =12−ð1Ω−λjγ∇ðuΩj2ju−j1μ−γjuxdj22x, d x −1ð 2∗ðsÞ Ω juj2∗ðsÞ jxjs dx ð2Þ for any u ∈ H10ðΩÞ
We study multiplicity of positive solutions for a class of semilinear elliptic equations with the nonlinearity containing singularity and Hardy-Sobolev exponents
A function u is called a weak solution of problem for all x

Summary

Introduction and Main Results

People have paid much attention to the existence of solutions for problems with the Sobolev critical exponent (the case that s = 0) (see [16,17,18,19,20,21] and the references therein); some authors considered the singular problems with the Hardy-Sobolev critical exponent (the case that s ≠ 0). We study problem (1) and obtain at least two solutions via the Nehari method It is well-known that the singular term leads to the nondifferentiability of the functional. The paper is organized as follows: in Section 2, we give some preliminaries; in Section 3, we prove Theorem 1. This idea is essentially introduced in 20]. (iii) The norm in LpðΩÞ is denoted by j·jp (iv) C, C0, C1, C2, ⋯ denote positive constants

Preliminaries

Proof of Theorem 1