Existence and Multiplicity of Nontrivial Solutions for Semilinear Elliptic Equations Involving Hardy–Sobolev Critical Exponents

Abstract

A class of semi-linear elliptic equations with the critical Hardy–Sobolev exponent has been considered. This model is widely used in hydrodynamics and glaciology, gas combustion in thermodynamics, quantum field theory, and statistical mechanics, as well as in gravity balance problems in galaxies. The PSc sequence of energy functional was investigated, and then the mountain pass lemma was used to prove the existence of at least one nontrivial solution. Also a multiplicity result was obtained. Some known results were generalized.

Keywords

• semilinear elliptic equation
• Hardy–Sobolev critical exponent
• mountain pass lemma
• (<i>PS</i>)_<i>c</i> condition