Existence and blow-up behavior of solutions for a class of degenerate elliptic equations(一类退化椭圆方程解的存在性与爆破行为)

Abstract

A class of conical degenerate elliptic equations with potential functions is studied. The existence of the ground state solution for the equation is established by studying the constrained minimization problem. The blow-up behavior is analyzed as well. Specifically, we prove that there exists at least one minimizer for the constraint minimization problem under certain conditions, but the minimizer does not exist when these conditions are unsatisfied. The blow-up behavior of the minimizer is analyzed in detail when the parameter approaches the critical value.(研究了一类具有位势函数的锥形退化椭圆方程。通过探究约束极小化问题，建立了方程基态解的存在性定理，并分析了其爆破行为。证明了当参数满足一定条件时，约束极小化问题至少存在1个可达元，但当参数不满足此条件时，不存在可达元。详细分析了当参数趋近于临界值时，可达元的爆破行为。)

Keywords

• degenerate elliptic equation(退化椭圆方程)
• constraint minimization problem(约束极小化问题)
• ground state solution(基态解)