Multiple solutions of a parameter-dependent quasilinear elliptic equation

作者：Jing, Yongtao; Liu, Zhaoli^*; Wang, Zhi-Qiang^*

来源：Calculus of Variations and Partial Differential Equations, 2016, 55(6): 150.

DOI：10.1007/s00526-016-1067-7

• CRITICAL-POINT THEORY
• P-LAPLACIAN EQUATION
• SCHRODINGER-EQUATIONS
• SOLITON-SOLUTIONS
• PERTURBATION METHOD
• INVARIANT-SETS
• EXISTENCE
• WAVES
• FLOW

Consider the quasilinear elliptic equation @@@ {Sigma(N)(i,j=1) D-j (a(ij)(x,u) + 1/2 Sigma(N)(i,j=1) D-s (a(ij)(x,u)DiuDju = lambda f (x,u) in Omega, @@@ u is an element of H-1(0) ( Omega) @@@ where Omega subset of R-N (N >= 2) is a bounded domain with smooth boundary and lambda > 0 is a parameter. For the quasilinear term, we assume that a(ij) = a(ji) and a(ij)(x,s)s growth is like (1 + s(2))delta(ij). The nonlinearity of power growth f(x,s)=|s|(r-2)s with 2 < r < 4 acts as a typical example of the nonlinear term f, a case in which less results are known compared with the cases 1 < r < 2 and 4 < r < 4N/(N-2)(+). We show the structure of solutions depends keenly upon the parameter ?. More precisely, while such an equation has no nontrivial solution for lambda small, we prove that both the number of solutions with positive energies and the number of solutions with negative energies tend to infinity as lambda -> +infinity boolean AND. Nodal properties are determined for six solutions among all of them.

• 出版日期2016-12
• 单位首都师范大学; 天津大学