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Non-existence and existence of entire solutions for a quasi-linear problem with singular and super-linear terms. (English) Zbl 1189.35104

Summary: We establish results concerning non-existence and existence of entire positive solutions for the nonlinear elliptic problem

-Δ p u=a(x)u m +λb(x)u n in N ,u>0in N ,u(x)@>|x|>>0,

where -<m<p-1<n with 1<p<N; a,b0, a,b0 are locally Hölder continuous functions and λ0 is a real parameter. The main purpose of this paper, in short, is to complement the principal theorem of B. Xu and Z. Yang [Bound. Value Probl. 2007, Article ID 16407, 8 p. (2007; Zbl 1139.35348)] showing existence and non-existence of solutions for the above problem for λ>0 appropriately.

35J61Semilinear elliptic equations
35J91Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian
35J25Second order elliptic equations, boundary value problems
35J20Second order elliptic equations, variational methods
35J67Boundary values of solutions of elliptic equations
35B09Positive solutions of PDE