Structure of boundary blow-up solutions for quasi-linear elliptic problems II: small and intermediate solutions

Abstract

The structure of positive boundary blow-up solutions to quasi-linear elliptic problems of the form −Δ p u= λf( u), u=∞ on ∂Ω, 1< p<∞, is studied in a bounded smooth domain Ω⊂ R N(N⩾2) , for a class of nonlinearities f∈ C 1((0,∞)⧹{ z 2})∩ C 0[0,∞) satisfying f(0)= f( z 1)= f( z 2)=0 with 0< z 1< z 2, f<0 in (0, z 1)∪( z 2,∞), f>0 in ( z 1, z 2). Large, small and intermediate solutions are obtained for λ sufficiently large. It is known from Part I (see Structure of boundary blow-up solutions for quasilinear elliptic problems, part (I): large and small solutions, preprint), that the large solution is the unique large solution to the problem. We will see that the small solution is also the unique small solution to the problem while there are infinitely many intermediate solutions. Our results are new even for the case p=2.