Nodal solutions for p-Laplace equations with critical growth Original Research Article

This paper is concerned with the existence and nodal character of the nontrivial solutions for the following quasilinear elliptic equations involving critical Sobolev exponents: equation(1) View the MathML source where p⩾2 and p∗=Np/(N−p) is the critical Sobolev exponent for the embedding View the MathML source. The function f satisfies some conditions given by (f1),(f2),(f3) in the paper. The main results obtained in this paper are that there exists at least a pair of nontrivial solutions for Eq. (1) provided that N⩾p2 and there exists at least a pair of nontrivial solutions uk+, uk− of Eq. (1) for each View the MathML source such that both uk+, and uk− possess exactly k nodes provided that (f(t))″⩾0 for t⩾0 and N⩾p2(p−1)+p.