Multiple Minimal Nodal Solutions For A Quasilinear Schrödinger Equation With Symmetric Potential

We deal with the quasilinear\ Schrödinger equation -div( ∇u p-2∇u) + (λa(x) + 1) u p-2u = u q-2u, u ∈ W1,p (ℝN), where 2 ≤ p < N, λ > 0, and p < q < p* = Np/(N - p). The potential a ≥ 0 has a potential well and is invariant under an orthogonal involution of ℝN. We apply variational methods to obtain, for λ large, existence of solutions which change sign exactly once. We study the concentration behavior of these solutions as λ → ∞. By taking q close p*, we also relate the number of solutions which change sign exactly once with the equivariant topology of the set where the potential a vanishes. © 2004 Elsevier Inc. All rights reserved.3041170188Alves, C.O., Existence of positive solutions for a problem with lack of compactness involving the p-Laplacian (2002) Nonlinear Anal., 51, pp. 1187-1206Alves, C.O., Ding, Y.H., Existence, multiplicity and concentration of positive solutions for a class of quasilinear problems preprintBartsch, T., Wang, Z.Q., Multiple positive solutions for a nonlinear Schrödinger equation (2000) Z. Angew. Math. Phys., 51, pp. 366-384Benci, V., Cerami, G., The effect of the domain topology on the number of positive solutions of nonlinear elliptic problems (1991) Arch. Rational Mech. Anal., 114, pp. 79-93Brézis, H., Lieb, E., A relation between pointwise convergence of functions and convergence of functionals (1983) Proc. Amer. Math. Soc., 88, pp. 486-490Castro, A., Clapp, M., The effect of the domain topology on the number of minimal nodal solutions of an elliptic equation at critical growth in a symmetric domain (2003) Nonlinearity, 16, pp. 579-590Clapp, M., On the number of positive symmetric solutions of a nonautonomous semilinear elliptic problem (2000) Nonlinear Anal., 42, pp. 405-422Clapp, M., Ding, Y.H., Positive solutions of a Schrödinger equation with critical nonlinearity Z. Angew. Math. Phys., , in pressClapp, M., Puppe, D., Critical point theory with symmetries (1991) J. Reine Angew. Math., 418, pp. 1-29Ekeland, I., On the variational principle (1974) J. Math. Anal. Appl., 47, pp. 324-353Furtado, M.F., A relation between the domain topology and the number of minimal nodal solutions for a quasilinear elliptic problem preprintLions, P.L., The concentration compactness principle in the calculus of variations. The limit case. I (1985) Rev. Mat. Iberoamericana, 1, pp. 145-201de Morais Filho, D.C., Souto, M.A.S., Do O, J.M., A compactness embedding lemma, a principle of symmetric criticality and applications to elliptic problems (2000) Proyecciones, 19, pp. 1-17Palais, R.S., The principle of symmetric criticality (1979) Comm. Math. Phys., 69, pp. 19-30Simon, J., Regularit de la solution d'une equation non lineaire dans ℝN Springer-Verlag Berlin (1978) Lecture Notes in Math., 665. , P. Benilan (Ed.)Smets, D., A concentration-compactness lemma with applications to singular eigenvalues problems (1999) J. Funct. Anal., 167, pp. 463-480Struwe, M., (1990) Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, , Berlin: Springer-VerlagWillem, M., (1996) Minimax Theorems, , Basel: Birkhäuse