Ground State Solutions for Quasilinear Schrödinger Equations with Critical Growth and Lower Power Subcritical Perturbation

We study the following generalized quasilinear Schrödinger equation: -(g^{2}(u)\nabla u)+g(u)g^{\prime}(u)|\nabla u|^{2}+V(x)u=h(u),\quad x\in% \mathbb{R}^{N}, where {N\geq 3} , {g\colon\mathbb{R}\rightarrow\mathbb{R}^{+}} is an even differentiable function such that {g^{\prime}(t)\geq 0} for all {t\geq 0} , {h\in C^{1}(\mathbb{R},\mathbb{R})} is a nonlinear function including critical growth and lower power subcritical perturbation, and the potential {V(x)\colon\mathbb{R}^{N}\rightarrow\mathbb{R}} is positive. Since the subcritical perturbation does not satisfy the (AR) condition, the standard variational method cannot be used directly. Combining the change of variables and the monotone method developed by Jeanjean in [L. Jeanjean, On the existence of bounded Palais–Smale sequences and application to a Landesman–Lazer-type problem set on {\mathbf{R}}^{N} , Proc. Roy. Soc. Edinburgh Sect. A 129 1999, 4, 787–809], we obtain the existence of positive ground state solutions for the given problem.

CONTINUITY OF THE VARIATIONAL EIGENVALUES OF THE p-LAPLACIAN WITH RESPECT TO p

In this note it is shown that a result of Champion and De Pascale [‘Asymptotic behavior of nonlinear eigenvalue problems involving p-Laplacian type operators’, Proc. Roy. Soc. Edinburgh Sect. A 137 (2007), 1179–1195] implies that the variational eigenvalues of the p-Laplacian are continuous with respect to p.

Erratum: Comparison principles for strongly coupled reaction-diffusion equations

[Proc. Roy. Soc. Edinburgh Sect. A106 (1987), 209–219]

Corrigendum: Enumeration of smooth labelled graphs

[Proc. Roy. Soc. Edinburgh Sect. A91 (1982), 205–212]

Erratum: Multiparameter definiteness conditions II [Proc. Roy. Soc. Edinburgh Sect. A 93 (1982), 47-61]

Page 58: Lemma 7.1 is incorrect. It is claimed that Ao =cocl for c =k−l when in fact the result is proved for c = (dim H)−1. Thus while Corollaries 7.4 and 7.6 still stand (as does Corollary 7.3 by alternative arguments) the remaining results in Section 7 must be withdrawn.