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We consider an eigenvalue problem for a certain type of quasi-linear second-order di®erential equation on the interval ð0; 1Þ. Using an appropriate version of the mountain pass theorem, we establish the existence of a positive solution in H 1 0 ð0; 1Þ for a range of values of the eigenvalue. It is shown that these solutions generate solutions of Maxwell's… (More)

- C A Stuart
- 2012

Let X be a real Banach space and Φ ∈ C 1 (X, R) a function with a mountain pass geometry. This ensures the existence of a Palais-Smale, and even a Cerami, sequence {u n } of approximate critical points for the mountain pass level. We obtain information about the location of such a sequence by estimating the distance of u n from S for certain types of set S… (More)

Under general hypotheses, we prove the existence of a nontrivial solution for the equation Lu = N(u), where u belongs to a Hilbert space H , L is an invertible continuous selfadjoint operator, and N is superlinear. We are particularly interested in the case where L is strongly indefinite and N is not compact. We apply the result to the Choquard-Pekar… (More)

- C. A. Stuart
- 2012

For a class of second order quasilinear elliptic equations we establish the existence of two non-negative weak solutions of the Dirichlet problem on a bounded domain, Ω. Solutions of the boundary value problem are critical points of C 1 −functional on H 1 0 (Ω). One solution is a local minimum and the other is of mountain pass type.

- C. A. Stuart
- 2010

We consider a class of nonlinear eigenvalue problems including equations such as −Δu(x) + q(x)u(x) + γ u(x) 2 ξ(x) 2 + u(x) 2 u = λu(x) for x ∈ R N , where γ > 0, q ∈ L ∞ (R N) and ξ ∈ L 2 (R N) are given and we are interested in eigenvalues λ ∈ R for which this equation admits a bound state, that is, a non-trivial solution in L 2 (R N). The formal… (More)

- Nathan R. Daczko, Sandra Piazolo, Uvana Meek, Catherine A. Stuart, Victoria Elliott
- Scientific reports
- 2016

Geochemical signatures throughout the layered Earth require significant mass transfer through the lower crust, yet geological pathways are under-recognized. Elongate bodies of basic to ultrabasic rocks are ubiquitous in exposures of the lower crust. Ultrabasic hornblendite bodies hosted within granulite facies gabbroic gneiss of the Pembroke Valley,… (More)

The paper considers the eigenvalue problem −∆u − αu + λg(x)u = 0 with u ∈ H 1 (R N), u = 0, where α, λ ∈ R and g(x) ≡ 0 on Ω, g(x) ∈ (0, 1] on R N \ Ω and lim |x |→+∞ g(x) = 1 for some bounded open set Ω ∈ R N. Given α > 0, does there exist a value of λ > 0 for which the problem has a positive solution? It is shown that this occurs if and only if α lies in… (More)

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