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Estimates of the principal eigenvalue of the p -Laplacian and the p -biharmonic operator

Jiří Benedikt (2015)

Mathematica Bohemica

We survey recent results concerning estimates of the principal eigenvalue of the Dirichlet p -Laplacian and the Navier p -biharmonic operator on a ball of radius R in ℝ N and its asymptotics for p approaching 1 and ∞ . Let p tend to ∞ . There is a critical radius R C of the ball such that the principal eigenvalue goes to ∞ for 0 < R ≤ R C and to 0 for R > R C . The critical radius is R C = 1 for any N ∈ ℕ for the p -Laplacian and R C = 2 N in the case of the p -biharmonic operator. When p approaches 1 , the principal eigenvalue of the Dirichlet...

Existence and multiplicity results for a nonlinear stationary Schrödinger equation

Danila Sandra Moschetto (2010)

Annales Polonici Mathematici

We revisit Kristály’s result on the existence of weak solutions of the Schrödinger equation of the form -Δu + a(x)u = λb(x)f(u), x ∈ ℝ N , u ∈ H ¹ ( ℝ N ) , where λ is a positive parameter, a and b are positive functions, while f : ℝ → ℝ is sublinear at infinity and superlinear at the origin. In particular, by using Ricceri’s recent three critical points theorem, we show that, under the same hypotheses, a much more precise conclusion can be obtained.

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