Petr Girg

Basis properties of eigenfunctions of the p-laplacian

For p ≥ 12 11 the eigenfunctions of the non-linear eigenvalue problem for the p-Laplacian on the interval (0,1) are shown to form a Riesz basis of L 2 (0,1) and a Schauder basis of Lq(0,1) whenever 1 < q < oo.

Dirichlet problem with nonlinearity depending only on the derivative

Abstract We study the existence of solutions for the following problem: where ƒ ϵ C[0, π], g ϵ C( R ) is bounded and has limits limu → ± ∞ g(u). We also give information on the set of ƒ for those that solution exists, relating it with the corresponding linear problem.

Superlinear elliptic systems with reaction terms involving product of powers

Abstract We consider a system of the form − Δ u = λ g 1 ( x , u , v ) in Ω ; − Δ v = λ g 2 ( x , u , v ) in Ω ; u = 0 = v on ∂ Ω , where λ > 0 is a parameter, Ω ⊂ R N ( N ≥ 2 ) is a bounded domain with sufficiently smooth boundary ∂ Ω (a bounded open interval if N = 1 ). Here g i ( x , s , t ) : Ω × [ 0 , + ∞ ) × [ 0 , + ∞ ) → R ( i = 1 , 2 ) are Caratheodory functions that exhibit superlinear growth at infinity involving product of powers of u and v . Using re-scaling argument combined with Leray–Schauder degree theory and a version of Leray–Schauder continuation theorem, we show that the system has a connected set of positive solutions for λ small.

-Trigonometric and -Hyperbolic Functions in Complex Domain

We study extension of -trigonometric functions and and of -hyperbolic functions and to complex domain. Our aim is to answer the question under what conditions on these functions satisfy well-known relations for usual trigonometric and hyperbolic functions, such as, for example, . In particular, we prove in the paper that for the -trigonometric and -hyperbolic functions satisfy very analogous relations as their classical counterparts. Our methods are based on the theory of differential equations in the complex domain using the Maclaurin series for -trigonometric and -hyperbolic functions.

The first nontrivial curve in the fučĺk spectrum of the dirichlet laplacian on the ball consists of nonradial eigenvalues

It is well-known that the second eigenvalue λ2 of the Dirichlet Laplacian on the ball is not radial. Recently, Bartsch, Weth and Willem proved that the same conclusion holds true for the so-called nontrivial (sign changing) Fučík eigenvalues on the first curve of the Fučík spectrum which are close to the point (λ2, λ2). We show that the same conclusion is true in dimensions 2 and 3 without the last restriction.