Radial solutions of Dirichlet problems with concave–convex nonlinearities

Dirichlet Problems with Asymmetric Nonlinearities

Nodal solutions of nonlinear elliptic Dirichlet problems on radial domains

Let Ω ⊂ R be a ball or an annulus and f : R → R absolutely continuous, superlinear, subcritical, and such that f(0) = 0. We prove that the least energy nodal solution of −∆u = f(u), u ∈ H 0 (Ω), is not radial. We also prove that Fučik eigenfunctions, i. e. solutions u ∈ H 0 (Ω) of −∆u = λu − μu−, with eigenvalue (λ, μ) on the first nontrivial curve of the Fučik spectrum, are not radial. A relat...

On the Number of Radially Symmetric Solutions to Dirichlet Problems with Jumping Nonlinearities of Superlinear Order

This paper is concerned with the multiplicity of radially symmetric solutions u(x) to the Dirichlet problem ∆u+ f(u) = h(x) + cφ(x) on the unit ball Ω ⊂ RN with boundary condition u = 0 on ∂Ω. Here φ(x) is a positive function and f(u) is a function that is superlinear (but of subcritical growth) for large positive u, while for large negative u we have that f ′(u) < μ, where μ is the smallest po...

Non-radial Solutions with Orthogonal Subgroup Invariance for Semilinear Dirichlet Problems

A semilinear elliptic equation, −∆u = λf(u), is studied in a ball with the Dirichlet boundary condition. For a closed subgroup G of the orthogonal group, it is proved that the number of non-radial G invariant solutions diverges to infinity as λ tends to ∞ if G is not transitive on the unit sphere.

Existence Theory for Single and Multiple Solutions to Semipositone Discrete Dirichlet Boundary Value Problems with Singular Dependent Nonlinearities

In this paper we establish the existence of single and multiple solutions to the semiposi-tone discrete Dirichlet boundary value problem ∆ 2 y(i − 1) + µf (i, y(i)) = 0, i ∈ {1, 2, ..., T } y(0) = y(T + 1) = 0, where µ > 0 is a constant and our nonlinear term f (i, u) may be singular at u = 0. .