17,789 research outputs found
Multiple Positive solutions of a -Laplacian system with nonlinear BCs
Using the theory of fixed point index, we discuss existence, non-existence,
localization and multiplicity of positive solutions for a -Laplacian
system with nonlinear Robin and/or Dirichlet type boundary conditions. We give
an example to illustrate our theory.Comment: arXiv admin note: text overlap with arXiv:1408.017
The p-Laplace equation in domains with multiple crack section via pencil operators
The p-Laplace equation
\n \cdot (|\n u|^n \n u)=0 \whereA n>0, in a bounded domain \O \subset
\re^2, with inhomogeneous Dirichlet conditions on the smooth boundary \p \O
is considered. In addition, there is a finite collection of curves
\Gamma = \Gamma_1\cup...\cup\Gamma_m \subset \O, \quad \{on which we assume
homogeneous Dirichlet boundary conditions} \quad u=0, modeling a multiple
crack formation, focusing at the origin 0 \in \O. This makes the above
quasilinear elliptic problem overdetermined. Possible types of the behaviour of
solution at the tip 0 of such admissible multiple cracks, being a
"singularity" point, are described, on the basis of blow-up scaling techniques
and a "nonlinear eigenvalue problem". Typical types of admissible cracks are
shown to be governed by nodal sets of a countable family of nonlinear
eigenfunctions, which are obtained via branching from harmonic polynomials that
occur for . Using a combination of analytic and numerical methods,
saddle-node bifurcations in are shown to occur for those nonlinear
eigenvalues/eigenfunctions.Comment: arXiv admin note: substantial text overlap with arXiv:1310.065
Multiple positive solutions for a class of p-Laplacian Neumann problems without growth conditions
For , we consider the following problem where
is either a ball or an annulus. The nonlinearity
is possibly supercritical in the sense of Sobolev embeddings; in particular our
assumptions allow to include the prototype nonlinearity
for every . We use the shooting method to get existence and multiplicity
of non-constant radial solutions. With the same technique, we also detect the
oscillatory behavior of the solutions around the constant solution .
In particular, we prove a conjecture proposed in [D. Bonheure, B. Noris, T.
Weth, {\it Ann. Inst. H. Poincar\'e Anal. Non Lin\'aire} vol. 29, pp. 573-588
(2012)], that is to say, if and , there exists
a radial solution of the problem having exactly intersections with
for a large class of nonlinearities.Comment: 22 pages, 4 figure
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