7,013 research outputs found
Existence and multiplicity results for resonant fractional boundary value problems
We study a Dirichlet-type boundary value problem for a pseudo-differential
equation driven by the fractional Laplacian, with a non-linear reaction term
which is resonant at infinity between two non-principal eigenvalues: for such
equation we prove existence of a non-trivial solution. Under further
assumptions on the behavior of the reaction at zero, we detect at least three
non-trivial solutions (one positive, one negative, and one of undetermined
sign). All results are based on the properties of weighted fractional
eigenvalues, and on Morse theory
Nonlinear second-order multivalued boundary value problems
In this paper we study nonlinear second-order differential inclusions
involving the ordinary vector -Laplacian, a multivalued maximal monotone
operator and nonlinear multivalued boundary conditions. Our framework is
general and unifying and incorporates gradient systems, evolutionary
variational inequalities and the classical boundary value problems, namely the
Dirichlet, the Neumann and the periodic problems. Using notions and techniques
from the nonlinear operator theory and from multivalued analysis, we obtain
solutions for both the `convex' and `nonconvex' problems. Finally, we present
the cases of special interest, which fit into our framework, illustrating the
generality of our results.Comment: 26 page
Positive and nodal solutions for nonlinear nonhomogeneous parametric neumann problems
We consider a parametric Neumann problem driven by a nonlinear nonhomogeneous differential operator plus an indefinite potential term. The reaction term is superlinear but does not satisfy the Ambrosetti-Rabinowitz condition. First we prove a bifurcation-type result describing in a precise way the dependence of the set of positive solutions on the parameter λ > 0. We also show the existence of a smallest positive solution. Similar results hold for the negative solutions and in this case we have a biggest negative solution. Finally using the extremal constant sign solutions we produce a smooth nodal solution
Constant sign and nodal solutions for nonhomogeneous Robin boundary value problems with asymmetric reactions
We study a nonlinear, nonhomogeneous elliptic equation with an asymmetric
reaction term depending on a positive parameter, coupled with Robin boundary
conditions. Under appropriate hypotheses on both the leading differential
operator and the reaction, we prove that, if the parameter is small enough, the
problem admits at least four nontrivial solutions: two of such solutions are
positive, one is negative, and one is sign-changing. Our approach is
variational, based on critical point theory, Morse theory, and truncation
techniques.Comment: 22 page
Multiplicity of nontrivial solutions for elliptic equations with nonsmooth potential and resonance at higher eigenvalues
We consider a semilinear elliptic equation with a nonsmooth, locally
\hbox{Lipschitz} potential function (hemivariational inequality). Our
hypotheses permit double resonance at infinity and at zero (double-double
resonance situation). Our approach is based on the nonsmooth critical point
theory for locally Lipschitz functionals and uses an abstract multiplicity
result under local linking and an extension of the Castro--Lazer--Thews
reduction method to a nonsmooth setting, which we develop here using tools from
nonsmooth analysis.Comment: 23 page
Perturbations of nonlinear eigenvalue problems
We consider perturbations of nonlinear eigenvalue problems driven by a
nonhomogeneous differential operator plus an indefinite potential. We consider
both sublinear and superlinear perturbations and we determine how the set of
positive solutions changes as the real parameter varies. We also show
that there exists a minimal positive solution and
determine the monotonicity and continuity properties of the map
. Special attention is given to the
particular case of the -Laplacian.Comment: arXiv admin note: text overlap with arXiv:1804.1000
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