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### Open Access Journals

CPAA

In this paper we conduct a detailed study of Neumann problems driven by a nonhomogeneous differential operator plus an indefinite potential and with concave contribution in the reaction. We deal with both superlinear and sublinear (possibly resonant) problems and we produce constant sign and nodal solutions. We also examine semilinear equations resonant at higher parts of the spectrum and equations with a negative concavity.

DCDS

In this paper, we study the relationship between flow-invariant sets for
an vector field $-f'(x)$ in a Banach space, and the critical
points of the functional $f(x)$. The Mountain-Pass
Lemma, for functionals defined on a Banach
space, is generalized to a more general setting where the domain of the
functional $f$ can
be any flow-invariant set for $-f'(x)$. Furthermore, the
intuitive approach taken in the proofs provides a new
technique in proving multiple critical points.

PROC

The Tenth AIMS International Conference on Dynamical Systems, Dieren-
tial Equations and Applications took place in the magnicent Madrid, Spain,
July 7 - 11, 2014. The present volume is the Proceedings, consisting of some
carefully selected submissions after a rigorous refereeing process.

For more information please click the “Full Text” above.

For more information please click the “Full Text” above.

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CPAA

We consider a nonlinear Dirichlet problem driven by the sum of a $p$-Laplacian ($p>2$) and a Laplacian, with a reaction that is ($p-1$)-sublinear and exhibits an asymmetric behavior near $\infty$ and $-\infty$, crossing $\hat{\lambda}_1>0$, the principal eigenvalue of $(-\Delta_p, W^{1,p}_0(\Omega))$ (crossing nonlinearity). Resonance with respect to $\hat{\lambda}_1(p)>0$ can also occur. We prove two multiplicity results. The first for a Caratheodory reaction producing two nontrivial solutions and the second for a reaction $C^1$ in the $x$-variable producing three nontrivial solutions. Our approach is variational and uses also the Morse theory.

CPAA

We consider a nonlinear Neumann problem driven by a nonhomogeneous
nonlinear differential operator and with a reaction which is
$(p-1)$-superlinear without necessarily satisfying the
Ambrosetti-Rabinowitz condition. A particular case of our
differential operator is the $p$-Laplacian. By combining
variational methods based on critical point theory with truncation
techniques and Morse theory, we show that the problem has at least
three nontrivial smooth solutions, two of which have constant sign
(one positive and the other negative).

DCDS

In this paper we study an eigenvalue boundary value problem which
arises when seeking radial convex solutions of the Monge-Ampère
equations. We shall establish several criteria for the existence,
multiplicity and nonexistence of strictly convex solutions for the
boundary value problem with or without an eigenvalue parameter.

CPAA

We consider nonlinear Neumann and Dirichlet problems driven by a nonhomogeneous
differential operator and a Caratheodory reaction. Our framework incorporates $p$-Laplacian
equations and equations with the $(p,q)$-differential operator and with
the generalized $p$-mean curvature operator. Using variational methods, together with
truncation and comparison techniques and Morse theory, we prove multiplicity theorems, producing three, five or six nontrivial smooth solutions, all with sign information.

CPAA

We study a semilinear parametric Dirichlet equation with an
indefinite and unbounded potential. The reaction is the sum of a
sublinear (concave) term and of an asymptotically linear resonant
term. The resonance is with respect to any nonprincipal
nonnegative eigenvalue of the differential operator. Using
variational methods based on the critical point theory and Morse
theory (critical groups), we show that when the parameter
$\lambda>0$ is small, the problem has at least three nontrivial
smooth solutions.

CPAA

We consider a semilinear Robin problem driven by the negative Laplacian plus an indefinite and unbounded potential and a superdiffusive lotistic-type reaction. We prove bifurcation results describing the dependence of the set of positive solutions on the parameter of the problem. We also establish the existence of extreme positive solutions and determine their properties.

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