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### Open Access Journals

CPAA

In this paper we deal with a nonlinear Neumann problem
driven by the $p$--Laplacian
and with a potential function which asymptotically at infinity is
$p$--linear. Using variational methods based on critical point
theory coupled with suitable truncation techniques, we prove a
theorem establishing the existence of at least three nontrivial
smooth solutions for the Neumann problem. For the semilinear
case (i.e., $p=2$) using Morse theory, we produce one more
nontrivial smooth solution.

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.

CPAA

We consider a semilinear Neumann problem driven by the negative Laplacian plus an indefinite and
unbounded potential and with a Carathéodory reaction term.
Using variational methods based on the critical point theory,
combined with Morse theory
(critical groups),
we prove two multiplicity theorems.

PROC

We consider a nonlinear Dirichlet problem driven by the

*p*-Laplacian differential operator, with a nonlinearity concave near the origin and a nonlinear perturbation of it. We look for multiple positive solutions. We consider two distinct cases. One when the perturbation is*p*-linear and resonant with respect to $\lambda_1 > 0$ (the principal eigenvalue of (-$\Delta_p,W_0^(1,p)(Z)$)) at infinity and the other when the perturbation is*p*-superlinear at infinity. In both cases we obtain two positive smooth solutions. The approach is variational, coupled with the method of upper-lower solutions and with suitable truncation techniques.
PROC

In this paper we consider an eigenvalue problem for
a quasilinear hemivariational inequality of the type
$-\Delta_p x(z) -\lambda f(z,x(z))\in \partial j(z,x(z))$
with null boundary condition,
where $f$ and $j$ satisfy ``$p-1$-growth condition''.
We prove the existence of a nontrivial
solution for $\lambda$ sufficiently close to zero.
Our approach is variational and is based on the critical
point theory for
nonsmooth, locally Lipschitz functionals
due to Chang [4].

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).

CPAA

A nonlinear elliptic equation with $p$-Laplacian, concave-convex
reaction term depending on a parameter $\lambda>0$, and
homogeneous boundary condition, is investigated. A bifurcation
result, which describes the set of positive solutions as $\lambda$
varies, is obtained through variational methods combined with
truncation and comparison techniques.

DCDS

We consider a parametric nonlinear Dirichlet equation driven by the sum of a $p$-Laplacian and
a $q$-Laplacian
($1 < q < p < +\infty$, $p ≥ 2$)
and with a Carathéodory reaction which at $\pm\infty$
is resonant with respect to the principal eigenvalue
$\widehat{\lambda}_1(p) > 0$ of
$(-\Delta_p, W^{1,p}_0(\Omega))$.
Using critical point theory, truncation and comparison techniques and critical groups
(Morse theory), we show that for all small values of the parameter $\lambda>0$,
the problem has at least five nontrivial solutions,
four of constant sign
(two positive and two negative)
and the fifth nodal
(sign-changing).

DCDS

We consider a nonlinear Dirichlet problem driven by the
$p$-Laplace differential operator. We assume that the
Carathéodory reaction term $f(z,x)$ exhibits an asymmetric
behavior on the two semiaxes of $\mathbb{R}$. Namely, $f(z,\cdot)$ is
$(p-1)$-linear near $-\infty$ and $(p-1)$-superlinear near
$+\infty$, but without satisfying the well-known
Ambrosetti--Rabinowitz condition (AR-condition). Combining
variational methods based on critical point theory, with suitable
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, the other negative).

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