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### Open Access Journals

CPAA

We consider a nonlinear periodic problem driven by the scalar
$p$-Laplacian and a nonlinearity that exhibits a $p$-superlinear growth
near $\pm\infty$, but need not satisfy the Ambrosetti-Rabinowitz
condition.
Using minimax methods, truncations techniques and Morse theory,
we show that the problem has at least three nontrivial solutions,
two of which are of fixed sign.

keywords:
Poincaré-Hopf formula.
,
mountain pass theorem
,
Morse
theory
,
Scalar p-Laplacian
,
critical groups

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

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].

PROC

In this paper we consider quasilinear hemivariational inequality at resonance. We obtain two existence theorems using a Landesman-Lazer type condition. The method of the proof is based on the nonsmooth critical point theory for locally Lipschitz functions.

DCDS

In this paper we study boundary value problem with one
dimensional $p$-Laplacian. Assuming complete resonance at
$+\infty$ and partial resonance at $0^+$, an existence of at
least one positive solution is proved. By strengthening our
assumptions we can guarantee strict positivity of the obtained
solution.

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).

CPAA

We consider a parametric nonlinear equation driven by the Neumann $p$-Laplacian.
Using variational methods we show that when the parameter
$\lambda > \widehat{\lambda}_1$
(where $\widehat{\lambda}_1$ is the first nonzero eigenvalue of the negative Neumann $p$-Laplacian),
then the problem has at least three nontrivial smooth solutions,
two of constant sign
(one positive and one negative)
and the third nodal.
In the semilinear case
(i.e., $p=2$),
using Morse theory and flow invariance argument,
we show that the problem has three nodal solutions.

PROC

In this paper we study an optimal control problem of Bolza-type described
by evolution hemivariational inequality of second order. Sufficient conditions
for obtaining an existence result for such problem are given.

DCDS-B

We study periodic problems for nonlinear evolution inclusions defined in the framework of an evolution triple $(X,H,X^*)$ of spaces. The operator $A(t,x)$ representing the spatial differential operator is not in general monotone. The reaction (source) term $F(t,x)$ is defined on $[0,b]× X$ with values in $2^{X^*}\setminus\{\emptyset\}$. Using elliptic regularization, we approximate the problem, solve the approximation problem and pass to the limit. We also present some applications to periodic parabolic inclusions.

keywords:
Evolution triple
,
pseudomonotone map
,
coercivity
,
elliptic regularization
,
parabolic inclusion

CPAA

We consider a parametric nonlinear Dirichlet problem driven by the $(p,q)$-differential
operator,
with a reaction consisting of a ``concave'' term perturbed by a $(p-1)$-superlinear
perturbation,
which need not satisfy the Ambrosetti-Rabinowitz condition
(problem with combined or competing nonlinearities).
Using variational methods we show that for small values of the parameter the problem has at
least two nontrivial positive smooth solutions.

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