Djairo G. de Figueiredo - Selected Papers

Let

X

be a real Banach space,

X*

its conjugate space, and let (

w

,

u

) denote the pairing between

w

in

X

*

and

u

in

X

.

Let

X

be a real normed space with norm || ||,

T

the

radial projection mapping

defined by

$$ Tx = x,\quad {\text{if}}\left\| x \right\| \leqq 1,\quad {\text{and}}\quad Tx = x/\left\| x \right\|,\quad {\text{if}}\left\| x \right\| \geqq 1. $$

A nonlinear integral equation of Hammerstein type is one of the form

$$ u(x) + \int_{G} {K(x,\,y)f(y,\,u(y))dy} = w(x). $$

Ler

X

be a normed space. A mapping

T

:

$$ D \to X $$

is called a contraction if

$$ \left\| {Tx - Ty} \right\| \le \left\| {x - y} \right\| $$

for all

x

and

y

in D.

Let

X

be a real Banach space and

X

* its conjugate Banach space. Let

A

be an unbounded monotone linear mapping from

X

to

X

* and

N

a potential mapping from

X

* to

X

. In this paper we establish the existence of a solution of the equation

u

+

ANu

=

v

for a given v in

X

* using variational method. Our method consists in using a splitting of

A

via an auxiliary Hilbert space and solving an equivalent equation in this auxiliary Hilbert space. In §2, we prove the same result in the case when

X

is a Hilbert space using the natural splitting of

A

in terms of its square root. We do this to compare and contrast the proofs in the two cases.

Let

$$ \Upomega $$

be a bounded domain in R

N

, and

$$ {\text{Lu}} = \sum\nolimits_{\left| \upalpha \right| \le \text{m}\;\left| \upbeta \right| \le \text{m}} {( - 1)^{\left| \upbeta \right|} \text{D}^{\upbeta } } (\text{a}_{\upalpha \upbeta } (\text{x})\text{D}^{\upalpha }\text{u}) $$

be a uniformly strongly elliptic operator acting on functions defined in

$$ \Upomega $$

.

In this paper we investigate the existence of solutions for the Dirichlet problem Lu = f(x, u), in

$$ \varOmega . $$

L

et

ℒ be a second order symmetric uniformly elliptic operator with smooth coefficients acting on real valued functions defined in a bounded smooth domain Ω in R

N.

In this paper we establish some existence results for problems of the type.

1. Let L be a uniformly strongly elliptic operator of order 2 m with smooth coefficients acting on real–valued functions defined in a bounded domain Ω in R

N

.

We are interested here in the existence of positive solutions of the problem.

L

et

be

a smooth bounded domain in

R

N

. We consider the semilinear elliptic boundary value problem.

This paper contains a variational treatment of the Ambrosetti-Prodi problem, including the superlinear case. The main result extends previous ones by Kazdan-Warner, Amann-Hess, Dancer, K. C. Chang and de Figueiredo. The required abstract results on critical point theory of functionals in Hilbert space are all proved using Ekeland’s variational principle. These results apply as well to other superlinear elliptic problems provided an ordered pair of a sub- and a supersolution is exhibited.

The Ambrosetti-Prodi boundary value problem with an asymptotically linear nonlinearity is considered. Under general conditions on the nonlinearity it is shown that there exist positive and negative solutions. In the case when the domain is a ball in

Rn

and the nonlinearity “crosses” the first

n

eigenvalues, corresponding to radial eigenfunctions, it is proved that there are at least

n

+ 1 radial solution.

The question of existence of positive solutions for semilinear elliptic problems of the type −Δ

u

=

f

(

u

) in Ω,

u

= 0 on ∂Ω, depends very strongly on the behavior of the function

f: R

+

→

R

at 0 and at +∞.

Let us consider the Dirichlet problem

$$ - \Updelta u = f\left( u \right) $$

and

$$ u > 0 $$

in

$$ \Upomega $$

, u

= 0 on

$$ \partial \Upomega $$

.

The Dirichlet problem in a bounded region for elliptic systems of the form (*)

$$ - \Updelta u = f\left( {x,u} \right) - v,\quad - \Updelta v = \delta u - \gamma v $$

is studied. For the question of existence of positive solutions the key ingredient is a maximum principle for a linear elliptic system associated with (*). A priori bounds for the solutions of (*) are proved under various types of growth conditions on f. Variational methods are used to establish the existence of pairs of solutions for (*).

We shall discuss here the uniqueness of solution of the Dirichlet problem

$$ - \Updelta u = f(u) + \rho h(x)\quad {\text{in}}\,\Upomega , {\text{ u = 0}}\quad {\text{on }}\partial \Upomega , $$

for large values of the real parameter

$$ \rho $$

.

Consider the Dirichlet problem

$$ - \Updelta u = g(x,u)\,{\text{in}}\Upomega ,\mu = {\text{ 0 on}}\partial \Upomega $$

where Q is a bounded smooth domain in

R

N

, N

> 2.

Let

Ω

be a bounded open subset in

$$ \hbox{I\!R}^{N} $$

with smooth boundary ∂

Ω

and let

f

(

x

,

s

) be a real-valued Caratheodory function on

$$ \Upomega\,\times\,\hbox{I\!R} $$

. We consider the semilinear Dirichlet problem.

In this paper we obtain some sufficient (necessary) conditionsfor the validity of the maximum principle for cooperative and non-cooperative elliptic systems.

In this paper we study the solvability of the Neumann problem

We consider a class of second order ordinary differential equations with

jumping nonlinearities

that

cross

the first

k

eigenvalues, under zero boundary conditions and, by means of a Morse Index computation, show the existence of

exactly 2

k

solutions.

In this paper, we show that the strict monotonicity of the eigenvalues of an uniformly elliptic operator of second order is equivalent to a unique continuation property.

We investigate the existence of positive solutions of a Dirich let problem for the system

$$ - \Updelta u = f(v),\,\Updelta v = g(u) $$

in a bounded convex domain Ω of

$$ {\mathbb{R}}^{N} $$

with smooth boundary. In particular

L

∞

a priori bounds are obtained in the same spirit as in De Figueiredo—Lions—Nussbaum [7].

In this article we study the existence of solutions for the elliptic system

$$ - \Updelta u = \frac{\partial H}{\partial v}\left( {u,v,x} \right)\,{\text{in}}\Upomega , $$

$$ - \Updelta v = \frac{\partial H}{\partial v}\left( {u,v,x} \right)\,{\text{in}}\Upomega , $$

$$ u = 0,v = 0\,{\text{on}}\,\partial \Upomega $$

where

$$ \Upomega $$

is a bounded open subset of

$$ {\mathbb{R}}^{N} $$

with smooth boundary

$$ \partial \Upomega $$

and the function

H

:

$$ {\mathbb{R}}^{2} \times \overline{\Upomega } \to {\mathbb{R}} $$

, is of class

C

1

.

A priori estimates for solutions of superlinear elliptic problems can be established by a blow up technique. Such a method has been used by Gidas-Spruck [GS1] for the case of a single equation. Similar arguments can also be used in the case of systems. We refer to the work of Jie Qing [J] and M.A. Souto [S]. As in the scalar case, the treatment of systems poses the question of the validity of a result which is referred as a Liouville-type theorem for solutions of systems of elliptic equations in

$$ {\mathbb{R}}^{N} . $$

A new Maximum Principle for elliptic equations has appeared recently in works of Berestycki, Nirenberg and Varadhan, see for instance. The aim of this note is to extend this Maximum Principle to cooperative elliptic systems and then to apply it to monotonicity and symmetry properties of nonlinear elliptic systems. In this way we get more general results than the ones in and, with even simpler proofs.

In this paper we study the solvability of problems of the type

$$ - \Updelta u = f(x,u)\;{\text{in}}\;\Upomega ,\;u = 0\;{\text{on}}\;\partial \Upomega , $$

where

$$ \Upomega $$

is some bounded domain in

R

2

, and the function

f

(

x

,

s

) has the maximal growth on

s

which allows to treat problem variationally in

$$ H_{0}^{1} (\Upomega ) $$

.

Our utmost aim was originally to establish the existence of solutions of system. However, in this effort we got sidetracked to consider other questions. In this way we come to interesting results on asymptotic behavior of solutions, on symmetry properties of such solutions, and the existence of ground states.

In this paper we study elliptic systems of the form

$$ \left\{ {\begin{array}{*{20}l} { - \Updelta u = H_{v} (x,u,v)\;{\text{in}}\;\Upomega } \\ { - \Updelta v = H_{u} (x,u,v)\;{\text{in}}\;\Upomega } \\ \end{array} } \right. $$

where

$$ \Upomega \subset {\mathbb{R}}^{N} $$

,

N

≥ 3, is a smooth bounded domain and

$$ H{:} \, \overline{\Upomega } \times {\mathbb{R}}\times{\mathbb{R}}\to{\mathbb{R}} $$

is a

C

1

-function.

A proof is given of an a priori bound for positive solutions of semilinear elliptic systems with nonlinearities depending on the gradients.

In this paper we study the functional

$$ \Upphi (u,v) = \frac{1}{p}\int_{\Upomega } {\left| {\nabla u} \right|^{p} } + \frac{1}{q}\int_{\Upomega } {\left| {\nabla u} \right|}^{q} - \int_{\Upomega } {F(x,u,v),} $$

where p and q rae real numbers larger than 1

This paper deals with existence and multiplicity of solutions to the nonlinear Schrödinger equation of the type

$$ - \Updelta u + (\lambda a(x) + a_{0} (x))u = f(x,\,u),\quad u\; \in \;H^{1} ({\mathbb{R}}^{N} ). $$

We improve some previous results in two respects: we do not require

a

0

to be positive on one hand, and allow

f

(

x

,

u

) to be critical nonlinear on the other hand.

We study existence and multiplicity of solutions of the elliptic system

$$ \left\{{\begin{array}{*{20}l} {- \Updelta u = H_{u} (x,u,v)} \hfill & {{\text{in}}\,\Upomega,} \hfill \\ {- \Updelta v = H_{v} (x,u,v)} \hfill & {{\text{in}}\,\Upomega,\quad u(x) = v(x) = 0\quad {\text{on}}\,\partial \Upomega,} \hfill \\ \end{array}} \right. $$

where

$$ \Upomega \subset {\mathbb{R}}^{N},\,N \ge 3, $$

is a smooth bounded domain and

$$ H \in {\mathcal{C}}^{1} (\overline{\Upomega} \times {\mathbb{R}}^{2},{\mathbb{R}}). $$

We assume that the nonlinear term

$$ H(x,\,u,\,v)\sim \left| u \right|^{p} + \left| v \right|^{q} + R(x,\,u,\,v)\,{\text{with}}\,\mathop {\lim}\limits_{{\left| {(u,v)} \right| \to \infty}} \frac{R(x,\,u,\,v)}{{\left| u \right|^{p} + \left| v \right|^{q}}} = 0, $$

where

$$ p \in (1,\,2^{*}),\,2^{*} : = 2N/(N - 2),\,{\text{and}}\,q \in (1,\,\infty). $$

So some supercritical systems are included. Nontrivial solutions are obtained. When

H

(

x, u, v

) is even in (

u

,

v

), we show that the system possesses a sequence of solutions associated with a sequence of positive energies (resp. negative energies) going toward infinity (resp. zero) if

p

> 2 (resp.

p

< 2). All results are proved using variational methods. Some new critical point theorems for strongly indefinite functionals are proved.

In this paper the usual notions of superlinearity and sublinearity for semilinear problems like _Du ¼ f ðx; uÞ are given a local form and extended to indefinite nonlinearities. Here f ðx; sÞ is allowed to change sign or to vanish for s near zero as well as for s near infinity. Some of the well-known results of Ambrosetti–Bre′zis–Cerami are partially extended to this context.

In this paper we study superlinear elliptic systems in Hamiltonian form. Using an Orlicz-space setting, we extend the notion of critical growth to superlinear nonlinearities which do not have a polynomial growth. Existence of nontrivial solutions is proved for superlinear nonlinearities which are subcritical in this generalized sense.

This paper is a contribution to the study of boundary value problems for systems of elliptic partial differential equations.

We study the Ambrosetti–Prodi problem for nonlinear elliptic equations and systems, with uniformly elliptic operators in non-divergence form and non-smooth coefficients, and with non-linearities with linear or power growth.

In this paper we study the existence of radially symmetric positive solutions in

$$ H_{\text{rad}}^{1} ({\mathbb{R}}^{N} )\; \times \;H_{\text{rad}}^{1} ({\mathbb{R}}^{N} ) $$

of the elliptic system:

$$ - \Updelta u + u - (\alpha u^{2} + \beta v^{2} )u\, = \,0, $$

$$ - \Updelta v + \omega^{2} v - (\beta u^{2} + \gamma v^{2} )v\, = \,0, $$

N = 1, 2, 3, where

α

and

γ

are positive constants ( will be allowed to be negative). This system has trivial solutions of the form (

ϕ

, 0) and (0,) where

ϕ

and are nontrivial solutions of scalar equations. The existence of nontrivial solutions for some values of the parameters

α

,

β

,

γ

,

ω

has been studied recently by several authors [A. Ambrosetti, E. Colorado, Bound and ground states of coupled nonlinear Schrödinger equations, C. R. Acad. Sci. Paris, Ser. I 342 (2006) 453–458; T.C. Lin, J. Wei, Ground states of

N

coupled nonlinear Schrödinger equations in

R

n

,

n

≤ 3, Comm. Math. Phys. 255 (2005) 629–653; T.C. Lin, J. Wei, Ground states of

N

coupled nonlinear Schrödinger equations in

R

n

,

n

≤ 3, Comm. Math. Phys., Erratum, in press; L. Maia, E.Montefusco, B. Pellacci, Positive solutions for a weakly coupled nonlinear Schrödinger system, preprint; B. Sirakov, Least energy solitary waves for a system of nonlinear Schrödinger equations in

R

N

, preprint; J. Yang, Classification of the solitary waves in coupled nonlinear Schrödinger equations, Physica D 108 (1997) 92–112]. For

N

= 2, 3, perhaps the most general existence result has been proved in [A. Ambrosetti, E. Colorado, Bound and ground states of coupled nonlinear Schrödinger equations, C. R. Acad. Sci. Paris, Ser. I 342 (2006) 453–458] under conditions which are equivalent to ours. Motivated by some numerical computations, we return to this problem and, using our approach, we give a more detailed description of the regions of parameters for which existence can be proved. In particular, based also on numerical evidence, we show that the shape of the region of the parameters for which existence of solution can be proved, changes drastically when we pass from dimensions

N

= 1, 2 to dimension

N

= 3. Our approach differs from the ones used before. It relies heavily on the spectral theory for linear elliptic operators. Furthermore, we also consider the case

N

= 1 which has to be treated more extensively due to some lack of compactness for even functions. This case has not been treated before.

We establish a priori bounds for positive solutions of semilinear elliptic systems of the form

$$ \left\{ {\begin{array}{*{20}c} { - \Updelta u = g(x,u,v)} & {{\text{in}}\;\;\Upomega ,} \\ { - \Updelta v = f(x,u,v)} & {{\text{in}}\;\;\Upomega ,} \\ {u\; > \;0,\quad v\; > \;0} & {{\text{in}}\;\;\Upomega ,} \\ {u = v = 0} & {{\text{on}}\;\;\partial \Upomega ,} \\ \end{array} } \right. $$

where Ω is a bounded and smooth domain in ℝ

2

. We obtain results concerning such bounds when

f

and

g

depend exponentially on

u

and

v

. Based on these bounds, existence of positive solutions is proved.

In this paper we look for existence results for nontrivial solutions to the system,

$$ \left\{ {\begin{array}{*{20}c} { - \Updelta u = \frac{{v^{p} }}{{\left| x \right|^{\alpha } }}} & {{\text{in}}\,\Upomega ,} \\ { - \Updelta v = \frac{{u^{p} }}{{\left| x \right|^{\beta } }}} & {{\text{in}}\,\Upomega ,} \\ \end{array} } \right. $$

with Dirichlet boundary conditions,

u

=

v

= 0 on ∂Ω and α, β <

N.

We find the existence of a critical hyperbola in the (

p, q)

plane (depending on α, β and

N

) below which there exists nontrivial solutions. For the proof we use a variational argument (a linking theorem).

We study the existence, nonexistence and multiplicity of positive solutions for a family of problems

$$ - \Updelta_{p} u = f_{\lambda } \,(x,\,u),\,u \in \,W_{0}^{1,p} (\Upomega ) $$

, where Ω is a bounded domain in

$$ {\mathbb{R}}^{N} ,\,N > p $$

, and λ > 0 is a parameter. The family we consider includes the well-known nonlinearities of Ambrosetti–Brezis–Cerami type in a more general form, namely

$$ \lambda a(x)u^{q} + b(x)u^{r} $$

, where

$$ 0 \leqslant q < p - 1 < r \leqslant p* - 1 $$

. Here the coefficient

a

(

x

) is assumed to be nonnegative but

b

(

x

) is allowed to change sign, even in the critical case. Preliminary results of independent interest include the extension to the

p

-Laplacian context of the Brezis–Nirenberg result on local minimization in

$$ W_{0}^{1,p} \,{\text{and}}\,C_{0}^{1} ,\,a\,C^{1,\alpha } $$

estimate for equations of the form

$$ - \Updelta_{p} u = h(x,u) $$

with

h

of critical growth, a strong comparison result for the

p

-Laplacian, and a variational approach to the method of upper–lower solutions for the

p

-Laplacian.