Djairo G. de Figueiredo

On Superquadratic Elliptic Systems

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.

Some Remarks on a System of Quasilinear Elliptic Equations

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

Decay, symmetry and existence of solutions of semilinear elliptic systems

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.

Local superlinearity and sublinearity for indefinite semilinear elliptic problems

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.

A Liouville-Type Theorem for Elliptic Systems

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 \( {\mathbb{R}}^{N} . \)

Strongly Indefinite Functionals and Multiple Solutions of Elliptic Systems

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.

Quasilinear elliptic equations with critical exponents

has no solution if Ω ⊂ R , N ≥ 3, is bounded and starshaped with respect to some point, and 2∗ = 2N/(N − 2). In (P0) the nonlinear term is a power of u with the critical exponent (N + 2)/(N − 2). This terminology comes from the fact that the continuous Sobolev imbeddings H 0 (Ω) ⊂ L(Ω), for p ≤ 2∗ and Ω bounded, are also compact except when p = 2∗. This loss of compactness reflects in that the functional whose Euler–Lagrange equation is (P0) fails to satisfy the Palais–Smale condition. Later Brezis and Nirenberg [BN] observed that the Palais–Smale condition fails at certain levels only. Then they proved that if the nonlinear term is slightly perturbed, the new problem has a solution.

Infinitely Many Solutions of Nonlinear Elliptic Systems

In this paper we study elliptic systems of the form

Maximum Principles for Linear Elliptic Systems

\left\{ {_{\Delta _v = H_{u(x,u,v)in\Omega } }^{ - \Delta _u = H_v (x,u,v)in\Omega } } \right.