On a class of variational systems in unbounded domains
Paulo Cesar Carrião Olimpio Hiroshi Miyagaki
Conference Publications 2001, 2001(Special): 74-79 doi: 10.3934/proc.2001.2001.74
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keywords: Critical exponents p-Laplacian. elliptic systems
Existence results for the Klein-Gordon-Maxwell equations in higher dimensions with critical exponents
Paulo Cesar Carrião Patrícia L. Cunha Olímpio Hiroshi Miyagaki
Communications on Pure & Applied Analysis 2011, 10(2): 709-718 doi: 10.3934/cpaa.2011.10.709
In this paper we study the existence of radially symmetric solitary waves in $R^N$ for the nonlinear Klein-Gordon equations coupled with the Maxwell's equations when the nonlinearity exhibits critical growth. The main feature of this kind of problem is the lack of compactness arising in connection with the use of variational methods.
keywords: critical growth. Klein-Gordon-Maxwell system radially symmetric solution
Nonlinear Biharmonic Problems with Singular Potentials
Paulo Cesar Carrião R. Demarque Olímpio H. Miyagaki
Communications on Pure & Applied Analysis 2014, 13(6): 2141-2154 doi: 10.3934/cpaa.2014.13.2141
We deal with the problem \begin{eqnarray} \Delta^2 u +V(|x|)u = f(u), u\in D^{2,2}(R^N) \end{eqnarray} where $\Delta^2$ is biharmonic operator and the potential $V > 0 $ is measurable, singular at the origin and may also have a continuous set of singularities. The nonlinearity is continuous and has a super-linear power-like behaviour; both sub-critical and super-critical cases are considered. We prove the existence of nontrivial radial solutions. If $f$ is odd, we show that the problem has infinitely many radial solutions.
keywords: critical potential Biharmonic operator radially symmetric solution.
Signed solution for a class of quasilinear elliptic problem with critical growth
Claudianor Oliveira Alves Paulo Cesar Carrião Olímpio Hiroshi Miyagaki
Communications on Pure & Applied Analysis 2002, 1(4): 531-545 doi: 10.3934/cpaa.2002.1.531
In this paper we will study the existence of signed solutions for problems of the type

$-L u=\lambda h(x)|x|^{\delta}(u_{+})^q-|x|^{\gamma}(u_{-})^p, \quad $ in $\Omega$,

$u_{\pm}$ ≠0, $\quad u\in E,$ $\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$ (P)

where $\Omega$ is either a whole space $\mathbb R^N$ or a bounded smooth domain, $Lu =:$ div$(|x|^{\alpha}|\nabla u|^{m-2}\nabla u), $ $\lambda >0, \quad0 < q < m-1 < p \leq m$*$-1,$ $\alpha, $ $\delta $ and $\gamma $ are real numbers, $ N> m-\alpha, $ $m$*$=\frac{(\gamma+N)m}{(\alpha+N-m)}$, $h:\Omega \rightarrow \mathbb R$ is a positive continuous function, $u_{\pm}=\max \{\pm u,0\}$ and $E$ is a Banach space that will be defined later on. We will show that (P) has a solution that changes sign in several situations. The proof of the main results are done by using variational methods applied to the energy functional associated to $(P)$.

keywords: Signed solutions elliptic equations. critical Sobolev exponents

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