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
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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