indefinite weight functions
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2017 ◽  
Vol 37 (3) ◽  
pp. 67-74
Author(s):  
Ghasem A. Afrouzi ◽  
S. Shakeri ◽  
N. T. Chung

In this work, we study the existence of positive solutions to the singular system$$\left\{\begin{array}{ll}-\Delta_{p}u = \lambda a(x)f(v)-u^{-\alpha} & \textrm{ in }\Omega,\\-\Delta_{p}v = \lambda b(x)g(u)-v^{-\alpha} & \textrm{ in }\Omega,\\u = v= 0 & \textrm{ on }\partial \Omega,\end{array}\right.$$where $\lambda $ is positive parameter, $\Delta_{p}u=\textrm{div}(|\nabla u|^{p-2} \nabla u)$, $p>1$, $ \Omega \subset R^{n} $ some for $ n >1 $, is a bounded domain with smooth boundary $\partial \Omega $ , $ 0<\alpha< 1 $, and $f,g: [0,\infty] \to\R$ are continuous, nondecreasing functions which are asymptotically $ p $-linear at $\infty$. We prove the existence of a positive solution for a certain range of $\lambda$ using the method of sub-supersolutions.


2014 ◽  
Vol 144 (6) ◽  
pp. 1113-1126 ◽  
Author(s):  
Jussi Behrndt ◽  
Shaozhu Chen ◽  
Friedrich Philipp ◽  
Jiangang Qi

Regular Sturm–Liouville problems with indefinite weight functions may possess finitely many non-real eigenvalues. In this paper we prove explicit bounds on the real and imaginary parts of these eigenvalues in terms of the coefficients of the differential expression.


Author(s):  
Jussi Behrndt

The spectral properties of a class of non-self-adjoint second-order elliptic operators with indefinite weight functions on unbounded domains Ω are investigated. It is shown, under an abstract regularity assumption, that the non-real spectrum of the associated elliptic operators in L2(Ω) is bounded. In the special case where Ω = ℝn decomposes into subdomains Ω+ and Ω− with smooth compact boundaries and the weight function is positive on Ω+ and negative on Ω−, it turns out that the non-real spectrum consists only of normal eigenvalues that can be characterized with a Dirichlet-to-Neumann map.


2012 ◽  
Vol 17 (2) ◽  
pp. 194-209
Author(s):  
Somayeh Khademloo ◽  
Shapur Heidarkhani

In this paper we consider a semilinear elliptic system with nonlinearities, indefinite weight functions and critical growth terms in bounded domains. The existence result of nontrivial nonnegative solutions is obtained by variational methods. 


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