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Rym Chemmam 《Mediterranean Journal of Mathematics》2013,10(3):1259-1272
We consider in this paper the existence and the asymptotic behavior of positive ground state solutions of the boundary value problem $${-}\Delta u = a_{1}(x)u^{\alpha_{1}} + a_{2}(x) u^{\alpha_{2}}\,\, {\rm in}\,\, \mathbb{R}^{n}, \lim_{|x| \rightarrow \infty} u(x) = 0$$ , where α 1, α 2 < 1 and a 1, a 2 are nonnegative functions in ${C^{\gamma}_{loc}} (\mathbb{R}^{n})$ , ${0 < \gamma < 1}$ , satisfying some appropriate assumptions related to Karamata regular variation theory. 相似文献
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Positivity - We are concerned with the existence of positive solutions to the following boundary value problem in $$(0,\infty ),$$ $$\begin{aligned} \frac{1}{A}\left( A\phi \left( \left| u^{\prime... 相似文献
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For p > 1, we establish existence and asymptotic behavior of a positive continuous solution to the following boundary value problem $$\left\{\begin{array}{ll}\frac{1}{A} \left( A\Phi _{p}(u^{\prime})\right) ^{\prime}+a_{1}(r)u^{\alpha _{1}}+a_{2}(r)u^{\alpha _{2}}=0, \, {\rm in}\, (0,\infty ),\\ {\rm lim}_{r\rightarrow 0} A\Phi _{p}(u^{\prime})(r)=0, {\rm lim}_{r\rightarrow \infty } u(r)=0,\end{array}\right.$$ where \({\alpha _{1}, \alpha _{2} < p -1, \Phi _{p}(t) = t|t| ^{p-2},A}\) is a positive differentiable function and a 1, a 2 are two positive functions in \({C_{\rm loc}^{\gamma}((0, \infty )), 0 < \gamma < 1,}\) satisfying some appropriate assumptions related to Karamata regular variation theory. Also, we obtain an uniqueness result when \({\alpha _{1}, \alpha _{2} \in (1-p,p-1)}\) . Our arguments combine a method of sub and supersolutions with Karamata regular variation theory. 相似文献
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Rym Chemmam Habib Mâagli Syrine Masmoudi 《Nonlinear Analysis: Theory, Methods & Applications》2011,74(5):1555-1576
The purpose of this paper is to extend some results of the potential theory of an elliptic operator to the fractional Laplacian (−Δ)α/2, 0<α<2, in a bounded C1,1 domain D in Rn. In particular, we introduce a new Kato class Kα(D) and we exploit the properties of this class to study the existence of positive solutions of some Dirichlet problems for the fractional Laplacian. 相似文献
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