scholarly journals Existence Results for Nonlinear Anisotropic Elliptic Equation

2017 ◽  
Vol 2 (5) ◽  
pp. 160-166 ◽  
Author(s):  
Youssef Akdim ◽  
Mostafa El moumni ◽  
Abdelhafid Salmani
Keyword(s):  
Elliptic Equation ◽  
Existence Results ◽  
Anisotropic Elliptic Equation

scholarly journals Radial quasilinear elliptic problems with singular or vanishing potentials

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Marino Badiale ◽  
Michela Guida ◽  
Sergio Rolando
Keyword(s):  
Elliptic Equation ◽  
Continuous Function ◽  
Quasilinear Elliptic Equation ◽  
Existence Results ◽  
Lebesgue Spaces ◽  
Linear Behavior ◽  
Nonnegative Solutions ◽  
Quasilinear Elliptic Problems ◽  
Vanishing Potentials ◽  
Quasilinear Elliptic

<p style='text-indent:20px;'>In this paper we continue the work that we began in [<xref ref-type="bibr" rid="b6">6</xref>]. Given <inline-formula><tex-math id="M1">\begin{document}$ 1&lt;p&lt;N $\end{document}</tex-math></inline-formula>, two measurable functions <inline-formula><tex-math id="M2">\begin{document}$ V\left(r \right)\geq 0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ K\left(r\right)&gt; 0 $\end{document}</tex-math></inline-formula>, and a continuous function <inline-formula><tex-math id="M4">\begin{document}$ A(r) &gt;0 $\end{document}</tex-math></inline-formula> (<inline-formula><tex-math id="M5">\begin{document}$ r&gt;0 $\end{document}</tex-math></inline-formula>), we consider the quasilinear elliptic equation</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ -\mathrm{div}\left(A(|x| )|\nabla u|^{p-2} \nabla u\right) +V\left( \left| x\right| \right) |u|^{p-2}u = K(|x|) f(u) \quad \text{in }\mathbb{R}^{N}, $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where all the potentials <inline-formula><tex-math id="M6">\begin{document}$ A,V,K $\end{document}</tex-math></inline-formula> may be singular or vanishing, at the origin or at infinity. We find existence of nonnegative solutions by the application of variational methods, for which we need to study the compactness of the embedding of a suitable function space <inline-formula><tex-math id="M7">\begin{document}$ X $\end{document}</tex-math></inline-formula> into the sum of Lebesgue spaces <inline-formula><tex-math id="M8">\begin{document}$ L_{K}^{q_{1}}+L_{K}^{q_{2}} $\end{document}</tex-math></inline-formula>. The nonlinearity has a double-power super <inline-formula><tex-math id="M9">\begin{document}$ p $\end{document}</tex-math></inline-formula>-linear behavior, as <inline-formula><tex-math id="M10">\begin{document}$ f(t) = \min \left\{ t^{q_1 -1}, t^{q_2 -1} \right\} $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M11">\begin{document}$ q_1,q_2&gt;p $\end{document}</tex-math></inline-formula> (recovering the power case if <inline-formula><tex-math id="M12">\begin{document}$ q_1 = q_2 $\end{document}</tex-math></inline-formula>). With respect to [<xref ref-type="bibr" rid="b6">6</xref>], in the present paper we assume some more hypotheses on <inline-formula><tex-math id="M13">\begin{document}$ V $\end{document}</tex-math></inline-formula>, and we are able to enlarge the set of values <inline-formula><tex-math id="M14">\begin{document}$ q_1 , q_2 $\end{document}</tex-math></inline-formula> for which we get existence results.</p>

Multiplicity and Existence Results for a Nonlinear Elliptic Equation With Sobolev Exponent

2010 ◽  
Vol 10 (3) ◽  
Author(s):  
Zakaria Bouchech ◽  
Hichem Chtioui
Keyword(s):  
Elliptic Equation ◽  
Boundary Conditions ◽  
Nonlinear Elliptic Equation ◽  
Dirichlet Boundary ◽  
Existence Results ◽  
Dirichlet Boundary Conditions ◽  
Nonlinear Elliptic ◽  
Sobolev Exponent

AbstractIn this paper we consider the following nonlinear elliptic equation with Dirichlet boundary conditions: -Δu = K(x)u

Least energy radial sign-changing solutions for a singular elliptic equation in lower dimensions

2014 ◽  
Vol 16 (04) ◽  
pp. 1350048
Author(s):  
Shuangjie Peng ◽  
Yanfang Peng
Keyword(s):  
Boundary Condition ◽  
Elliptic Equation ◽  
Variational Method ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Nehari Manifold ◽  
Existence Results ◽  
Singular Elliptic Equation ◽  
Sign Changing Solutions ◽  
Least Energy

We study the following singular elliptic equation [Formula: see text] with Dirichlet boundary condition, which is related to the well-known Caffarelli–Kohn–Nirenberg inequalities. By virtue of variational method and Nehari manifold, we obtain least energy sign-changing solutions in some ranges of the parameters μ and λ. In particular, our result generalizes the existence results of sign-changing solutions to lower dimensions 5 and 6.

scholarly journals Multiple solutions to a nonlinear elliptic equation involving Paneitz type operators

2004 ◽  
Vol 2004 (46) ◽  
pp. 2453-2472
Author(s):  
Abdallah El Hamidi
Keyword(s):  
Elliptic Equation ◽  
Positive Solutions ◽  
Riemannian Manifolds ◽  
Multiple Solutions ◽  
Nonlinear Elliptic Equation ◽  
Existence Results ◽  
Real Parameter ◽  
Nonlinear Elliptic ◽  
Existence Of Positive Solutions ◽  
Characteristic Values

This paper deals with an elliptic equation involving Paneitz type operators on compact Riemannian manifolds with concave-convex nonlinearities and a real parameter. Nonlocal and multiple existence results are established. Characteristic values of the real parameter are introduced and their role in the change of the energy sign and the existence of positive solutions are highlighted.

scholarly journals On singular quasilinear elliptic equations with data measures

2021 ◽  
Vol 10 (1) ◽  
pp. 1284-1300
Author(s):  
Nour Eddine Alaa ◽  
Fatima Aqel ◽  
Laila Taourirte
Keyword(s):  
Elliptic Equation ◽  
Weak Solutions ◽  
Elliptic Equations ◽  
Quasilinear Elliptic Equation ◽  
Sufficient Conditions ◽  
Existence Results ◽  
Quasilinear Elliptic Equations ◽  
Main Ingredient ◽  
Uniqueness Results ◽  
Quasilinear Elliptic

Abstract The aim of this work is to study a quasilinear elliptic equation with singular nonlinearity and data measure. Existence and non-existence results are obtained under necessary or sufficient conditions on the data, where the main ingredient is the isoperimetric inequality. Finally, uniqueness results for weak solutions are given.

scholarly journals Existence and multiplicity of solutions for anisotropic elliptic equation

2022 ◽  
Vol 40 ◽  
pp. 1-12
Author(s):  
El Amrouss Abdelrachid ◽  
Ali El Mahraoui
Keyword(s):  
Elliptic Equation ◽  
Variational Method ◽  
Nonlinear Problem ◽  
Multiplicity Of Solutions ◽  
Anisotropic Elliptic Equation ◽  
Existence And Multiplicity

In this article we study the nonlinear problem $$\left\{ \begin{array}{lr} -\sum_{i=1}^{N}\partial_{x_{i}}a_{i}(x,\partial_{x_{i}}u)+ b(x)~|u|^{P_{+}^{+}-2}u =\lambda f(x,u) \quad in \quad \Omega\\ u=0 \qquad on \qquad \partial\Omega \end{array} \right.$$ Using the variational method, under appropriate assumptions on $f$, we obtain a result on existence and multiplicity of solutions.

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