scholarly journals Existence of Solutions for Degenerate Elliptic Problems in Weighted Sobolev Space

2015 ◽  
Vol 2015 ◽  
pp. 1-9
Author(s):  
Lili Dai ◽  
Wenjie Gao ◽  
Zhongqing Li
Keyword(s):  
Sobolev Space ◽  
Growth Condition ◽  
Elliptic Problem ◽  
Nonlinear Term ◽  
Existence Of Solutions ◽  
Weighted Sobolev Space ◽  
Elliptic Problems ◽  
Weight Functions ◽  
General Elliptic ◽  
Degenerate Elliptic

This paper is devoted to the study of the existence of solutions to a general elliptic problemAu+g(x,u,∇u)=f-div⁡F, withf∈L1(Ω)andF∈∏i=1NLp'(Ω,ωi*), whereAis a Leray-Lions operator from a weighted Sobolev space into its dual andg(x,s,ξ)is a nonlinear term satisfyinggx,s,ξsgn⁡(s)≥ρ∑i=1Nωi|ξi|p,|s|≥h>0, and a growth condition with respect toξ. Here,ωi,ωi*are weight functions that will be defined in the Preliminaries.

scholarly journals Existence results for some nonlinear degenerate problems in the anisotropic spaces

2021 ◽  
Vol 39 (6) ◽  
pp. 53-66
Author(s):  
Mohamed Boukhrij ◽  
Benali Aharrouch ◽  
Jaouad Bennouna ◽  
Ahmed Aberqi
Keyword(s):  
Elliptic Problem ◽  
Weak Solutions ◽  
Nonlinear Term ◽  
Existence Of Solutions ◽  
Existence Results ◽  
Existence Of Weak Solutions ◽  
Sign Condition ◽  
Degenerate Elliptic ◽  
Anisotropic Spaces ◽  
Nonlinear Degenerate

Our goal in this study is to prove the existence of solutions for the following nonlinear anisotropic degenerate elliptic problem:- \partial_{x_i} a_i(x,u,\nabla u)+ \sum_{i=1}^NH_i(x,u,\nabla u)= f- \partial_{x_i} g_i \quad \mbox{in} \ \ \Omega,where for $i=1,...,N$ $ a_i(x,u,\nabla u)$ is allowed to degenerate with respect to the unknown u, and $H_i(x,u,\nabla u)$ is a nonlinear term without a sign condition. Under suitable conditions on $a_i$ and $H_i$, we prove the existence of weak solutions.

scholarly journals Multiplicity results for Kirchhoff type elliptic problems with Hardy potential

2019 ◽  
Vol 38 (4) ◽  
pp. 31-50
Author(s):  
M. Bagheri ◽  
Ghasem A. Afrouzi
Keyword(s):  
Local Minimum ◽  
Nonlinear Term ◽  
Existence Of Solutions ◽  
Mountain Pass Theorem ◽  
Elliptic Problems ◽  
Mountain Pass ◽  
Hardy Potential ◽  
Multiplicity Results ◽  
Kirchhoff Type ◽  
Minimum Theorem

In this paper, we are concerned with the existence of solutions for fourth-order Kirchhoff type elliptic problems with Hardy potential. In fact, employing a consequence of the local minimum theorem due to Bonanno and mountain pass theorem we look into the existence results for the problem under algebraic conditions with the classical Ambrosetti-Rabinowitz (AR) condition on the nonlinear term. Furthermore, by combining two algebraic conditions on the nonlinear term using two consequences of the local minimum theorem due to Bonanno we ensure the existence of two solutions, applying the mountain pass theorem given by Pucci and Serrin we establish the existence of third solution for our problem.

scholarly journals Existence of Solutions for a Class of Quasilinear Parabolic Equations with Superlinear Nonlinearities

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Zhong-Xiang Wang ◽  
Gao Jia ◽  
Xiao-Juan Zhang
Keyword(s):  
Sobolev Space ◽  
Boundary Value Problem ◽  
Parabolic Equations ◽  
Existence Of Solutions ◽  
Weighted Sobolev Space ◽  
Boundary Value ◽  
Growth Conditions ◽  
Quasilinear Parabolic Equations ◽  
Superlinear Growth ◽  
Quasilinear Parabolic

Working in a weighted Sobolev space, this paper is devoted to the study of the boundary value problem for the quasilinear parabolic equations with superlinear growth conditions in a domain of RN. Some conditions which guarantee the solvability of the problem are given.

On the regularity of solutions to quasi-linear variational inequalities of degenerate elliptic type

1977 ◽  
Vol 77 (3-4) ◽  
pp. 251-263
Author(s):  
Frances Cooper
Keyword(s):  
Variational Inequality ◽  
Sobolev Space ◽  
Variational Inequalities ◽  
Differential Inequality ◽  
Weighted Sobolev Space ◽  
Growth Conditions ◽  
Regularity Of Solutions ◽  
Elliptic Type ◽  
Degenerate Elliptic ◽  
Image Position

SynopsisUsing the theory of the weighted Sobolev space H1,2(μ), on a bounded domain Ω, in Rn, the existence and regularity of solutions u in K to the variational inequalityis established for various convex subsets K of H1, 2(μ). The growth conditions imposed on the functions A and B give the differential inequality degenerate elliptic structure, extending the results on regularity for inequalities of elliptic type.

Infinitely Many Solutions for Quasilinear Elliptic Problems with Broken Symmetry

2013 ◽  
Vol 13 (3) ◽  
Author(s):  
Rossella Bartolo
Keyword(s):  
Bounded Domain ◽  
Elliptic Problem ◽  
Existence Of Solutions ◽  
Elliptic Problems ◽  
Broken Symmetry ◽  
Infinitely Many Solutions ◽  
Quasilinear Elliptic Problem ◽  
Quasilinear Elliptic Problems ◽  
Quasilinear Elliptic

AbstractThe aim of this paper is investigating the existence of solutions of the quasilinear elliptic Problemwhere Ω is an open bounded domain of R

scholarly journals Existence of Solutions for Choquard Type Elliptic Problems with Doubly Critical Nonlinearities

2019 ◽  
Vol 21 (1) ◽  
pp. 77-93
Author(s):  
Yansheng Shen
Keyword(s):  
Variational Methods ◽  
Minimization Problem ◽  
Existence Of Solutions ◽  
Elliptic Problems ◽  
Type Equation ◽  
Nontrivial Solutions ◽  
Critical Nonlinearities

Abstract In this article, we first study the existence of nontrivial solutions to the nonlocal elliptic problems in ℝ N {\mathbb{R}^{N}} involving fractional Laplacians and the Hardy–Sobolev–Maz’ya potential. Using variational methods, we investigate the attainability of the corresponding minimization problem, and then obtain the existence of solutions. We also consider another Choquard type equation involving the p-Laplacian and critical nonlinearities in ℝ N {\mathbb{R}^{N}} .

scholarly journals A class of quasilinear degenerate elliptic problems

2003 ◽  
Vol 189 (1) ◽  
pp. 71-98 ◽  
Author(s):  
Sunčica Čanić ◽  
Eun Heui Kim
Keyword(s):  
Elliptic Problems ◽  
Degenerate Elliptic

Optimal L 2 -decay of solutions to a cubic dissipative nonlinear Schrödinger equation

2021 ◽  
pp. 1-13
Author(s):  
Kita Naoyasu ◽  
Sato Takuya
Keyword(s):  
Sobolev Space ◽  
Global Solution ◽  
Initial Value Problem ◽  
Trivial Solution ◽  
Decay Estimate ◽  
Weighted Sobolev Space ◽  
The Other ◽  
Initial Value ◽  
Decay Of Solutions ◽  
Dissipative Nonlinearity

This paper presents the optimality of decay estimate of solutions to the initial value problem of 1D Schrödinger equations containing a long-range dissipative nonlinearity, i.e., λ | u | 2 u. Our aim is to obtain the two results. One asserts that, if the L 2 -norm of a global solution, with an initial datum in the weighted Sobolev space, decays at the rate more rapid than ( log t ) − 1 / 2 , then it must be a trivial solution. The other asserts that there exists a solution decaying just at the rate of ( log t ) − 1 / 2 in L 2 .

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