nonlinear elliptic
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2022 ◽  
Vol 2022 (1) ◽  
Author(s):  
Imed Bachar ◽  
Entesar Aljarallah

AbstractWe consider the following singular semilinear problem $$ \textstyle\begin{cases} \Delta u(x)+p(x)u^{\gamma }=0,\quad x\in D ~(\text{in the distributional sense}), \\ u>0,\quad \text{in }D, \\ \lim_{ \vert x \vert \rightarrow 0} \vert x \vert ^{n-2}u(x)=0, \\ \lim_{ \vert x \vert \rightarrow \infty }u(x)=0,\end{cases} $$ { Δ u ( x ) + p ( x ) u γ = 0 , x ∈ D ( in the distributional sense ) , u > 0 , in  D , lim | x | → 0 | x | n − 2 u ( x ) = 0 , lim | x | → ∞ u ( x ) = 0 , where $\gamma <1$ γ < 1 , $D=\mathbb{R}^{n}\backslash \{0\}$ D = R n ∖ { 0 } ($n\geq 3$ n ≥ 3 ) and p is a positive continuous function in D, which may be singular at $x=0$ x = 0 . Under sufficient conditions for the weighted function $p(x)$ p ( x ) , we prove the existence of a positive continuous solution on D, which could blow-up at the origin. The global asymptotic behavior of this solution is also obtained.


2022 ◽  
Vol 32 (3) ◽  
Author(s):  
Shengda Zeng ◽  
Yunru Bai ◽  
Leszek Gasiński

AbstractIn this paper, a nonlinear elliptic obstacle problem is studied. The nonlinear nonhomogeneous partial differential operator generalizes the notions of p-Laplacian while on the right hand side we have a multivalued convection term (i.e., a multivalued reaction term may depend also on the gradient of the solution). The main result of the paper provides existence of the solutions as well as bondedness and closedness of the set of weak solutions of the problem, under quite general assumptions on the data. The main tool of the paper is the surjectivity theorem for multivalued functions given by the sum of a maximal monotone multivalued operator and a bounded multivalued pseudomonotone one.


2022 ◽  
Author(s):  
Rong Zhang

Abstract In this paper, we study the problem for a nonlinear elliptic system involving fractional Laplacion: (equation 1.1) where 0 < α, β < 2, p, q > 0 and max{p, q} ≥ 1, α + γ > 0, β + τ > 0, n ≥ 2. First of all, while in the subcritical case, i.e. n + α + γ − p(n − α) − (q + 1)(n − β) > 0, n + β + τ − (p + 1)(n − α) − q(n − β) > 0, we prove the nonexistence of positive solution for the above system in R n . Moreover, though Doubling Lemma to obtain the singularity estimates of the positive solution on bounded domain Ω. In addition, while in the critical case, i.e. n+α+γ −p(n−α)−(q + 1)(n−β) = 0, n+β +τ −(p+ 1)(n−α)−q(n−β) = 0, we show that the positive solution of above system are radical symmetric and decreasing about some point by using the method of Moving planes in Rn Mathematics Subject Classification (2020): 35R11, 35A10, 35B06.


Author(s):  
Abdelaziz Hellal

This paper is concerned with the study of the nonlinear elliptic equations in a bounded subset Ω ⊂ RN Au = f, where A is an operator of Leray-Lions type acted from the space W1,p(·)0(Ω) into its dual. when the second term f belongs to Lm(·), with m(·) > 1 being small. we prove existence and regularity of weak solutions for this class of problems p(x)-growth conditions. The functional framework involves Sobolev spaces with variable exponents as well as Lebesgue spaces with variable exponents.


Author(s):  
Sunghan Kim ◽  
Ki-Ahm Lee

AbstractThis article is concerned with uniform $$C^{1,\alpha }$$ C 1 , α and $$C^{1,1}$$ C 1 , 1 estimates in periodic homogenization of fully nonlinear elliptic equations. The analysis is based on the compactness method, which involves linearization of the operator at each approximation step. Due to the nonlinearity of the equations, the linearized operators involve the Hessian of correctors, which appear in the previous step. The involvement of the Hessian of the correctors deteriorates the regularity of the linearized operator, and sometimes even changes its oscillating pattern. These issues are resolved with new approximation techniques, which yield a precise decomposition of the regular part and the irregular part of the homogenization process, along with a uniform control of the Hessian of the correctors in an intermediate level. The approximation techniques are even new in the context of linear equations. Our argument can be applied not only to concave operators, but also to certain class of non-concave operators.


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