Blow-up rates of large solutions for a ϕ-Laplacian problem with gradient term

2014 ◽  
Vol 144 (4) ◽  
pp. 669-689 ◽  
Author(s):  
W. Arriagada ◽  
J. Huentutripay
Keyword(s):  
Bounded Domain ◽  
Elliptic Problem ◽  
Blow Up ◽  
Smooth Boundary ◽  
Gradient Term ◽  
Regularly Varying Functions ◽  
Large Solutions ◽  
Regularly Varying ◽  
Increasing Function

We study the behaviour of solutions of a boundary blow-up elliptic problem on a bounded domain Ω with smooth boundary in ℝN. The data of the problem consist of an increasing function f : ℝ+ → ℝ+ and two real regularly varying functions ϕ and g.

scholarly journals A weak solution for a $(p(x),q(x))$-Laplacian elliptic problem with a singular term

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
MirKeysaan Mahshid ◽  
Abdolrahman Razani
Keyword(s):  
Boundary Condition ◽  
Weak Solution ◽  
Variational Method ◽  
Bounded Domain ◽  
Elliptic Problem ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Smooth Boundary ◽  
Singular Term

AbstractHere, we consider the following elliptic problem with variable components: $$ -a(x)\Delta _{p(x)}u - b(x) \Delta _{q(x)}u+ \frac{u \vert u \vert ^{s-2}}{|x|^{s}}= \lambda f(x,u), $$ − a ( x ) Δ p ( x ) u − b ( x ) Δ q ( x ) u + u | u | s − 2 | x | s = λ f ( x , u ) , with Dirichlet boundary condition in a bounded domain in $\mathbb{R}^{N}$ R N with a smooth boundary. By applying the variational method, we prove the existence of at least one nontrivial weak solution to the problem.

LAYERED SOLUTIONS WITH CONCENTRATION ON LINES IN THREE-DIMENSIONAL DOMAINS

2014 ◽  
Vol 12 (02) ◽  
pp. 161-194 ◽  
Author(s):  
WEIWEI AO ◽  
JUN YANG
Keyword(s):  
Small Parameter ◽  
Bounded Domain ◽  
Elliptic Problem ◽  
Smooth Boundary ◽  
Three Dimensional ◽  
Singularly Perturbed ◽  
Existence Of A Solution ◽  
Straight Line ◽  
Dimensional Domain ◽  
Exponentially Small

We consider the following singularly perturbed elliptic problem [Formula: see text] where Ω is a bounded domain in ℝ3with smooth boundary, ε is a small parameter, 1 < p < ∞, ν is the outward normal of ∂Ω. We employ techniques already developed in [39] to extend their result to three-dimensional domain. More precisely, let Γ be a straight line intersecting orthogonally with ∂Ω at exactly two points and satisfying a non-degenerate condition. We establish the existence of a solution uεconcentrating along a curve [Formula: see text] near Γ, exponentially small in ε at any positive distance from the curve, provided ε is small and away from certain critical numbers. The concentrating curve [Formula: see text] will collapse to Γ as ε → 0.

scholarly journals Existence and Nonexistence Results for Classes of Singular Elliptic Problem

2010 ◽  
Vol 2010 ◽  
pp. 1-10
Author(s):  
Peng Zhang ◽  
Jia-Feng Liao
Keyword(s):  
Bounded Domain ◽  
Elliptic Problem ◽  
Smooth Boundary ◽  
Semilinear Elliptic Problem ◽  
Semilinear Elliptic ◽  
Existence And Nonexistence

The singular semilinear elliptic problem-Δu+k(x)u-γ=λupinΩ,u>0inΩ,u=0on∂Ω, is considered, whereΩis a bounded domain with smooth boundary inRN,k∈Clocα(Ω)∩C(Ω¯), andγ,p,λare three positive constants. Some existence or nonexistence results are obtained for solutions of this problem by the sub-supersolution method.

Nodal Blow-Up Solutions to Slightly Subcritical Elliptic Problems with Hardy-Critical Term

2017 ◽  
Vol 17 (1) ◽  
Author(s):  
Thomas Bartsch ◽  
Qianqiao Guo
Keyword(s):  
Bounded Domain ◽  
Elliptic Problem ◽  
Blow Up ◽  
Elliptic Problems ◽  
Critical Term

AbstractThe paper is concerned with the slightly subcritical elliptic problem with Hardy-critical termin a bounded domain

scholarly journals Multiplicity of Solutions for a Class of Elliptic Problem of p-Laplacian Type with a p-Gradient Term

2019 ◽  
Vol 2019 ◽  
pp. 1-10
Author(s):  
Zakariya Chaouai ◽  
Soufiane Maatouk
Keyword(s):  
Elliptic Problem ◽  
Smooth Boundary ◽  
Mountain Pass Theorem ◽  
Gradient Term ◽  
Main Difficulty ◽  
Bounded Solutions ◽  
Equivalent Problem ◽  
Bounded Set ◽  
Variational Structure ◽  
Sign Condition

We consider the following problem: -Δpu=c(x)|u|q-1u+μ|∇u|p+h(x)  in  Ω,  u=0  on  ∂Ω, where Ω is a bounded set in RN (N≥3) with a smooth boundary, 1<p<N, q>0, μ∈R⁎, and c and h belong to Lk(Ω) for some k>N/p. In this paper, we assume that c≩0 a.e. in Ω and h without sign condition and then we prove the existence of at least two bounded solutions under the condition that ck and hk are suitably small. For this purpose, we use the Mountain Pass theorem, on an equivalent problem to (P) with variational structure. Here, the main difficulty is that the nonlinearity term considered does not satisfy Ambrosetti and Rabinowitz condition. The key idea is to replace the former condition by the nonquadraticity condition at infinity.

Very singular problems with critical nonlinearities in two dimensions

2017 ◽  
Vol 20 (02) ◽  
pp. 1650067 ◽  
Author(s):  
S. Prashanth ◽  
Sweta Tiwari ◽  
K. Sreenadh
Keyword(s):  
Bounded Domain ◽  
Elliptic Problem ◽  
Smooth Boundary ◽  
Two Dimensions ◽  
Bifurcation Parameter ◽  
Singular Problems ◽  
Exponential Nonlinearity ◽  
Critical Nonlinearities ◽  
Existence And Multiplicity ◽  
Multiplicity Of Positive Solutions

In this paper, we consider the following singular elliptic problem involving an exponential nonlinearity in two dimensions: [Formula: see text] [Formula: see text] where [Formula: see text] is a bounded domain with smooth boundary, [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text]. We show the existence and multiplicity of positive solutions globally with respect to the bifurcation parameter [Formula: see text].

scholarly journals Implicit Equations Involving the p-Laplace Operator

2021 ◽  
Vol 18 (2) ◽  
Author(s):  
Greta Marino ◽  
Andrea Paratore
Keyword(s):  
Bounded Domain ◽  
Laplace Operator ◽  
Elliptic Problem ◽  
Existence Of Solutions ◽  
Smooth Boundary ◽  
Implicit Equations

AbstractIn this work we study the existence of solutions $$u \in W^{1,p}_0(\Omega )$$ u ∈ W 0 1 , p ( Ω ) to the implicit elliptic problem $$ f(x, u, \nabla u, \Delta _p u)= 0$$ f ( x , u , ∇ u , Δ p u ) = 0 in $$ \Omega $$ Ω , where $$ \Omega $$ Ω is a bounded domain in $$ {\mathbb {R}}^N $$ R N , $$ N \ge 2 $$ N ≥ 2 , with smooth boundary $$ \partial \Omega $$ ∂ Ω , $$ 1< p< \infty $$ 1 < p < ∞ , and $$ f:\Omega \times {\mathbb {R}}\times {\mathbb {R}}^N \times {\mathbb {R}}\rightarrow {\mathbb {R}}$$ f : Ω × R × R N × R → R . We choose the particular case when the function f can be expressed in the form $$ f(x, z, w, y)= \varphi (x, z, w)- \psi (y) $$ f ( x , z , w , y ) = φ ( x , z , w ) - ψ ( y ) , where the function $$ \psi $$ ψ depends only on the p-Laplacian $$ \Delta _p u $$ Δ p u . We also present some applications of our results.

Two Classes of Nonlinear Singular Dirichlet Problems with Natural Growth: Existence and Asymptotic Behavior

2020 ◽  
Vol 20 (1) ◽  
pp. 77-93 ◽  
Author(s):  
Zhijun Zhang
Keyword(s):  
Asymptotic Behavior ◽  
Bounded Domain ◽  
Smooth Boundary ◽  
Classical Solutions ◽  
Natural Growth ◽  
Dirichlet Problems ◽  
Nonlinear Transformations

AbstractThis paper is concerned with the existence, uniqueness and asymptotic behavior of classical solutions to two classes of models {-\triangle u\pm\lambda\frac{|\nabla u|^{2}}{u^{\beta}}=b(x)u^{-\alpha}}, {u>0}, {x\in\Omega}, {u|_{\partial\Omega}=0}, where Ω is a bounded domain with smooth boundary in {\mathbb{R}^{N}}, {\lambda>0}, {\beta>0}, {\alpha>-1}, and {b\in C^{\nu}_{\mathrm{loc}}(\Omega)} for some {\nu\in(0,1)}, and b is positive in Ω but may be vanishing or singular on {\partial\Omega}. Our approach is largely based on nonlinear transformations and the construction of suitable sub- and super-solutions.

NONNEGATIVE SOLUTIONS FOR INDEFINITE SUBLINEAR ELLIPTIC PROBLEMS

2012 ◽  
Vol 14 (03) ◽  
pp. 1250021 ◽  
Author(s):  
FRANCISCO ODAIR DE PAIVA
Keyword(s):  
Bounded Domain ◽  
Positive Solutions ◽  
Elliptic Problem ◽  
Solution Method ◽  
Mountain Pass Theorem ◽  
Semilinear Elliptic Problem ◽  
Semilinear Elliptic ◽  
Nonnegative Solutions ◽  
Carathéodory Function ◽  
Multiplicity Of Positive Solutions

This paper is devoted to the study of existence, nonexistence and multiplicity of positive solutions for the semilinear elliptic problem [Formula: see text] where Ω is a bounded domain of ℝN, λ ∈ ℝ and g(x, u) is a Carathéodory function. The obtained results apply to the following classes of nonlinearities: a(x)uq + b(x)up and c(x)(1 + u)p (0 ≤ q < 1 < p). The proofs rely on the sub-super solution method and the mountain pass theorem.

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