Existence of solution for elliptic equations with supercritical Trudinger–Moser growth

2021 ◽  
pp. 1-20
Author(s):  
Luiz F. O. Faria ◽  
Marcelo Montenegro
Keyword(s):  
Boundary Condition ◽  
Unit Ball ◽  
Positive Solution ◽  
Dirichlet Boundary Condition ◽  
Elliptic Equations ◽  
Dirichlet Boundary ◽  
Existence Of Solutions ◽  
Existence Of Solution

This paper is concerned with the existence of solutions for a class of elliptic equations on the unit ball with zero Dirichlet boundary condition. The nonlinearity is supercritical in the sense of Trudinger–Moser. Using a suitable approximating scheme we obtain the existence of at least one positive solution.

scholarly journals Existence of a positive solution for a singular system

2010 ◽  
Vol 140 (2) ◽  
pp. 435-447 ◽  
Author(s):  
Marcelo Montenegro ◽  
Antonio Suárez
Keyword(s):  
Boundary Condition ◽  
Positive Solution ◽  
Positive Solutions ◽  
Dirichlet Boundary Condition ◽  
Elliptic Equations ◽  
Dirichlet Boundary ◽  
Singular System ◽  
Singular Elliptic Equations ◽  
Existence Of Positive Solutions

We show the existence and non-existence of positive solutions to a system of singular elliptic equations with the Dirichlet boundary condition.

On the Dispersive Estimate for the Dirichlet Schrödinger Propagator and Applications to Energy Critical NLS

2014 ◽  
Vol 66 (5) ◽  
pp. 1110-1142
Author(s):  
Dong Li ◽  
Guixiang Xu ◽  
Xiaoyi Zhang
Keyword(s):  
Boundary Condition ◽  
Fundamental Solution ◽  
Unit Ball ◽  
Dirichlet Boundary Condition ◽  
Obstacle Problem ◽  
Dirichlet Boundary ◽  
Radial Symmetry ◽  
Well Posedness ◽  
Robust Algorithm ◽  
Dispersive Estimate

AbstractWe consider the obstacle problem for the Schrödinger evolution in the exterior of the unit ball with Dirichlet boundary condition. Under radial symmetry we compute explicitly the fundamental solution for the linear Dirichlet Schrödinger propagator and give a robust algorithm to prove sharp L1 → L∞ dispersive estimates. We showcase the analysis in dimensions n = 5, 7. As an application, we obtain global well–posedness and scattering for defocusing energy-critical NLS on with Dirichlet boundary condition and radial data in these dimensions.

Existence and Regularizing Effect of Degenerate Lower Order Terms in Elliptic Equations Beyond the Hardy Constant

2018 ◽  
Vol 18 (4) ◽  
pp. 775-783 ◽  
Author(s):  
David Arcoya ◽  
Alexis Molino ◽  
Lourdes Moreno-Mérida
Keyword(s):  
Differential Equation ◽  
Boundary Condition ◽  
Lower Order ◽  
Dirichlet Boundary Condition ◽  
Elliptic Equations ◽  
Dirichlet Boundary ◽  
Model Problem ◽  
Elliptic Problems ◽  
Hardy Potential ◽  
Lower Order Terms

AbstractIn this paper, we study the regularizing effect of lower order terms in elliptic problems involving a Hardy potential. Concretely, our model problem is the differential equation-\Delta u+h(x)|u|^{p-1}u=\lambda\frac{u}{|x|^{2}}+f(x)\quad\text{in }\Omega,with Dirichlet boundary condition on {\partial\Omega}, where {p>1} and {f\in L^{m}_{h}(\Omega)} (i.e. {|f|^{m}h\in L^{1}(\Omega)}) with {m\geq\frac{p+1}{p}}. We prove that there is a solution of the above problem even for λ greater than the Hardy constant; i.e., {\lambda\geq\mathcal{H}=\frac{(N-2)^{2}}{4}} and nonnegative functions {h\in L^{1}(\Omega)} which could vanish in a subset of Ω. Moreover, we show that all the solutions are in {L^{pm}_{h}(\Omega)}. These results improve and generalize the case {h(x)\equiv h_{0}} treated in [2, 10].

The Bahri–Coron Theorem for Fractional Yamabe-Type Problems

2018 ◽  
Vol 18 (2) ◽  
pp. 393-407 ◽  
Author(s):  
Wael Abdelhedi ◽  
Hichem Chtioui ◽  
Hichem Hajaiej
Keyword(s):  
Boundary Condition ◽  
Bounded Domain ◽  
Dirichlet Boundary Condition ◽  
Fractional Laplacian ◽  
Dirichlet Boundary ◽  
Existence Of Solutions ◽  
Type Equation ◽  
Laplacian Operator ◽  
Fractional Laplacian Operator

AbstractWe study the following fractional Yamabe-type equation:\left\{\begin{aligned} \displaystyle A_{s}u&\displaystyle=u^{\frac{n+2s}{n-2s}% },\\ \displaystyle u&\displaystyle>0&&\displaystyle\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\text{on }\partial\Omega,\end{% aligned}\right.Here Ω is a regular bounded domain of{\mathbb{R}^{n}},{n\geq 2}, and{A_{s}},{s\in(0,1)}, represents the fractional Laplacian operator{(-\Delta)^{s}}in Ω with zero Dirichlet boundary condition. We investigate the effect of the topology of Ω on the existence of solutions. Our result can be seen as the fractional counterpart of the Bahri–Coron theorem [3].

scholarly journals Multiplicity of solutions for singular semilinear elliptic equations with critical Hardy-Sobolev exponents

2008 ◽  
Vol 2 (2) ◽  
pp. 158-174 ◽  
Author(s):  
Qianqiao Guo ◽  
Pengcheng Niu ◽  
Jingbo Dou
Keyword(s):  
Boundary Condition ◽  
Variational Methods ◽  
Elliptic Problem ◽  
Dirichlet Boundary Condition ◽  
Elliptic Equations ◽  
Dirichlet Boundary ◽  
Nontrivial Solutions ◽  
Existence And Multiplicity ◽  
Semilinear Elliptic Problem ◽  
Semilinear Elliptic

We consider the semilinear elliptic problem with critical Hardy-Sobolev exponents and Dirichlet boundary condition. By using variational methods we obtain the existence and multiplicity of nontrivial solutions and improve the former results.

Non-stabilizing solutions of semilinear hyperbolic and elliptic equations with damping

2003 ◽  
Vol 133 (5) ◽  
pp. 1137-1153 ◽  
Author(s):  
M. A. Jendoubi ◽  
P. Poláčik
Keyword(s):  
Boundary Condition ◽  
Wave Equation ◽  
Elliptic Equation ◽  
Dirichlet Boundary Condition ◽  
Elliptic Equations ◽  
Dirichlet Boundary ◽  
Cylindrical Domain ◽  
Damped Wave Equation ◽  
Bounded Solutions ◽  
Single Function

We consider two types of equations on a cylindrical domain Ω × (0, ∞), where Ω is a bounded domain in RN, N ≥ 2. The first type is a semilinear damped wave equation, in which the unbounded direction of Ω × (0, ∞) is reserved for time t. The second type is an elliptic equation with a singled-out unbounded variable t. In both cases, we consider solutions that are defined and bounded on Ω × (0, ∞) and satisfy a Dirichlet boundary condition on ∂Ω × (0, ∞). We show that, for some nonlinearities, the equations have bounded solutions that do not stabilize to any single function φ: Ω → R, as t → ∞; rather, they approach a continuum of such functions. This happens despite the presence of damping in the equation that forces the t derivative of bounded solutions to converge to 0 as t → ∞. Our results contrast with known stabilization properties of solutions of such equations in the case N = 1.

Estimate, existence and nonexistence of positive solutions of Hardy–Hénon equations

2021 ◽  
pp. 1-24
Author(s):  
Xiyou Cheng ◽  
Lei Wei ◽  
Yimin Zhang
Keyword(s):  
Boundary Condition ◽  
Dirichlet Problem ◽  
Positive Solution ◽  
Positive Solutions ◽  
Dirichlet Boundary Condition ◽  
Existence And Uniqueness ◽  
Dirichlet Boundary ◽  
Existence And Nonexistence ◽  
Hénon Equation ◽  
Image Position

We consider the boundary Hardy–Hénon equation \[ -\Delta u=(1-|x|)^{\alpha} u^{p},\ \ x\in B_1(0), \] where $B_1(0)\subset \mathbb {R}^{N}$   $(N\geq 3)$ is a ball of radial $1$ centred at $0$ , $p>0$ and $\alpha \in \mathbb {R}$ . We are concerned with the estimate, existence and nonexistence of positive solutions of the equation, in particular, the equation with Dirichlet boundary condition. For the case $0< p<({N+2})/({N-2})$ , we establish the estimate of positive solutions. When $\alpha \leq -2$ and $p>1$ , we give some conclusions with respect to nonexistence. When $\alpha >-2$ and $1< p<({N+2})/({N-2})$ , we obtain the existence of positive solution for the corresponding Dirichlet problem. When $0< p\leq 1$ and $\alpha \leq -2$ , we show the nonexistence of positive solutions. When $0< p<1$ , $\alpha >-2$ , we give some results with respect to existence and uniqueness of positive solutions.

scholarly journals Existence and non-existence of positive solution for (p, q)-Laplacian with singular weights

2015 ◽  
Vol 34 (2) ◽  
pp. 147-167 ◽  
Author(s):  
Abdellah Ahmed Zerouali ◽  
Belhadj Karim
Keyword(s):  
Boundary Condition ◽  
Eigenvalue Problem ◽  
Positive Solution ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Existence Results ◽  
Singular Weight ◽  
Quasilinear Elliptic Problem ◽  
Quasilinear Elliptic ◽  
Singular Weights

We use the Hardy-Sobolev inequality to study existence and non-existence results for a positive solution of the quasilinear elliptic problem -\Delta{p}u − \mu \Delta{q}u = \limda[mp(x)|u|p−2u + \mu mq(x)|u|q−2u] in \Omega driven by nonhomogeneous operator (p, q)-Laplacian with singular weights under the Dirichlet boundary condition. We also prove that in the case where μ > 0 and with 1 < q < p < \infinity the results are completely different from those for the usual eigenvalue for the problem p-Laplacian with singular weight under the Dirichlet boundary condition, which is retrieved when μ = 0. Precisely, we show that when μ > 0 there exists an interval of eigenvalues for our eigenvalue problem.

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