Stationary Kirchhoff equations involving critical growth and vanishing potential

2020 ◽  
Vol 26 ◽  
pp. 74
Author(s):  
João Marcos do Ó ◽  
Marco Souto ◽  
Pedro Ubilla
Keyword(s):  
Positive Solutions ◽  
A Priori ◽  
A Priori Estimate ◽  
Critical Growth ◽  
Sobolev Embedding ◽  
Priori Estimate ◽  
Kirchhoff Equations ◽  
Kirchhoff Type ◽  
Local Case ◽  
Existence Of Positive Solutions

We establish the existence of positive solutions for a class of stationary Kirchhoff-type equations defined in the whole ℝ3 involving critical growth in the sense of the Sobolev embedding and potentials, which may decay to zero at infinity. We use minimax techniques combined with an appropriate truncated argument and a priori estimate. These results are new even for the local case, which corresponds to nonlinear Schrödinger equations.

A priori bounds and existence of positive solutions to a p-kirchhoff equations

2021 ◽  
pp. 2150082
Author(s):  
Pengfei Li ◽  
Junhui Xie
Keyword(s):  
Positive Solutions ◽  
Blow Up ◽  
A Priori Estimates ◽  
A Priori ◽  
Auxiliary Problem ◽  
Degree Theory ◽  
Kirchhoff Equations ◽  
Priori Estimates ◽  
Existence Of Positive Solutions ◽  
Dirichlet Boundary Problem

In this paper, we consider a [Formula: see text]-Kirchhoff problem with Dirichlet boundary problem in a bounded domain. Under suitable conditions, we get a priori estimates for positive solutions to an auxiliary problem by the well-known blow-up argument. As an application, a existence result for positive solutions is proved by the topological degree theory.

scholarly journals Regularity and existence of positive solutions for a fractional system

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Ran Zhuo ◽  
Yan Li
Keyword(s):  
Positive Solutions ◽  
A Priori Estimates ◽  
A Priori ◽  
Degree Theory ◽  
A Priori Estimate ◽  
Technical Difficulty ◽  
Priori Estimates ◽  
Existence Of Positive Solutions ◽  
Regularity And Existence ◽  
Fractional Orders

<p style='text-indent:20px;'>We consider the nonlinear fractional elliptic system</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \left\{\begin{array}{ll} (- \Delta)^{\frac{\alpha_1}{2}}u(x) = f(x, u, v), &amp; \text{in}\, \, \, \Omega, \\ (- \Delta)^{\frac{\alpha_2}{2}}v(x) = g(x, u, v), &amp; \text{in}\, \, \, \Omega, \\ u = v = 0, &amp; \text{in}\, \, \, \mathbb{R}^n\setminus\Omega, \end{array} \right. \label{a-1.2} \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ 0&lt;\alpha_1, \alpha_2&lt;2 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> is a bounded domain with <inline-formula><tex-math id="M3">\begin{document}$ C^2 $\end{document}</tex-math></inline-formula> boundary in <inline-formula><tex-math id="M4">\begin{document}$ \mathbb{R}^n $\end{document}</tex-math></inline-formula>. To overcome the technical difficulty due to the different fractional orders, we employ two distinct methods and derive the a priori estimates for <inline-formula><tex-math id="M5">\begin{document}$ 0&lt;\alpha_1, \alpha_2&lt;1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ 1&lt;\alpha_1, \alpha_2 &lt;2 $\end{document}</tex-math></inline-formula> respectively. Moreover, combining the a priori estimate with the topological degree theory, we prove the existence of positive solutions.</p>

scholarly journals The existence of positive solutions for Kirchhoff-type problems via the sub-supersolution method

2018 ◽  
Vol 26 (1) ◽  
pp. 5-41 ◽  
Author(s):  
Baoqiang Yan ◽  
Donal O’Regan ◽  
Ravi P. Agarwal
Keyword(s):  
Positive Solutions ◽  
Bifurcation Theory ◽  
Sufficient Conditions ◽  
Global Bifurcation ◽  
Existence Of A Solution ◽  
Kirchhoff Type ◽  
Existence Of Positive Solutions ◽  
Global Bifurcation Theory ◽  
Necessary And Sufficient ◽  
Kirchhoff Type Problems

Abstract In this paper we discuss the existence of a solution between wellordered subsolution and supersolution of the Kirchhoff equation. Using the sub-supersolution method together with a Rabinowitz-type global bifurcation theory, we establish the existence of positive solutions for Kirchhoff-type problems when the nonlinearity is singular or sign-changing. Moreover, we obtain some necessary and sufficient conditions for the existence of positive solutions for the problem when N = 1.

scholarly journals Multiple positive solutions for a class of Kirchhoff type equations with indefinite nonlinearities

2021 ◽  
Vol 11 (1) ◽  
pp. 598-619
Author(s):  
Guofeng Che ◽  
Tsung-fang Wu
Keyword(s):  
Variational Principle ◽  
Positive Solutions ◽  
Type Equation ◽  
Nehari Manifold ◽  
Sobolev Embedding ◽  
Multiple Positive Solutions ◽  
Kirchhoff Type ◽  
Indefinite Nonlinearities ◽  
The Nehari Manifold ◽  
Constraint Method

Abstract We study the following Kirchhoff type equation: − a + b ∫ R N | ∇ u | 2 d x Δ u + u = k ( x ) | u | p − 2 u + m ( x ) | u | q − 2 u     in     R N , $$\begin{equation*}\begin{array}{ll} -\left(a+b\int\limits_{\mathbb{R}^{N}}|\nabla u|^{2}\mathrm{d}x\right)\Delta u+u =k(x)|u|^{p-2}u+m(x)|u|^{q-2}u~~\text{in}~~\mathbb{R}^{N}, \end{array} \end{equation*}$$ where N=3, a , b > 0 $ a,b \gt 0 $ , 1 < q < 2 < p < min { 4 , 2 ∗ } $ 1 \lt q \lt 2 \lt p \lt \min\{4, 2^{*}\} $ , 2≤=2N/(N − 2), k ∈ C (ℝ N ) is bounded and m ∈ L p/(p−q)(ℝ N ). By imposing some suitable conditions on functions k(x) and m(x), we firstly introduce some novel techniques to recover the compactness of the Sobolev embedding H 1 ( R N ) ↪ L r ( R N ) ( 2 ≤ r < 2 ∗ ) $ H^{1}(\mathbb{R}^{N})\hookrightarrow L^{r}(\mathbb{R}^{N}) (2\leq r \lt 2^{*}) $ ; then the Ekeland variational principle and an innovative constraint method of the Nehari manifold are adopted to get three positive solutions for the above problem.

scholarly journals A PRIORI ESTIMATE FOR AN EQUATION WITH FRACTIONAL DERIVATIVES WITH DIFFERENT ORIGINS

2019 ◽  
pp. 41-47
Author(s):  
Л.М. Энеева
Keyword(s):  
Fractional Derivatives ◽  
Model Equation ◽  
A Priori ◽  
A Priori Estimate ◽  
Equation Of Motion ◽  
Point Boundary ◽  
Priori Estimate ◽  
Fractal Media ◽  
Different Origins ◽  
Variable Potential

В работе исследуется обыкновенное дифференциальное уравнение дробного порядка, содержащее композицию дробных производных с различными началами, с переменным потенциалом. Рассматриваемое уравнение выступает модельным уравнением движения во фрактальной среде. Для исследуемого уравнения доказана априорная оценка решения смешанной двухточечной краевой задачи. We consider an ordinary differential equation of fractional order with the composition of leftand right-sided fractional derivatives, and with variable potential. The considered equation is a model equation of motion in fractal media. We prove an a priori estimate for solutions of a mixed two-point boundary value problem for the equation under study.

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