scholarly journals Existence of nonnegative solutions of nonlinear fractional parabolic inequalities

2021 ◽  
Author(s):  
Steven D. Taliaferro
Keyword(s):  
Nonnegative Solutions ◽  
Parabolic Inequalities

scholarly journals Fujita type results for quasilinear parabolic inequalities with nonlocal terms

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Roberta Filippucci ◽  
Marius Ghergu
Keyword(s):  
Curvature Operator ◽  
Maximum Principles ◽  
Nonlinear Capacity ◽  
Nonnegative Solutions ◽  
Comparison Results ◽  
Generalized Mean Curvature ◽  
Mean Curvature Operator ◽  
Nonlocal Terms ◽  
Conditions At Infinity ◽  
Parabolic Inequalities

<p style='text-indent:20px;'>In this paper we investigate the nonexistence of nonnegative solutions of parabolic inequalities of the form</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{cases} &amp;u_t \pm L_\mathcal A u\geq (K\ast u^p)u^q \quad\mbox{ in } \mathbb R^N \times \mathbb (0,\infty),\, N\geq 1,\\ &amp;u(x,0) = u_0(x)\ge0 \,\, \text{ in } \mathbb R^N,\end{cases} \qquad (P^{\pm}) $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ u_0\in L^1_{loc}({\mathbb R}^N) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ L_{\mathcal{A}} $\end{document}</tex-math></inline-formula> denotes a weakly <inline-formula><tex-math id="M3">\begin{document}$ m $\end{document}</tex-math></inline-formula>-coercive operator, which includes as prototype the <inline-formula><tex-math id="M4">\begin{document}$ m $\end{document}</tex-math></inline-formula>-Laplacian or the generalized mean curvature operator, <inline-formula><tex-math id="M5">\begin{document}$ p,\,q&gt;0 $\end{document}</tex-math></inline-formula>, while <inline-formula><tex-math id="M6">\begin{document}$ K\ast u^p $\end{document}</tex-math></inline-formula> stands for the standard convolution operator between a weight <inline-formula><tex-math id="M7">\begin{document}$ K&gt;0 $\end{document}</tex-math></inline-formula> satisfying suitable conditions at infinity and <inline-formula><tex-math id="M8">\begin{document}$ u^p $\end{document}</tex-math></inline-formula>. For problem <inline-formula><tex-math id="M9">\begin{document}$ (P^-) $\end{document}</tex-math></inline-formula> we obtain a Fujita type exponent while for <inline-formula><tex-math id="M10">\begin{document}$ (P^+) $\end{document}</tex-math></inline-formula> we show that no such critical exponent exists. Our approach relies on nonlinear capacity estimates adapted to the nonlocal setting of our problems. No comparison results or maximum principles are required.</p>

scholarly journals Measure Data Problems for a Class of Elliptic Equations with Mixed Absorption-Reaction

2021 ◽  
Vol 21 (2) ◽  
pp. 261-280
Author(s):  
Marie-Françoise Bidaut-Véron ◽  
Marta Garcia-Huidobro ◽  
Laurent Véron
Keyword(s):  
Dirichlet Problem ◽  
Compact Subset ◽  
Elliptic Equations ◽  
Radon Measure ◽  
Measure Data ◽  
Nonnegative Solutions

Abstract In the present paper, we study the existence of nonnegative solutions to the Dirichlet problem ℒ p , q M ⁢ u := - Δ ⁢ u + u p - M ⁢ | ∇ ⁡ u | q = μ {{\mathcal{L}}^{{M}}_{p,q}u:=-\Delta u+u^{p}-M|\nabla u|^{q}=\mu} in a domain Ω ⊂ ℝ N {\Omega\subset\mathbb{R}^{N}} where μ is a nonnegative Radon measure, when p > 1 {p>1} , q > 1 {q>1} and M ≥ 0 {M\geq 0} . We also give conditions under which nonnegative solutions of ℒ p , q M ⁢ u = 0 {{\mathcal{L}}^{{M}}_{p,q}u=0} in Ω ∖ K {\Omega\setminus K} , where K is a compact subset of Ω, can be extended as a solution of the same equation in Ω.

scholarly journals Existence and Convergence of Solutions to Fractional Pure Critical Exponent Problems

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Víctor Hernández-Santamaría ◽  
Alberto Saldaña
Keyword(s):  
Critical Exponent ◽  
Convergence Properties ◽  
Scaling Invariance ◽  
Convergence Of Solutions ◽  
Nonexistence Result ◽  
Homogeneous Sobolev Space ◽  
Similar Characterization ◽  
Nonnegative Solutions ◽  
Least Energy ◽  
Local Transition

Abstract We study existence and convergence properties of least-energy symmetric solutions (l.e.s.s.) to the pure critical exponent problem ( - Δ ) s ⁢ u s = | u s | 2 s ⋆ - 2 ⁢ u s , u s ∈ D 0 s ⁢ ( Ω ) ,  2 s ⋆ := 2 ⁢ N N - 2 ⁢ s , (-\Delta)^{s}u_{s}=\lvert u_{s}\rvert^{2_{s}^{\star}-2}u_{s},\quad u_{s}\in D^% {s}_{0}(\Omega),\,2^{\star}_{s}:=\frac{2N}{N-2s}, where s is any positive number, Ω is either ℝ N {\mathbb{R}^{N}} or a smooth symmetric bounded domain, and D 0 s ⁢ ( Ω ) {D^{s}_{0}(\Omega)} is the homogeneous Sobolev space. Depending on the kind of symmetry considered, solutions can be sign-changing. We show that, up to a subsequence, a l.e.s.s. u s {u_{s}} converges to a l.e.s.s. u t {u_{t}} as s goes to any t > 0 {t>0} . In bounded domains, this convergence can be characterized in terms of an homogeneous fractional norm of order t - ε {t-\varepsilon} . A similar characterization is no longer possible in unbounded domains due to scaling invariance and an incompatibility with the functional spaces; to circumvent these difficulties, we use a suitable rescaling and characterize the convergence via cut-off functions. If t is an integer, then these results describe in a precise way the nonlocal-to-local transition. Finally, we also include a nonexistence result of nontrivial nonnegative solutions in a ball for any s > 1 {s>1} .

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.

scholarly journals Loop Type Subcontinua of Positive Solutions for Indefinite Concave-Convex Problems

2019 ◽  
Vol 19 (2) ◽  
pp. 391-412
Author(s):  
Uriel Kaufmann ◽  
Humberto Ramos Quoirin ◽  
Kenichiro Umezu
Keyword(s):  
Elliptic Equations ◽  
Bifurcation Theory ◽  
A Priori ◽  
Global Bifurcation ◽  
Neumann Boundary ◽  
Zero Solution ◽  
Regularization Scheme ◽  
Convex Problems ◽  
Nonnegative Solutions ◽  
Indefinite Weights

AbstractWe establish the existence of loop type subcontinua of nonnegative solutions for a class of concave-convex type elliptic equations with indefinite weights, under Dirichlet and Neumann boundary conditions. Our approach depends on local and global bifurcation analysis from the zero solution in a nonregular setting, since the nonlinearities considered are not differentiable at zero, so that the standard bifurcation theory does not apply. To overcome this difficulty, we combine a regularization scheme with a priori bounds, and Whyburn’s topological method. Furthermore, via a continuity argument we prove a positivity property for subcontinua of nonnegative solutions. These results are based on a positivity theorem for the associated concave problem proved by us, and extend previous results established in the powerlike case.

scholarly journals Pointwise estimates for solutions of semilinear parabolic inequalities with a potential

2018 ◽  
Vol 29 (2) ◽  
pp. 255-288
Author(s):  
Luigi Montoro ◽  
Fabio Punzo ◽  
Berardino Sciunzi
Keyword(s):  
Pointwise Estimates ◽  
Semilinear Parabolic ◽  
Parabolic Inequalities

scholarly journals On Critical p-Laplacian Systems

2017 ◽  
Vol 17 (4) ◽  
pp. 641-659
Author(s):  
Zhenyu Guo ◽  
Kanishka Perera ◽  
Wenming Zou
Keyword(s):  
Critical Sobolev Exponent ◽  
Laplacian Operator ◽  
Existence And Multiplicity ◽  
Nonnegative Solutions ◽  
Least Energy Solution ◽  
Existence And Nonexistence ◽  
Alpha Beta ◽  
Beta 2 ◽  
Sobolev Exponent ◽  
Least Energy

AbstractWe consider the critical p-Laplacian system\left\{\begin{aligned} &\displaystyle{-}\Delta_{p}u-\frac{\lambda a}{p}\lvert u% \rvert^{a-2}u\lvert v\rvert^{b}=\mu_{1}\lvert u\rvert^{p^{\ast}-2}u+\frac{% \alpha\gamma}{p^{\ast}}\lvert u\rvert^{\alpha-2}u\lvert v\rvert^{\beta},&&% \displaystyle x\in\Omega,\\ &\displaystyle{-}\Delta_{p}v-\frac{\lambda b}{p}\lvert u\rvert^{a}\lvert v% \rvert^{b-2}v=\mu_{2}\lvert v\rvert^{p^{\ast}-2}v+\frac{\beta\gamma}{p^{\ast}}% \lvert u\rvert^{\alpha}\lvert v\rvert^{\beta-2}v,&&\displaystyle x\in\Omega,\\ &\displaystyle u,v\text{ in }D_{0}^{1,p}(\Omega),\end{aligned}\right.where {\Delta_{p}u:=\operatorname{div}(\lvert\nabla u\rvert^{p-2}\nabla u)} is the p-Laplacian operator defined onD^{1,p}(\mathbb{R}^{N}):=\bigl{\{}u\in L^{p^{\ast}}(\mathbb{R}^{N}):\lvert% \nabla u\rvert\in L^{p}(\mathbb{R}^{N})\bigr{\}},endowed with the norm {{\lVert u\rVert_{D^{1,p}}:=(\int_{\mathbb{R}^{N}}\lvert\nabla u\rvert^{p}\,dx% )^{\frac{1}{p}}}}, {N\geq 3}, {1<p<N}, {\lambda,\mu_{1},\mu_{2}\geq 0}, {\gamma\neq 0}, {a,b,\alpha,\beta>1} satisfy {a+b=p}, {\alpha+\beta=p^{\ast}:=\frac{Np}{N-p}}, the critical Sobolev exponent, Ω is {\mathbb{R}^{N}} or a bounded domain in {\mathbb{R}^{N}} and {D_{0}^{1,p}(\Omega)} is the closure of {C_{0}^{\infty}(\Omega)} in {D^{1,p}(\mathbb{R}^{N})}. Under suitable assumptions, we establish the existence and nonexistence of a positive least energy solution of this system. We also consider the existence and multiplicity of the nontrivial nonnegative solutions.

Classification of nonnegative solutions to a bi-harmonic equation with Hartree type nonlinearity

2018 ◽  
Vol 149 (04) ◽  
pp. 979-994 ◽  
Author(s):  
Daomin Cao ◽  
Wei Dai
Keyword(s):  
Critical Case ◽  
Classical Solutions ◽  
Sobolev Inequalities ◽  
Extremal Functions ◽  
Method Of Moving Planes ◽  
Nonnegative Solutions ◽  
Moving Planes ◽  
Image Position ◽  
Harmonic Equation

AbstractIn this paper, we are concerned with the following bi-harmonic equation with Hartree type nonlinearity $$\Delta ^2u = \left( {\displaystyle{1 \over { \vert x \vert ^8}}* \vert u \vert ^2} \right)u^\gamma ,\quad x\in {\open R}^d,$$where 0 &lt; γ ⩽ 1 and d ⩾ 9. By applying the method of moving planes, we prove that nonnegative classical solutions u to (𝒫γ) are radially symmetric about some point x0 ∈ ℝd and derive the explicit form for u in the Ḣ2 critical case γ = 1. We also prove the non-existence of nontrivial nonnegative classical solutions in the subcritical cases 0 &lt; γ &lt; 1. As a consequence, we also derive the best constants and extremal functions in the corresponding Hardy-Littlewood-Sobolev inequalities.

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