{"title":"Hénon type equations with jumping nonlinearities involving critical growth","authors":"Eudes Barboza, Ó. JoãoMarcosdo, Bruno Ribeiro","doi":"10.57262/ade/1571731545","DOIUrl":"https://doi.org/10.57262/ade/1571731545","url":null,"abstract":"","PeriodicalId":53312,"journal":{"name":"Advances in Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2019-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45749721","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A note on deformation argument for $L^2$ normalized solutions of nonlinear Schrödinger equations and systems","authors":"N. Ikoma, Kazunaga Tanaka","doi":"10.57262/ade/1571731543","DOIUrl":"https://doi.org/10.57262/ade/1571731543","url":null,"abstract":"","PeriodicalId":53312,"journal":{"name":"Advances in Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2019-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48837939","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-08-15DOI: 10.57262/ade027-0304-147
Jingyi Dong, Jiamei Hu, Yibin Zhang
Let $Omega$ be a bounded domain in $mathbb{R}^2$ with smooth boundary, we study the following elliptic Dirichlet problem begin{equation*} aligned left{aligned &-Deltaupsilon= e^{upsilon}-sphi_1-4pialphadelta_p-h(x),,,, ,textrm{in},,,,,Omega,[2mm] &upsilon=0 quadquadquadquadquadquad quadqquadqquadquadquad, textrm{on}, ,partialOmega, endalignedright. endaligned end{equation*} where $s>0$ is a large parameter, $hin C^{0,gamma}(overline{Omega})$, $pinOmega$, $alphain(-1,+infty)setminusmathbb{N}$, $delta_p$ denotes the Dirac measure supported at point $p$ and $phi_1$ is a positive first eigenfunction of the problem $-Deltaphi=lambdaphi$ under Dirichlet boundary condition in $Omega$. If $p$ is a strict local maximum point of $phi_1$, we show that such a problem has a family of solutions $upsilon_s$ with arbitrary $m$ bubbles accumulating to $p$, and the quantity $int_{Omega}e^{upsilon_s}rightarrow8pi(m+1+alpha)phi_1(p)$ as $srightarrow+infty$.
{"title":"Bubbling solutions for a planar exponential nonlinear elliptic equation with a singular source","authors":"Jingyi Dong, Jiamei Hu, Yibin Zhang","doi":"10.57262/ade027-0304-147","DOIUrl":"https://doi.org/10.57262/ade027-0304-147","url":null,"abstract":"Let $Omega$ be a bounded domain in $mathbb{R}^2$ with smooth boundary, we study the following elliptic Dirichlet problem begin{equation*} aligned left{aligned &-Deltaupsilon= e^{upsilon}-sphi_1-4pialphadelta_p-h(x),,,, ,textrm{in},,,,,Omega,[2mm] &upsilon=0 quadquadquadquadquadquad quadqquadqquadquadquad, textrm{on}, ,partialOmega, endalignedright. endaligned end{equation*} where $s>0$ is a large parameter, $hin C^{0,gamma}(overline{Omega})$, $pinOmega$, $alphain(-1,+infty)setminusmathbb{N}$, $delta_p$ denotes the Dirac measure supported at point $p$ and $phi_1$ is a positive first eigenfunction of the problem $-Deltaphi=lambdaphi$ under Dirichlet boundary condition in $Omega$. If $p$ is a strict local maximum point of $phi_1$, we show that such a problem has a family of solutions $upsilon_s$ with arbitrary $m$ bubbles accumulating to $p$, and the quantity $int_{Omega}e^{upsilon_s}rightarrow8pi(m+1+alpha)phi_1(p)$ as $srightarrow+infty$.","PeriodicalId":53312,"journal":{"name":"Advances in Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2019-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42768465","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On the Kirchhoff type equations in $mathbb{R}^{N}$","authors":"Juntao Sun, Tsung‐fang Wu","doi":"10.57262/ade027-0304-97","DOIUrl":"https://doi.org/10.57262/ade027-0304-97","url":null,"abstract":"Consider a nonlinear Kirchhoff type equation as follows begin{equation*} left{ begin{array}{ll} -left( aint_{mathbb{R}^{N}}|nabla u|^{2}dx+bright) Delta u+u=f(x)leftvert urightvert ^{p-2}u & text{ in }mathbb{R}^{N}, uin H^{1}(mathbb{R}^{N}), & end{array}% right. end{equation*}% where $Ngeq 1,a,b>0,2<p<min left{ 4,2^{ast }right}$($2^{ast }=infty $ for $N=1,2$ and $2^{ast }=2N/(N-2)$ for $Ngeq 3)$ and the function $fin C(mathbb{R}^{N})cap L^{infty }(mathbb{R}^{N})$. Distinguishing from the existing results in the literature, we are more interested in the geometric properties of the energy functional related to the above problem. Furthermore, the nonexistence, existence, unique and multiplicity of positive solutions are proved dependent on the parameter $a$ and the dimension $N.$ In particular, we conclude that a unique positive solution exists for $1leq Nleq4$ while at least two positive solutions are permitted for $Ngeq5$.","PeriodicalId":53312,"journal":{"name":"Advances in Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2019-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47582051","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-05-23DOI: 10.57262/ade026-0304-163
Jia-Feng Cao, Wan-Tong Li, Jie Wang
This paper is concerned with a nonlocal diffusion Lotka-Volterra type competition model that consisting of a native species and an invasive species in a one-dimensional habitat with free boundaries. We prove the well-posedness of the system and get a spreading-vanishing dichotomy for the invasive species. We also provide some sufficient conditions to ensure spreading success or spreading failure for the case that the invasive species is an inferior competitor or a superior competitor, respectively.
{"title":"The Dynamics of a Lotka-Volterra competition model with nonlocal diffusion and free boundaries","authors":"Jia-Feng Cao, Wan-Tong Li, Jie Wang","doi":"10.57262/ade026-0304-163","DOIUrl":"https://doi.org/10.57262/ade026-0304-163","url":null,"abstract":"This paper is concerned with a nonlocal diffusion Lotka-Volterra type competition model that consisting of a native species and an invasive species in a one-dimensional habitat with free boundaries. We prove the well-posedness of the system and get a spreading-vanishing dichotomy for the invasive species. We also provide some sufficient conditions to ensure spreading success or spreading failure for the case that the invasive species is an inferior competitor or a superior competitor, respectively.","PeriodicalId":53312,"journal":{"name":"Advances in Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2019-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41336816","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-05-05DOI: 10.57262/ade028-0304-169
Tomasz Klimsiak
We study the asymptotics of solutions of logistic type equations with fractional Laplacian as time goes to infinity and as the exponent in nonlinear part goes to infinity. We prove strong convergence of solutions in the energy space and uniform convergence to the solution of an obstacle problem. As a by-product, we also prove the cut-off property for eigenvalues of the Dirichlet fractional Laplace operator perturbed by exploding potentials.
{"title":"Asymptotics for logistic-type equations with Dirichlet fractional Laplace operator","authors":"Tomasz Klimsiak","doi":"10.57262/ade028-0304-169","DOIUrl":"https://doi.org/10.57262/ade028-0304-169","url":null,"abstract":"We study the asymptotics of solutions of logistic type equations with fractional Laplacian as time goes to infinity and as the exponent in nonlinear part goes to infinity. We prove strong convergence of solutions in the energy space and uniform convergence to the solution of an obstacle problem. As a by-product, we also prove the cut-off property for eigenvalues of the Dirichlet fractional Laplace operator perturbed by exploding potentials.","PeriodicalId":53312,"journal":{"name":"Advances in Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2019-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47014833","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the Cauchy problem ∂tu + u∂xu + L(∂xu) = 0, u(0, x) = u0(x) on the torus and on the real line for a class of Fourier multiplier operators L, and prove that the solution map u0 7→ u(t) is not uniformly continuous in H(T) or H(R) for s > 3 2 . Under certain assumptions, the result also hold for s > 0. The class of equations considered includes in particular the Whitham equation and fractional Korteweg-de Vries equations and we show that, in general, the flow map cannot be uniformly continuous if the dispersion of L is weaker than that of the KdV operator. The result is proved by constructing two sequences of solutions converging to the same limit at the initial time, while the distance at a later time is bounded below by a positive constant.
{"title":"Non-uniform dependence on initial data for equations of Whitham type","authors":"Mathias Nikolai Arnesen","doi":"10.57262/ade/1554256825","DOIUrl":"https://doi.org/10.57262/ade/1554256825","url":null,"abstract":"We consider the Cauchy problem ∂tu + u∂xu + L(∂xu) = 0, u(0, x) = u0(x) on the torus and on the real line for a class of Fourier multiplier operators L, and prove that the solution map u0 7→ u(t) is not uniformly continuous in H(T) or H(R) for s > 3 2 . Under certain assumptions, the result also hold for s > 0. The class of equations considered includes in particular the Whitham equation and fractional Korteweg-de Vries equations and we show that, in general, the flow map cannot be uniformly continuous if the dispersion of L is weaker than that of the KdV operator. The result is proved by constructing two sequences of solutions converging to the same limit at the initial time, while the distance at a later time is bounded below by a positive constant.","PeriodicalId":53312,"journal":{"name":"Advances in Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2019-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45899588","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"$L^2$ representations of the Second Variation and Łojasiewicz-Simon gradient estimates for a decomposition of the Möbius energy","authors":"Katsunori Gunji","doi":"10.57262/ade/1554256827","DOIUrl":"https://doi.org/10.57262/ade/1554256827","url":null,"abstract":"","PeriodicalId":53312,"journal":{"name":"Advances in Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2019-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43854579","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Supersonic-sonic patches are ubiquitous in regions of transonic flows and they boil down to a family of degenerate hyperbolic problems in regions surrounded by a streamline, a characteristic curve and a possible sonic curve. This paper establishes the global existence of solutions in a whole supersonic-sonic patch characterized by the two-dimensional full system of steady Euler equations and studies solution behaviors near sonic curves, depending on the proper choice of boundary data extracted from the airfoil problem and related contexts. New characteristic decompositions are developed for the full system and a delicate local partial hodograph transformation is introduced for the solution estimates. It is shown that the solution is uniformly $C^{1,frac{1}{6}}$ continuous up to the sonic curve and the sonic curve is also $C^{1,frac{1}{6}}$ continuous.
{"title":"On a global supersonic-sonic patch characterized by 2-D steady full Euler equations","authors":"Yan-bo Hu, Jiequan Li","doi":"10.57262/ade/1589594418","DOIUrl":"https://doi.org/10.57262/ade/1589594418","url":null,"abstract":"Supersonic-sonic patches are ubiquitous in regions of transonic flows and they boil down to a family of degenerate hyperbolic problems in regions surrounded by a streamline, a characteristic curve and a possible sonic curve. This paper establishes the global existence of solutions in a whole supersonic-sonic patch characterized by the two-dimensional full system of steady Euler equations and studies solution behaviors near sonic curves, depending on the proper choice of boundary data extracted from the airfoil problem and related contexts. New characteristic decompositions are developed for the full system and a delicate local partial hodograph transformation is introduced for the solution estimates. It is shown that the solution is uniformly $C^{1,frac{1}{6}}$ continuous up to the sonic curve and the sonic curve is also $C^{1,frac{1}{6}}$ continuous.","PeriodicalId":53312,"journal":{"name":"Advances in Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2019-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41508427","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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