Existence of solutions for a perturbation sublinear elliptic equation in {\mathbb {R}^N}

In this paper we consider the following problem $\left\{\begin{array}{l} -\Delta u=u-\left|u\right|^{-2\theta}u+f \\u \in H^1(\mathbb{R}^N)\cap L^{2(1-\theta)}(\mathbb{R}^N)\end{array}\right.$ ${f \in L^2(\mathbb{R}^N)\cap L^\frac{2(1-\theta)}{1-2\theta}(\mathbb{R}^N),\, N\geq 3,\, f\geq 0,\, f \neq 0}In this paper we consider the following problem

{l -Du=u-|u|-2qu+f u ? H1(\mathbbRN)?L2(1-q)(\mathbbRN)\left\{\begin{array}{l} -\Delta u=u-\left|u\right|^{-2\theta}u+f \\u \in H^1(\mathbb{R}^N)\cap L^{2(1-\theta)}(\mathbb{R}^N)\end{array}\right.      2. Quasilinear elliptic equations on $$\mathbb{R}^N $$with infinitely many solutions      Patrick?J.?RabierEmail author 《NoDEA : Nonlinear Differential Equations and Applications》2004,11(3):311-333 We give verifiable conditions ensuring that second order quasilinear elliptic equations on have infinitely many solutions in the Sobolev space for generic right-hand sides. This amounts to translating in concrete terms the more elusive hypotheses of an abstract theorem. Salient points include the proof that a key denseness property is equivalent to the existence of nontrivial solutions to an auxiliary problem, and an estimate of the size of the set of critical points of nonlinear Schrödinger operators. Conditions for the real-analyticity of Nemytskii operators are also discussed.      3. A smooth global branch of solutions for a semilinear elliptic equation on {\mathbb{R}^N}      François Genoud 《Calculus of Variations and Partial Differential Equations》2010,38(1-2):207-232 The existence of a global branch of positive spherically symmetric solutions ${\{(\lambda,u(\lambda)):\lambda\in(0,\infty)\}}$ of the semilinear elliptic equation $$\Delta u - \lambda u + V(x)|u|^{p-1}u = 0 \quad \text{in}\,\mathbb{R}^N\,\text{with}\,N\geq3$$ is proved for ${1 < p < 1+\frac{4-2b}{N-2}}$ , where ${b\in(0,2)}$ is such that the radial function V vanishes at infinity like |x|?b . V is allowed to be singular at the origin but not worse than |x|?b . The mapping ${\lambda\mapsto u(\lambda)}$ is of class ${C^r((0,\infty),H^1(\mathbb{R}^N))}$ if ${V\in C^r(\mathbb{R}^N\setminus\{0\},\mathbb{R})}$ , for r = 0, 1. Further properties of regularity and decay at infinity of solutions are also established. This work is a natural continuation of previous results by Stuart and the author, concerning the existence of a local branch of solutions of the same equation for values of the bifurcation parameter λ in a right neighbourhood of λ = 0. The variational structure of the equation is deeply exploited and the global continuation is obtained via an implicit function theorem.      4. Some remarks on systems of elliptic equations doubly critical in the whole $${\mathbb{R}^N}$$      Boumediene Abdellaoui  Veronica Felli  Ireneo Peral 《Calculus of Variations and Partial Differential Equations》2009,34(1):97-137 We study the existence of different types of positive solutions to problem where , , and is the critical Sobolev exponent. A careful analysis of the behavior of Palais-Smale sequences is performed to recover compactness for some ranges of energy levels and to prove the existence of ground state solutions and mountain pass critical points of the associated functional on the Nehari manifold. A variational perturbative method is also used to study the existence of a non trivial manifold of positive solutions which bifurcates from the manifold of solutions to the uncoupled system corresponding to the unperturbed problem obtained for ν = 0. B. Abdellaoui and I. Peral supported by projects MTM2007-65018, MEC and CCG06-UAM/ESP-0340, Spain. V. Felli supported by Italy MIUR, national project Variational Methods and Nonlinear Differential Equations.      5. Branches of solutions to semilinear elliptic equations on mathbb{R}^N      H. Jeanjean  M. Lucia  C.A. Stuart 《Mathematische Zeitschrift》1999,230(1):79-105 For a large class of functions f, we consider the nonlinear elliptic eigenvalue problem We describe the behaviour of the branch of solutions emanating from an eigenvalue of odd multiplicity below the essential spectrum of the linearized problem. A sharper result is obtained in the case of the lowest eigenvalue. The discussion is based on the degree theory for proper Fredholm maps developed by P.M Fitzpatrick, J. Pejsachowicz and P.J. Rabier. Received November 13, 1996; in final form March 24, 1997      6. Multibubble analysis on N-Laplace equation in {\mathbb{R}^N}      Adimurthi  Yunyan Yang 《Calculus of Variations and Partial Differential Equations》2011,40(1-2):1-14 In this note, we describe the asymptotic behavior of sequences of solutions to N-Laplace equations with critical exponential growth in smooth bounded domain in ${\mathbb{R}^N}$ . Precisely we prove multibubble phenomena and obtain an energy inequality for those concentrating solutions. In fact we partly extend the corresponding two-dimensional results of Adimurthi and Struwe (J Funct Anal 175:125?C167, 2000) and Druet (Duke Math J 132:217?C269, 2006) to high dimensional case.      7. Multiple solutions for quasi-linear elliptic problems in $${\mathbb{R}} ^2$$ with exponential growth      M. Squassina  C. Tarsi 《manuscripta mathematica》2007,124(3):409-410       8. On the Rao modules of minimal curves in $${\mathbb{P}^N}$$      Roberta Di Gennaro 《Ricerche di matematica》2009,58(2):249-262 We study the Hartshorne-Rao modules M C of minimal curves C in \mathbbPN{\mathbb{P}^N} , with N ≥ 4, lying in the same liaison class of curves on a smooth rational scroll surface. We get a free minimal resolution of M C for some of such curves and an upper bound for Betti numbers of M C , for any C.      9. Existence and multiplicity of solutions of semi-linear elliptic equations in {mathbb R}^N      Boyan Sirakov 《Calculus of Variations and Partial Differential Equations》2000,11(2):119-142       10. Three radial positive solutions for semilinear elliptic problems in $\mathbb{R}^N}          下载免费PDF全文 Suyun Wang  Yanhong Zhang  Ruyun Ma 《Journal of Applied Analysis & Computation》2020,10(2):760-770 This paper is concerned with the semilinear elliptic problem $$ \left\{ \begin{aligned} &-\Delta u=\lambda h(|x|)f(u) \ \ \ \ \ \ \ \ \ \ \text{in}\ \mathbb{R}^N, \\~ & u(x)>0\hskip 3cm \ \text{in}\ \mathbb{R}^N, \\~ &u\to 0 \hskip 3cm \ \ \ \ \text{as}\ |x|\to \infty, \end{aligned} \right. $$ where $\lambda$ is a real parameter and $h$ is a weight function which is positive. We show the existence of three radial positive solutions under suitable conditions on the nonlinearity. Proofs are mainly based on the bifurcation technique.      11. Existence of positive solutions for a class of indefinite elliptic problems in {mathbb R}^N      D.G. Costa  H. Tehrani 《Calculus of Variations and Partial Differential Equations》2001,13(2):159-189 We consider the question of existence of positive solutions for a class of elliptic problems in all of having a noncoercive linear part and a sign-changing nonlinearity of the form . Received December 20, 1999 / Accepted July 17, 2000 /Published online November 9, 2000      12. Global attractor for a semilinear strongly degenerate parabolic equation on {\mathbb{R}^N}      Cung The Anh 《NoDEA : Nonlinear Differential Equations and Applications》2014,21(5):663-678 The aim of this paper is to prove the existence of the global attractor for a semilinear strongly degenerate parabolic equation on \({\mathbb{R}^N}\) with the locally Lipschitz nonlinearity satisfying a subcritical growth condition.      13. A note on uniqueness of entropy solutions to degenerate parabolic equations in {\mathbb{R}^N}      Boris Andreianov  Mohamed Maliki 《NoDEA : Nonlinear Differential Equations and Applications》2010,17(1):109-118 We study the Cauchy problem in \mathbbRN{\mathbb{R}^N} for the parabolic equation ut+div F(u)=Dj(u),u_t+{\rm div}\,F(u)=\Delta\varphi(u),      14. Random self-affine multifractal Sierpinski sponges in $${\mathbb{R}^d}$$      Lars Olsen 《Monatshefte für Mathematik》2011,162(1):89-117 In this paper we study the Hausdorff and packing dimensions and the Rényi dimensions of random self-affine multifractal Sierpinski sponges in \({\mathbb{R}^{d}}\).      15. Asymmetric positive solutions for a symmetric nonlinear problem in {mathbb R}^N      Silvia Cingolani  José Luis Gámez 《Calculus of Variations and Partial Differential Equations》2000,11(1):97-117 We study a symmetric semilinear elliptic problem in all and we prove existence of an asymmetric positive solution by using variational arguments. The corresponding problem in dimension N=2, which provides the motivation of this work, arises in Nonlinear Optics from the study of the behaviour of optical cylindrical waveguides. Received September 28, 1999/ Accepted January 14, 2000 / Published online June 28, 2000      16. Pointwise estimates for relative fundamental solutions of heat equations in $${\mathbb{R} \times \mathbb{C}}$$      Andrew Raich 《Mathematische Zeitschrift》2007,256(1):193-220 Let be a subharmonic, nonharmonic polynomial and a parameter. Define , a closed, densely defined operator on . If and , we solve the heat equations , u(0,z) = f(z) and , . We write the solutions via heat semigroups and show that the solutions can be written as integrals against distributional kernels. We prove that the kernels are C ∞ off of the diagonal {(s, z, w) : s = 0 and z = w} and find pointwise bounds for the kernels and their derivatives.       17. Foliations and polynomial diffeomorphisms of $${\mathbb{R}^{3}}$$      Carlos Gutierrez  Carlos Maquera 《Mathematische Zeitschrift》2009,262(3):613-626 Let be a C 2 map and let Spec(Y) denote the set of eigenvalues of the derivative DY p , when p varies in . We begin proving that if, for some ϵ > 0, then the foliation with made up by the level surfaces {k = constant}, consists just of planes. As a consequence, we prove a bijectivity result related to the three-dimensional case of Jelonek’s Jacobian Conjecture for polynomial maps of The first author was supported by CNPq-Brazil Grant 306992/2003-5. The first and second author were supported by FAPESP-Brazil Grant 03/03107-9.      18. The positive energy conjecture for a class of AHM metrics on R2× Tn-2      Zhuobin Liang  Xiao Zhang 《中国科学 数学(英文版)》2023,66(5):1041-1052 We prove the positive energy conjecture for a class of asymptotically Horowitz-Myers(AHM) metrics on R2× Tn-2. This generalizes the previous results of Barzegar et al.(2020) as well as Liang and Zhang(2020).      19. Multiple positive solutions of singular and critical elliptic problem in {\mathbb{R}^2} with discontinuous nonlinearities      K. Sreenadh  Sweta Tiwari 《NoDEA : Nonlinear Differential Equations and Applications》2013,20(6):1831-1850 Let Ω be a bounded domain in ${\mathbb{R}^2}$ with smooth boundary. We consider the following singular and critical elliptic problem with discontinuous nonlinearity: $$(P_\lambda)\left \{\begin{array}{ll} - \Delta u = \lambda \left(\frac{m(x, u) e^{\alpha{u}^2}}{|x|^{\beta}} + u^{q}g(u - a)\right),\quad{u} > 0 \quad {\rm in} \quad \Omega\\u \quad \quad = 0\quad {\rm on} \quad \partial \Omega \end{array}\right.$$ where ${0\leq q < 1 ,0< \alpha\leq4\pi}$ and ${\beta \in [0, 2)}$ such that ${\frac{\beta}{2} + \frac{\alpha}{4\pi} \leq 1}$ and ${{g(t - a) = \left\{\begin{array}{ll}1, t \leq a\\ 0, t > a.\end{array}\right.}}$ Under the suitable assumptions on m(x, t) we show the existence and multiplicity of solutions for maximal interval for λ.      20. Phylogenetic Invariants for $${\mathbb{Z}_3}$$ Scheme-Theoretically      Maria Donten-Bury 《Annals of Combinatorics》2016,20(3):549-568 We study phylogenetic invariants of general group-based models of evolution with group of symmetries \({\mathbb{Z}_3}\). We prove that complex projective schemes corresponding to the ideal I of phylogenetic invariants of such a model and to its subideal \({I'}\) generated by elements of degree at most 3 are the same. This is motivated by a conjecture of Sturmfels and Sullivant [14, Conj. 29], which would imply that \({I = I'}\).

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Quasilinear elliptic equations on $$\mathbb{R}^N $$with infinitely many solutions

We give verifiable conditions ensuring that second order quasilinear elliptic equations on have infinitely many solutions in the Sobolev space for generic right-hand sides. This amounts to translating in concrete terms the more elusive hypotheses of an abstract theorem. Salient points include the proof that a key denseness property is equivalent to the existence of nontrivial solutions to an auxiliary problem, and an estimate of the size of the set of critical points of nonlinear Schrödinger operators. Conditions for the real-analyticity of Nemytskii operators are also discussed.     

A smooth global branch of solutions for a semilinear elliptic equation on {\mathbb{R}^N}

The existence of a global branch of positive spherically symmetric solutions ${\{(\lambda,u(\lambda)):\lambda\in(0,\infty)\}}$ of the semilinear elliptic equation $$\Delta u - \lambda u + V(x)|u|^{p-1}u = 0 \quad \text{in}\,\mathbb{R}^N\,\text{with}\,N\geq3$$ is proved for ${1 < p < 1+\frac{4-2b}{N-2}}$ , where ${b\in(0,2)}$ is such that the radial function V vanishes at infinity like |x|^?b . V is allowed to be singular at the origin but not worse than |x|^?b . The mapping ${\lambda\mapsto u(\lambda)}$ is of class ${C^r((0,\infty),H^1(\mathbb{R}^N))}$ if ${V\in C^r(\mathbb{R}^N\setminus\{0\},\mathbb{R})}$ , for r = 0, 1. Further properties of regularity and decay at infinity of solutions are also established. This work is a natural continuation of previous results by Stuart and the author, concerning the existence of a local branch of solutions of the same equation for values of the bifurcation parameter λ in a right neighbourhood of λ = 0. The variational structure of the equation is deeply exploited and the global continuation is obtained via an implicit function theorem.     

Some remarks on systems of elliptic equations doubly critical in the whole $${\mathbb{R}^N}$$

We study the existence of different types of positive solutions to problem

Branches of solutions to semilinear elliptic equations on mathbb{R}^N

For a large class of functions f, we consider the nonlinear elliptic eigenvalue problem We describe the behaviour of the branch of solutions emanating from an eigenvalue of odd multiplicity below the essential spectrum of the linearized problem. A sharper result is obtained in the case of the lowest eigenvalue. The discussion is based on the degree theory for proper Fredholm maps developed by P.M Fitzpatrick, J. Pejsachowicz and P.J. Rabier. Received November 13, 1996; in final form March 24, 1997     

Multibubble analysis on N-Laplace equation in {\mathbb{R}^N}

In this note, we describe the asymptotic behavior of sequences of solutions to N-Laplace equations with critical exponential growth in smooth bounded domain in ${\mathbb{R}^N}$ . Precisely we prove multibubble phenomena and obtain an energy inequality for those concentrating solutions. In fact we partly extend the corresponding two-dimensional results of Adimurthi and Struwe (J Funct Anal 175:125?C167, 2000) and Druet (Duke Math J 132:217?C269, 2006) to high dimensional case.     

On the Rao modules of minimal curves in $${\mathbb{P}^N}$$

We study the Hartshorne-Rao modules M _C of minimal curves C in \mathbbP^N{\mathbb{P}^N} , with N ≥ 4, lying in the same liaison class of curves on a smooth rational scroll surface. We get a free minimal resolution of M _C for some of such curves and an upper bound for Betti numbers of M _C , for any C.     

Three radial positive solutions for semilinear elliptic problems in $\mathbb{R}^N}

This paper is concerned with the semilinear elliptic problem $$ \left\{ \begin{aligned} &-\Delta u=\lambda h(|x|)f(u) \ \ \ \ \ \ \ \ \ \ \text{in}\ \mathbb{R}^N, \\~ & u(x)>0\hskip 3cm \ \text{in}\ \mathbb{R}^N, \\~ &u\to 0 \hskip 3cm \ \ \ \ \text{as}\ |x|\to \infty, \end{aligned} \right. $$ where $\lambda$ is a real parameter and $h$ is a weight function which is positive. We show the existence of three radial positive solutions under suitable conditions on the nonlinearity. Proofs are mainly based on the bifurcation technique.     

Existence of positive solutions for a class of indefinite elliptic problems in {mathbb R}^N

We consider the question of existence of positive solutions for a class of elliptic problems in all of having a noncoercive linear part and a sign-changing nonlinearity of the form . Received December 20, 1999 / Accepted July 17, 2000 /Published online November 9, 2000     

Global attractor for a semilinear strongly degenerate parabolic equation on {\mathbb{R}^N}

The aim of this paper is to prove the existence of the global attractor for a semilinear strongly degenerate parabolic equation on \({\mathbb{R}^N}\) with the locally Lipschitz nonlinearity satisfying a subcritical growth condition.     

A note on uniqueness of entropy solutions to degenerate parabolic equations in {\mathbb{R}^N}

We study the Cauchy problem in \mathbbR^N{\mathbb{R}^N} for the parabolic equation

Random self-affine multifractal Sierpinski sponges in $${\mathbb{R}^d}$$

In this paper we study the Hausdorff and packing dimensions and the Rényi dimensions of random self-affine multifractal Sierpinski sponges in \({\mathbb{R}^{d}}\).     

Asymmetric positive solutions for a symmetric nonlinear problem in {mathbb R}^N

We study a symmetric semilinear elliptic problem in all and we prove existence of an asymmetric positive solution by using variational arguments. The corresponding problem in dimension N=2, which provides the motivation of this work, arises in Nonlinear Optics from the study of the behaviour of optical cylindrical waveguides. Received September 28, 1999/ Accepted January 14, 2000 / Published online June 28, 2000     

Pointwise estimates for relative fundamental solutions of heat equations in $${\mathbb{R} \times \mathbb{C}}$$

Let be a subharmonic, nonharmonic polynomial and a parameter. Define , a closed, densely defined operator on . If and , we solve the heat equations , u(0,z) = f(z) and , . We write the solutions via heat semigroups and show that the solutions can be written as integrals against distributional kernels. We prove that the kernels are C ^∞ off of the diagonal {(s, z, w) : s = 0 and z = w} and find pointwise bounds for the kernels and their derivatives.      

Foliations and polynomial diffeomorphisms of $${\mathbb{R}^{3}}$$

Let be a C ² map and let Spec(Y) denote the set of eigenvalues of the derivative DY _p , when p varies in . We begin proving that if, for some ϵ > 0, then the foliation with made up by the level surfaces {k = constant}, consists just of planes. As a consequence, we prove a bijectivity result related to the three-dimensional case of Jelonek’s Jacobian Conjecture for polynomial maps of The first author was supported by CNPq-Brazil Grant 306992/2003-5. The first and second author were supported by FAPESP-Brazil Grant 03/03107-9.     

The positive energy conjecture for a class of AHM metrics on R2× Tn-2

We prove the positive energy conjecture for a class of asymptotically Horowitz-Myers(AHM) metrics on R²× T^n-2. This generalizes the previous results of Barzegar et al.(2020) as well as Liang and Zhang(2020).     

Multiple positive solutions of singular and critical elliptic problem in {\mathbb{R}^2} with discontinuous nonlinearities

Let Ω be a bounded domain in ${\mathbb{R}^2}$ with smooth boundary. We consider the following singular and critical elliptic problem with discontinuous nonlinearity: $$(P_\lambda)\left \{\begin{array}{ll} - \Delta u = \lambda \left(\frac{m(x, u) e^{\alpha{u}^2}}{|x|^{\beta}} + u^{q}g(u - a)\right),\quad{u} > 0 \quad {\rm in} \quad \Omega\\u \quad \quad = 0\quad {\rm on} \quad \partial \Omega \end{array}\right.$$ where ${0\leq q < 1 ,0< \alpha\leq4\pi}$ and ${\beta \in [0, 2)}$ such that ${\frac{\beta}{2} + \frac{\alpha}{4\pi} \leq 1}$ and ${{g(t - a) = \left\{\begin{array}{ll}1, t \leq a\\ 0, t > a.\end{array}\right.}}$ Under the suitable assumptions on m(x, t) we show the existence and multiplicity of solutions for maximal interval for λ.     

Phylogenetic Invariants for $${\mathbb{Z}_3}$$ Scheme-Theoretically

We study phylogenetic invariants of general group-based models of evolution with group of symmetries \({\mathbb{Z}_3}\). We prove that complex projective schemes corresponding to the ideal I of phylogenetic invariants of such a model and to its subideal \({I'}\) generated by elements of degree at most 3 are the same. This is motivated by a conjecture of Sturmfels and Sullivant [14, Conj. 29], which would imply that \({I = I'}\).