Some properties of the Paneitz operator and nodal solutions to elliptic equations

We consider non-autonomous semilinear elliptic equations of the type [Formula: see text] where [Formula: see text] is either a ball or an annulus centered at the origin, [Formula: see text] and [Formula: see text] is [Formula: see text] on bounded sets of [Formula: see text]. We address the question of estimating the Morse index [Formula: see text] of a sign changing radial solution [Formula: see text]. We prove that [Formula: see text] for every [Formula: see text] and that [Formula: see text] if [Formula: see text] is even. If [Formula: see text] is superlinear the previous estimates become [Formula: see text] and [Formula: see text], respectively, where [Formula: see text] denotes the number of nodal sets of [Formula: see text], i.e. of connected components of [Formula: see text]. Consequently, every least energy nodal solution [Formula: see text] is not radially symmetric and [Formula: see text] as [Formula: see text] along the sequence of even exponents [Formula: see text].

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Nodal solutions for fractional elliptic equations involving exponential critical growth

Mathematical Methods in the Applied Sciences ◽

10.1002/mma.6145 ◽

2020 ◽

Vol 43 (6) ◽

pp. 3650-3672

Author(s):

Manassés Souza ◽

Uberlandio Batista Severo ◽

Thiago Luiz do Rêgo

Keyword(s):

Elliptic Equations ◽

Critical Growth ◽

Nodal Solutions ◽

Exponential Critical Growth

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Nonexistence of nodal solutions of elliptic equations with critical growth in $\mathbb{R}^2$

Transactions of the American Mathematical Society ◽

10.1090/s0002-9947-1992-1050083-3 ◽

1992 ◽

Vol 332 (1) ◽

pp. 449-458

Author(s):

Adimurthi ◽

S. L. Yadava

Keyword(s):

Elliptic Equations ◽

Critical Growth ◽

Nodal Solutions

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Existence of nodal solutions for quasilinear elliptic problems in ℝN

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210515000190 ◽

2015 ◽

Vol 145 (5) ◽

pp. 937-957

Author(s):

Ann Derlet ◽

François Genoud

Keyword(s):

Elliptic Equations ◽

Weighted Sobolev Space ◽

Point Theory ◽

Quasilinear Elliptic Equations ◽

Nodal Solutions ◽

Laplacian Equation ◽

Quasilinear Elliptic Problems ◽

Sign Changing Solutions ◽

Quasilinear Elliptic ◽

Constraint Minimization

We prove the existence of one positive, one negative and one sign-changing solution of a p-Laplacian equation on ℝN with a p-superlinear subcritical term. Sign-changing solutions of quasilinear elliptic equations set on the whole of ℝN have scarcely been investigated in the literature. Our assumptions here are similar to those previously used by some authors in bounded domains, and our proof uses fairly elementary critical point theory, based on constraint minimization on the nodal Nehari set. The lack of compactness due to the unbounded domain is overcome by working in a suitable weighted Sobolev space.

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Multiple nodal solutions for elliptic equations

Nonlinear Analysis ◽

10.1016/j.na.2004.03.010 ◽

2004 ◽

Vol 57 (4) ◽

pp. 615-632 ◽

Cited By ~ 6

Author(s):

Aixia Qian ◽

Shujie Li

Keyword(s):

Elliptic Equations ◽

Nodal Solutions

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Nodal Solutions for a Quasilinear Elliptic Equation Involving the p-Laplacian and Critical Exponents

Advanced Nonlinear Studies ◽

10.1515/ans-2017-6022 ◽

2018 ◽

Vol 18 (1) ◽

pp. 17-40

Author(s):

Yinbin Deng ◽

Shuangjie Peng ◽

Jixiu Wang

Keyword(s):

Elliptic Equations ◽

Order Term ◽

Quasilinear Elliptic Equations ◽

Nodal Solution ◽

Nodal Solutions ◽

Content Type ◽

Nodal Domains ◽

Change Of Variables ◽

Graphic Content ◽

Quasilinear Elliptic

AbstractThis paper is concerned with the following type of quasilinear elliptic equations in{\mathbb{R}^{N}}involving thep-Laplacian and critical growth:-\Delta_{p}u+V(|x|)|u|^{p-2}u-\Delta_{p}(|u|^{2})u=\lambda|u|^{q-2}u+|u|^{2p^{% *}-2}u,which arises as a model in mathematical physics, where{2<p<N},{p^{*}=\frac{Np}{N-p}}. For any given integer{k\geq 0}, by using change of variables and minimization arguments, we obtain, under some additional assumptions onpandq, a radial sign-changing nodal solution with{k+1}nodal domains. Since the critical exponent appears and the lower order term (obtained by a transformation) may change sign, we shall use delicate arguments.

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