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

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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Moser iteration applied to elliptic equations with critical growth on the boundary

Nonlinear Analysis ◽

10.1016/j.na.2018.10.002 ◽

2019 ◽

Vol 180 ◽

pp. 154-169 ◽

Cited By ~ 7

Author(s):

Greta Marino ◽

Patrick Winkert

Keyword(s):

Elliptic Equations ◽

Critical Growth ◽

Moser Iteration

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On a class of nonlinear elliptic equations with fast increasing weight and critical growth

Journal of Differential Equations ◽

10.1016/j.jde.2010.04.014 ◽

2010 ◽

Vol 249 (5) ◽

pp. 1035-1055 ◽

Cited By ~ 9

Author(s):

Marcelo F. Furtado ◽

Olímpio H. Myiagaki ◽

João Pablo P. da Silva

Keyword(s):

Elliptic Equations ◽

Nonlinear Elliptic Equations ◽

Critical Growth ◽

Nonlinear Elliptic

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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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