scholarly journals Multiplicity of multi-bump type nodal solutions for a class of elliptic problems in ${\Bbb R}^N$

2009 ◽  
Vol 34 (2) ◽  
pp. 231 ◽  
Author(s):  
Claudianor O. Alves
Keyword(s):  
Elliptic Problems ◽  
Nodal Solutions

scholarly journals Nodal solutions to critical growth elliptic problems under Steklov boundary conditions

2009 ◽  
Vol 8 (2) ◽  
pp. 533-557 ◽  
Author(s):  
Elvise Berchio ◽  
◽  
Filippo Gazzola ◽  
Dario Pierotti ◽  
◽  
...  
Keyword(s):  
Boundary Conditions ◽  
Elliptic Problems ◽  
Critical Growth ◽  
Nodal Solutions

scholarly journals Sign-Changing Solutions for Nonlinear Elliptic Problems Depending on Parameters

2010 ◽  
Vol 2010 ◽  
pp. 1-33
Author(s):  
Siegfried Carl ◽  
Dumitru Motreanu
Keyword(s):  
Multiple Solutions ◽  
Mountain Pass Theorem ◽  
Elliptic Problems ◽  
Point Theory ◽  
Energy Functional ◽  
Constant Sign ◽  
Nodal Solutions ◽  
Variational Characterization ◽  
Spectrum Of The Laplacian ◽  
Sign Changing Solutions

The study of multiple solutions for quasilinear elliptic problems under Dirichlet or nonlinear Neumann type boundary conditions has received much attention over the last decades. The main goal of this paper is to present multiple solutions results for elliptic inclusions of Clarke's gradient type under Dirichlet boundary condition involving the -Laplacian which, in general, depend on two parameters. Assuming different structure and smoothness assumptions on the nonlinearities generating the multivalued term, we prove the existence of multiple constant-sign and sign-changing (nodal) solutions for parameters specified in terms of the Fučik spectrum of the -Laplacian. Our approach will be based on truncation techniques and comparison principles (sub-supersolution method) for elliptic inclusions combined with variational and topological arguments for, in general, nonsmooth functionals, such as, critical point theory, Mountain Pass Theorem, Second Deformation Lemma, and the variational characterization of the “beginning”of the Fu\v cik spectrum of the -Laplacian. In particular, the existence of extremal constant-sign solutions and their variational characterization as global (resp., local) minima of the associated energy functional will play a key-role in the proof of sign-changing solutions.

Multiplicity of positive and nodal solutions for nonlinear elliptic problems in RN

1998 ◽  
Vol 128 (5) ◽  
pp. 1069-1097 ◽  
Author(s):  
Ziqing Xie
Keyword(s):  
Positive Solutions ◽  
Elliptic Problems ◽  
Nodal Solutions ◽  
Nonlinear Elliptic Problems ◽  
Nonlinear Elliptic ◽  
Image Position

We consider the following problem:wherefor x ∈ RN, f(x, t), ft (x, t) ∈ C(RN × R), and f (x, t) ≧ 0 for all x ∈ RN and t ∈ R+, f(x, t) is an odd function of t. We show that if the maximum of Q(x) is achieved at k different points of RN, then for μ large enough the above problem has at least k positive solutions and k nodal solutions.

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