Constant sign and nodal solutions for parametric anisotropic (p, 2) -equations

ABSTRACT

We consider an anisotropic $(p,2)$-equation, with a parametric and superlinear reaction term. We show that for all small values of the parameter the problem has at least five nontrivial smooth solutions, four with constant sign and the fifth nodal (sign-changing). The proofs use tools from critical point theory, truncation and comparison techniques, and critical groups.

KEYWORDS:

• Anisotropic operators
• regularity theory
• maximum principle
• constant sign and nodal solutions
• critical groups
• variable exponent
• electrorheological fluids

2010 Mathematics Subject Classifications:

• Primary: 35J20
• 35J60
• 35J92
• Secondary:47J15
• 58E05

Acknowledgements

The authors wish to thank a knowledgeable referee for his/her constructive remarks and criticisms.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The second author was supported by Slovenian Research Agency [grant numbers P1-0292, N1-0114, N1-0083, N1-0064, and J1-8131].