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On semilinear biharmonic equations with concave-convex nonlinearities involving weight functions

Lu Yang12* and Xuan Wang3

Author Affiliations

1 School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu, 730000, P.R. China

2 Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou, Gansu, 730000, P.R. China

3 College of Mathematics and Statistics, Northwest Normal University, Lanzhou, 730070, P.R. China

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Boundary Value Problems 2014, 2014:117  doi:10.1186/1687-2770-2014-117

Published: 14 May 2014

Abstract

In this paper, we consider semilinear biharmonic equations with concave-convex nonlinearities involving weight functions, where the concave nonlinear term is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M1">View MathML</a> and the convex nonlinear term is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M2">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M3">View MathML</a>. By use of the Nehari manifold and the direct variational methods, the existence of multiple positive solutions is established as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M4">View MathML</a>, here the explicit expression of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M5">View MathML</a> is provided.

MSC: 35J35, 35J40, 35J65.

Keywords:
biharmonic equations; concave-convex nonlinearities; weight functions