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The solvability of nonhomogeneous boundary value problems with ϕ-Laplacian operator

Sha-Sha Chen1 and Zhi-Hong Ma2*

Author Affiliations

1 Department of Mathematics, Ningbo University, Ningbo, 315211, P.R. China

2 Basic Science Department, Tianjin Agricultural University, Tianjin, 300384, P.R. China

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Boundary Value Problems 2014, 2014:82  doi:10.1186/1687-2770-2014-82

Published: 10 April 2014

Abstract

We treat the nonhomogeneous boundary value problems with ϕ-Laplacian operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M1">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M4">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M5">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M6">View MathML</a>) is an increasing homeomorphism such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M8">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M9">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M10">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M11">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M12">View MathML</a> is continuous. We will show that even if some of the τ and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/82/mathml/M13">View MathML</a> are negative, the boundary value problem with singular ϕ-Laplacian operator is always solvable, and the problem with a bounded ϕ-Laplacian operator has at least one positive solution.

Keywords:
nonhomogeneous; ϕ-Laplacian; negative coefficient; positive solution