Applications of Sub-Supersolution Theorems to Singular Nonlinear Elliptic Problems

2011 ◽  
Vol 11 (3) ◽  
Author(s):  
Nguyen Hoang Loc ◽  
Klaus Schmitt
Keyword(s):  
Boundary Value Problems ◽  
Nonlinear Equations ◽  
Elliptic Equations ◽  
Semilinear Elliptic Equations ◽  
Singular Boundary ◽  
Singular Boundary Value Problems ◽  
Nonlinear Elliptic Problems ◽  
Nonlinear Elliptic ◽  
Semilinear Elliptic ◽  
Singular Nonlinear Equations

AbstractThe results presented here were motivated by several recent papers on singular boundary value problems for semilinear elliptic equations with convection terms. We present extensions which cover singular nonlinear equations (mainly equations involving the p−Laplacian) containing convection terms. The results obtained are proved using sub- and supersolution theorems (motivated by the results in [18, 19, 20, 23]) and the construction of a well-ordered pair of such using a principal eigenfunction of the p−Laplacian.

Positive solutions for a class of nonlinear elliptic problems in RN

2000 ◽  
Vol 130 (1) ◽  
pp. 141-166 ◽  
Author(s):  
R. Molle ◽  
M. Musso ◽  
D. Passaseo
Keyword(s):  
Positive Solutions ◽  
Elliptic Equations ◽  
Critical Value ◽  
Nonlinear Elliptic Problems ◽  
Decay Condition ◽  
Nonlinear Elliptic ◽  
Semilinear Elliptic ◽  
Minimization Methods ◽  
Subcritical Nonlinearity ◽  
Positive Functions

We are concerned with positive solutions decaying at infinity for a class of semilinear elliptic equations in all of RN having superlinear subcritical nonlinearity. The corresponding variational problem lacks compactness because of the unboundedness of the domain and, in particular, it cannot be solved by minimization methods. However, we prove the existence of a positive solution, corresponding to a higher critical value of the related functional, under a suitable fast decay condition on the coefficient of the linear term. Moreover, we analyse the behaviour of the solution as this coefficient goes to infinity and show that the solution tends to split as the sum of two positive functions sliding to infinity in opposite directions. Finally, we use this property to prove the existence of at least 2k − 1 distinct positive solutions, when this coefficient splits as the sum of k bumps sufficiently far apart.

Points of Spherical Maxima and Solvability of Semilinear Elliptic Equations

1991 ◽  
Vol 43 (4) ◽  
pp. 825-831 ◽  
Author(s):  
Martin Schechter ◽  
Kyril Tintarev
Keyword(s):  
Boundary Value Problems ◽  
Elliptic Equations ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Semilinear Elliptic Equations ◽  
Nonlinear Functional ◽  
Semilinear Elliptic

AbstractWe give mild sufficient conditions on a nonlinear functional to have eigenvalues. These results are intended for the study of boundary value problems for semilinear elliptic equations.

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