Unilateral global bifurcation phenomena and nodal solutions for p-Laplacian ✩

In this paper, we establish a Dancer-type unilateral global bifurcation result for one-dimensional p-Laplacian problem {−(φp(u′))′=μm(t)φp(u)+g(t,u;μ),t∈(0,1),u(0)=u(1)=0. Under some natural hypotheses on the perturbation function g:(0,1)×R×R→R, we show that μk(p) is a bifurcation point of the above problem and there are two distinct unbounded continua, Ck+ and Ck−, consisting of the bifurcation branch Ck from (μk(p),0), where μk(p) is the k-th eigenvalue of the linear problem corresponding to the above problem. As the applications of the above result, we study the existence of nodal solutions for the following problem {(φp(u′))′+f(t,u)=0,t∈(0,1),u(0)=u(1)=0. Moreover, based on the bifurcation result of Girg and Takac (2008) [13], we prove that there exist at least a positive solution and a negative one for the following problem {−div(φp(∇u))=f(x,u),in Ω,u=0,on ∂Ω.

[1]  Jean Mawhin,et al.  Periodic solutions for nonlinear systems with p-Laplacian-like operators , 1998 .

[2]  J.-L. Lions,et al.  Simplicit'e et isolation de la premi`ere valeur propre du p-Laplacien avec poids , 1987 .

[3]  Haiyan Wang,et al.  On the existence of positive solutions of ordinary differential equations , 1994 .

[4]  Johnny Henderson,et al.  Positive Solutions for Nonlinear Eigenvalue Problems , 1997 .

[5]  B. Thompson,et al.  Global behavior of positive solutions of nonlinear three-point boundary value problems , 2005 .

[6]  Yong-Hoon Lee,et al.  Global bifurcation phenomena for singular one-dimensional p-Laplacian , 2006 .

[7]  P. Drábek.  The least eigenvalues of nonhomogeneous degenerated quasilinear eigenvalue problems , 1995 .

[8]  P. Takáč,et al.  Bifurcations of Positive and Negative Continua in Quasilinear Elliptic Eigenvalue Problems , 2008 .

[9]  J. Schwartz.  Nonlinear Functional Analysis , 1969 .

[10]  Josep Blat,et al.  Bifurcation of steady-state solutions in predator-prey and competition systems , 1984, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[11]  Yong-Hoon Lee,et al.  Existence results of sign-changing solutions for singular one-dimensional p-Laplacian problems , 2008 .

[12]  Paul H. Rabinowitz,et al.  Some global results for nonlinear eigenvalue problems , 1971 .

[13]  K. J. Brown,et al.  Principal eigenvalues for problems with indefinite weight function on RN , 1990 .

[14]  Manuel del Pino,et al.  Global bifurcation from the eigenvalues of the p-Laplacian , 1991 .

[15]  Yong-Hoon Lee,et al.  A global bifurcation phenomenon for second order singular boundary value problems , 2005 .

[16]  P. Rabinowitz.  Some aspects of nonlinear eigenvalue problems , 1973 .

[17]  Meirong Zhang.  The Rotation Number Approach to Eigenvalues of the One‐Dimensional p‐Laplacian with Periodic Potentials , 2001 .

[18]  Yisheng Huang,et al.  Positive solution for -Deltapu = f (chi, u;) with f (chi, u) growing as up-1 at infinity , 2004, Appl. Math. Lett..

[19]  Gary M. Lieberman,et al.  Boundary regularity for solutions of degenerate elliptic equations , 1988 .

[20]  P. Rabinowitz.  Nonlinear sturm‐lionville problems for second order ordinary differential equations , 1970 .

[21]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[22]  Manuel del Pino,et al.  A homotopic deformation along p of a Leray-Schauder degree result and existence for (¦u′¦p − 2u′)′ + ƒ(t, u) = 0, u(0) = u(T) = 0, p > 1 , 1989 .

[23]  Edward Norman Dancer,et al.  Bifurcation from Simple Eigenvalues and Eigenvalues of Geometric Multiplicity One , 2002 .

[24]  Ruyun Ma,et al.  Multiplicity results for second-order two-point boundary value problems with superlinear or sublinear nonlinearities , 2005 .

[25]  O. Chakrone,et al.  Spectrum of one dimensional p-Laplacian operator with indefinite weight , 2002 .

[26]  B. Thompson,et al.  Nodal solutions for nonlinear eigenvalue problems , 2004 .

[27]  J. López-Gómez.  Spectral Theory and Nonlinear Functional Analysis , 2001 .

[28]  Xianling Fan,et al.  Existence of solutions for the p-Laplacian with crossing nonlinearity , 2002 .

[29]  Hiroshi Matano,et al.  Monotonicity and convergence results in order-preserving systems in the presence of symmetry , 1998 .