We consider the existence of positive solutions of one-dimensional prescribed mean curvature equation−(u′/1+u′2)′=λf(u),0<t<1,u(t)>0,t∈(0,1),u(0)=u(1)=0whereλ>0is a parameter, andf:[0,∞)→[0,∞)is continuous. Further, whenfsatisfiesmax{up,uq}≤f(u)≤up+uq,0<p≤q<+∞, we obtain the exact number of positive solutions. The main results are based upon quadrature method.
We consider the equations involving the one-dimensional p-Laplacian (P): (u′tp-2u ′(t))′+λf(u(t))=0, 0<t<1, and
u(0)=u(1)=0, where p>1,λ>0,f∈C1(R;R),f(s)s>0, and
s≠0. We show the existence of sign-changing solutions under the assumptions f∞=lim|s|→∞(fs/sp-1)=+∞ and f0=lim|s|→0(f(s)/sp-1)∈[0,∞]. We also show that (P) has exactly one solution having specified nodal properties for λ∈(0,λ*) for some λ*∈(0,∞). Our main results are based on quadrature method.
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