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Keywords:
nonlinear Dirichlet problem; classical solution; bifurcation point; ordinary differential equation
Summary:
We consider the nonlinear Dirichlet problem $$-u'' -r(x)|u|^\sigma u= \lambda u \text{ in } (0,\infty ), \, u(0)=0 \text{ and } \lim _{x\rightarrow \infty } u(x)=0,$$ and develop conditions for the function $r$ such that the considered problem has a positive classical solution. Moreover, we present some results showing that $\lambda =0$ is a bifurcation point in $W^{1,2} (0,\infty )$ and in $L^p(0,\infty )\, (2\leq p\leq \infty )$.
References:
[1] Adams R.A.: Sobolev Spaces. Academic Press, New York, 1975. MR 0450957 | Zbl 1098.46001
[2] Berger M.S.: On the existence and structure of stationary states for a nonlinear Klein-Gordon equation. J. Funct. Analysis 9 (1972), 249-261. MR 0299966 | Zbl 0224.35061
[3] Brezis H., Kato T.: Remarks on the Schrödinger operator with singular complex potentials. J. Math. pures et appl. 58 (1979), 137-151. MR 0539217 | Zbl 0408.35025
[4] Gilbarg D., Trudinger N.S.: Elliptic Partial Differential Equations of Second Order. SpringerVerlag, Berlin, Heidelberg, New York, 1983. MR 0737190 | Zbl 1042.35002
[5] Hörmander L.: Linear Partial Differential Operators. Springer-Verlag, Berlin, Heidelberg, New York, 1976. MR 0404822
[6] Stuart C.A.: Bifurcation for Dirichlet problems without eigenvalues. Proc. London Math. Soc. (3) 45 (1982), 169-192. MR 0662670 | Zbl 0505.35010

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