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Existence and nonexistence of nontrivial solutions of semilinear fourth- and sixth-order differential equations. (English) Zbl 1126.34316
The paper gives conditions for the existence of non-trivial solutions to the fourth-order boundary value problem $$ \gather u^{(4)}-pu''-a(x)u+b(x)u^3 =0, \quad 0 < x < L,\\ u(0) = u(L) = u''(0) = u''(L) =0 \endgather $$ as well as to the sixth-order one $$ \gather u^{(6)} + A u^{(4)} + Bu'' + Cu -b(x)u^3= 0, \quad 0 < x < L,\\ u(0) = u(L) = u''(0) = u''(L) = u^{(4)}(0) = u^{(4)}(L)=0\endgather $$ in which $A$, $B$, $C $ are constants, $C>0$ and $a(x)$ and $b(x)$ are even, continuous, positive, $2L$-periodic functions. The proof of the results uses the variational method introduced by {\it D. C. Clark} [“A variant of the Lusternik-Schnirelman theory”, Indiana Univ. Math. J. 22, 65--74 (1972; Zbl 0228.58006)].

34B15Nonlinear boundary value problems for ODE