Positive solutions for nonlinear elastic beam models (Q5957229)

Let \(\lambda_1\) be the first eigenvalue of the problem NEWLINE\[NEWLINEu''''=\lambda u,\quad 0< x< 1,\quad u(0)= u'(0)= u(1)= u'(1)= 0.NEWLINE\]NEWLINE The author studies the boundary value problem NEWLINE\[NEWLINEu''''= f(x,u),\quad 0< x< 1,\quad u(0)= \alpha,\;u'(0)=\beta,\;u(1)=\gamma,\;-u'(1)= \delta,\tag{1}NEWLINE\]NEWLINE where \(\alpha,\beta,\gamma,\delta\geq 0\), \(f\in (\langle 0,1\rangle\times \mathbb{R}^1_+,\mathbb{R}^1_+)\) is continuous and the following conditions are given NEWLINE\[NEWLINE\liminf_{u\to+\infty} \min_{x\in\langle 0,1\rangle} {f(x,u)\over u}> \lambda_1,\;\limsup_{u\to 0+} \max_{x\in\langle 0,1\rangle} {f(x,u+\xi(x))\over u}< \lambda_1,\tag{2}NEWLINE\]NEWLINE NEWLINE\[NEWLINE\limsup_{u\to+\infty} \max_{x\in\langle 0,1\rangle} {f(x,u)\over u}< \lambda_1,\;\liminf_{u\to 0+} \min_{x\in\langle 0,1\rangle} {f(x,u)\over u}> \lambda_1,\tag{3}NEWLINE\]NEWLINE NEWLINE\[NEWLINE\sup_{u> 0} \min_{x\in \langle 0,1\rangle} {f(x,u)\over u}= +\infty,\;\inf_{u\geq 2\|\xi\|} \max_{x\in \langle 0,1\rangle} {f(x,u)\over u}= 0.\tag{4}NEWLINE\]NEWLINE Here, the proofs of the following results are given:NEWLINENEWLINENEWLINE(A) If (3) is satisfied, then (1) has at least one positive solution.NEWLINENEWLINENEWLINE(B) Let (2) be satisfied. If \(\xi(x)\equiv 0\), \(x\in\langle 0,1\rangle\), then (1) has at least one positive solution. If \(\xi(x)\not\equiv 0\) \(x\in \langle 0,1\rangle\), then (1) has at least two positive solutions.NEWLINENEWLINENEWLINE(C) Let \(f(x,u)\) be increasing in \(u\). If (4) is satisfied, then (1) has at least one positive solution.