Non-resonance of the first two eigenvalues of a quasilinear problem. (Q1281196)

If \(\lambda_k(p)\) is the \(k\)th eigenvalue of the problem \[ -(\phi_p(u'))'= \lambda\phi_p(u),\quad a<x<b,\quad u(a)= u(b)= 0, \] and sign \(sf(s)\to+\infty\) when \(| s|\to+\infty\), \[ \lambda_1(p)< \limsup_{s\to\pm\infty} {pF(s)\over| s|^p},\;\limsup_{s\to\infty} {f(s)\over\phi_p(s)}\leq \lambda_2(p)\text{ and }\limsup_{s\to-\infty} {f(s)\over\phi_p(s)}< \lambda_2(p), \] then the problem \[ -(\phi_p(u'))'= f(u)+ h(x),\quad a< x<b,\quad u(a)= u(b)= 0 \] has at least one solution for each \(h\in L^1(a,b)\). Here \(F(s)= \int^s_0 f(t)dt\), \(f\in C(\mathbb{R},\mathbb{R})\) and \(\phi_p(s)= | s|^{p-2}s\), \(p>1\).