The article deals with the boundary value problem \[ (\varphi_p(u'))'+ \lambda_1\varphi_p(u)= h,\quad u(0)= u(T)= 0, \] with \(\varphi_p(s)= | s|^{p-2}s\), \(p>1\), \(h\in L^\infty\), and \(\lambda_1> 0\) is the first eigenvalue of the corresponding homogeneous problem with \(h= 0\). The existence of at least one solution and at least two distinct solutions is studied. A nonlinear analog of the Fredholm alternative is obtained for the considered case. The authors study the a priori boundedness of the set of all solutions and properties of the energy functional.