Solvability of nonlinear elliptic boundary value problem via its associated linear problem. (Q1426977)

The author deals with the following problem on a bounded domain \(\Omega\subset\mathbb{R}^n\) with smooth boundary \[ \begin{cases} \phi_u+ \lambda_1 u+ g(x,u)= h(x),\quad & x\in\Omega,\\ u= 0,\quad & x\in\partial\Omega,\end{cases}\tag{1} \] where \(h\in L^\infty(\Omega)\) and \(\lambda_1\) is the first eigenvalue for the uniformly strongly selfadjoint elliptic linear operator \[ -\phi:= -\sum_{i,j} {\partial\over\partial x_i} \Biggl(a_{ij}{\partial\over\partial x_j}\Biggr)+ c \] with zero Dirichlet boundray condition and real, smooth coefficients \(a_{ij}\), \(a_{ij}= a_{ji}\) and \(c\geq 0\). Under some natural assumptions to the associated linear boundary value problem (no sign-changing solution), the author proves the existence (even multiple-nontrivial) solutions for (1). To this end, the author uses corresponding versions of the Leray-Schauder principle.