Drábek, P.; Girg, P.; Takáč, P.; Ulm, M. The Fredholm alternative for the \(p\)-Laplacian: bifurcation from infinity, existence and multiplicity. (English) Zbl 1081.35031 Indiana Univ. Math. J. 53, No. 2, 433-482 (2004). Summary: This work is concerned with the existence and multiplicity of weak solutions \(u\in W_0^{1,p}(\Omega)\) to the quasilinear elliptic problem \[ \begin{cases} -\Delta_pu=\lambda|u|^{p-2}u+f(x)\quad &\text{in } \Omega;\\ u=0\quad &\text{on }\partial\Omega,\end{cases} \tag{P} \] with the spectral parameter \(\lambda\in\mathbb{R}\) near the (simple) principal eigenvalue \(\lambda_1\) of the positive Dirichlet \(p\)-Laplacian \(-\Delta_p\) in a bounded domain \(\Omega\subset\mathbb{R}^N\), for \(1<p< \infty\). Here, \(\Delta_pu\equiv\text{div}(|\nabla u|^{p-2}\nabla u)\) and \(f\in L^\infty(\Omega)\) is a given function. A priori bounds on the solutions are obtained from a rather precise description of possible “large solutions” investigated by bifurcations from infinity. They take the form \(u=t^{-1} (\varphi_1+v^\top)\) as \(t\to 0\), \(t\in\mathbb{R}\setminus \{0\}\), where \(\varphi_1\) stands for the (positive) eigenfunction associated with \(\lambda_1\), and \(v^\top\) is a relatively small perturbation of \(\varphi_1\) which is orthogonal to \(\varphi_1\). We also allow \(\lambda\) and \(f\) to vary with \(t\to 0\). Our method is based on the linearization of \(\Delta_p\) near \(\varphi_1\). As a result of our asymptotic formula for \(\lambda\) depending on \(t\) and \(f\), with the integral \(\int_\Omega f\varphi_1dx\) playing a major role, we are able to obtain a number of new results for problem (P). Some of these results for \(p\neq 2\) are quite different from the linear case \(p=2\). Cited in 3 ReviewsCited in 19 Documents MSC: 35J60 Nonlinear elliptic equations 47J15 Abstract bifurcation theory involving nonlinear operators 35J65 Nonlinear boundary value problems for linear elliptic equations 35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs PDFBibTeX XMLCite \textit{P. Drábek} et al., Indiana Univ. Math. J. 53, No. 2, 433--482 (2004; Zbl 1081.35031) Full Text: DOI