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The Fredholm alternative for the \(p\)-Laplacian: bifurcation from infinity, existence and multiplicity. (English) Zbl 1081.35031

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 \(p=2\).

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
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