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The beginning of the Fučik spectrum for the $$p$$-Laplacian. (English) Zbl 0947.35068
Let $$\Omega$$ be a bounded domain in $$\mathbb{R}^N$$, $$N\geq 1$$ and let $$p$$ be a real number greater than $$1$$. The Fučik spectrum of the $$p$$-Laplacian $$\Delta_p=\operatorname {div}(|\nabla u|^{p-2}\nabla u)$$ on $$W^{1,p}_0(\Omega)$$ is defined as the set $$\Sigma_p$$ of pairs $$(\alpha,\beta)\in \mathbb{R}^2$$ such that the Dirichlet problem $$-\Delta_pu=\alpha(u^+)^{p-1}-\beta(u^-)^{p-1}$$ in $$\Omega$$ and $$u=0$$ on $$\partial\Omega$$ has a nontrivial solution. The usual spectrum of $$-\Delta_p$$ corresponds to $$\alpha=\beta$$. Clearly, if $$\lambda_1$$ is the first eigenvalue of $$-\Delta_p$$ on $$W^{1,p}_0$$, the set $$\Sigma_p$$ contains the two lines $$(\lambda_1\times \mathbb{R})$$ and $$(\mathbb{R}\times\lambda_1)$$. The authors prove that these lines are isolated in $$\Sigma_p$$. Then, by using the mountain pass theorem, they construct a nontrivial curve in $$\Sigma_p$$ and prove that such a curve is in fact the “first nontrivial curve” in $$\Sigma_p$$. As a consequence, a variational characterization via a mountain pass argument of the second eigenvalue of $$-\Delta_p$$ follows. Also, the regularity, monotonicity and asymptotic behaviour of this curve is studied. As an application of the above results, the solvability of the homogeneous Dirichlet problem $$-\Delta_pu=f(x,u)$$ in $$\Omega$$ is studied assuming that $$f(x,u)/|u|^{p-2}u$$ lies asymptotically between $$(\lambda_1,\lambda_1)$$ and one point $$(\alpha,\beta)$$ of the first curve in $$\Sigma_p$$. To study this problem, the montain pass theorem is used again.
Reviewer: G.Porru (Cagliari)

##### MSC:
 35J65 Nonlinear boundary value problems for linear elliptic equations 35P05 General topics in linear spectral theory for PDEs 35J20 Variational methods for second-order elliptic equations 35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
##### Keywords:
$$p$$-Laplacian; Fucik spectrum; nonresonance
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