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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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##### References:
  Anane, A., Etude des valeurs propres et de la résonnance pour l’opérateur p-laplacien, (1987), Université Libre de Bruxelles  Anane, A.; Tsouli, N., On the second eigenvalue of the p-Laplacian, (), 1-9 · Zbl 0854.35081  A. Anane, and, N. Tsouli, On a nonresonnance condition between the first and the second eigenvalue for the p-laplacian, preprint, 1995. · Zbl 1200.35140  Arias, M.; Campos, J., Radial fučik spectrum of the Laplace operator, J. math. anal. appl., 190, 654-666, (1995) · Zbl 0824.34083  M. Arias, J. Campos, and, J.-P. Gossez, On the antimaximum principle and the Fučik spectrum for the Neumann p-laplacian, to appear. · Zbl 0979.35048  Arcoya, D.; Orsina, L., Landesman – lazer conditions and quasilinear elliptic equations, Nonlinear anal., 28, 1623-1632, (1997) · Zbl 0871.35037  Bonnet, A., A deformation lemma on a C1 manifold, Manuscripta math., 81, 339-359, (1993) · Zbl 0801.57023  Boccardo, L.; Drabek, P.; Giachetti, D.; Kucera, M., Generalization of Fredholm alternative for nonlinear differential operators, Nonlinear anal., 10, 1083-1103, (1986) · Zbl 0623.34031  Brezis, H., Analyse fonctionnelle, théorie et applications, (1983), Masson Paris  Browder, F.E., Nonlinear eigenvalue problems and group invariance, functional analysis and related fields, (1970), Springer-Verlag New York/Berlin · Zbl 0213.41304  Costa, D.; Cuesta, M., Existence results for resonant perturbations of the fučik spectrum, Topol. methods nonlinear anal., 8, 295-314, (1996) · Zbl 0896.35047  Costa, D.; Oliveira, A., Existence of solutions for a class of semilinear problems at double resonance, Boll. soc. brasil. mat., 19, 21-37, (1988) · Zbl 0704.35048  Cuesta, M., Etude de la résonnance et du spectre de fučik des opérateurs laplacien et p-laplacien, (1993), Université Libre de Bruxelles  Cuesta, M.; Gossez, J.-P., A variational approach to nonresonance with respect to the fučik spectrum, Nonlinear anal., 19, 487-500, (1992) · Zbl 0768.34025  Dancer, N., On the Dirichlet problem for weakly nonlinear elliptic partial differential equation, Proc. roy. soc. Edinburgh, 76, 283-300, (1977) · Zbl 0351.35037  Dancer, N., Generic domain dependance for nonsmooth equations and the open set problem for jumping nonlinearities, Topol. methods nonlinear anal., 1, 139-150, (1993) · Zbl 0817.35026  de Figueiredo, D., Lectures on the Ekeland variational principle with applications and Détours, (1989), Springer-VerlagTATA Institute New York/Berlin  de Figueiredo, D.; Gossez, J.-P., On the first curve of the fučik spectrum of an elliptic operator, Differential integral equations, 7, 1285-1302, (1994) · Zbl 0797.35032  DiBenedetto, E., C1+α local regularity of weak solutions of degenerate elliptic equations, Nonlinear anal., 7, 827-850, (1983) · Zbl 0539.35027  Drabek, P., Solvability and bifurcations of nonlinear equations, Pitman research notes in mathematics, (1992), Longman Harlow/New York · Zbl 0753.34002  Fučik, S., Solvability of nonlinear equations and boundary value problems, (1980), Reidel Dordrecht · Zbl 0453.47035  Ghoussoub, N., Duality and perturbation methods in critical point theory, (1993), Cambridge Univ. Press Cambridge · Zbl 0790.58002  Gossez, J.-P.; Omari, P., Nonresonnace with respect to the fučik spectrum for periodic solutions of second order ordinary differential equations, Nonlinear anal., 14, 1079-1104, (1990) · Zbl 0709.34037  Lindqvist, P., On the equation div(|≠u|p−2≠u)+λ|u|p−2u=0, Proc. amer. math. soc., 109, 157-164, (1990) · Zbl 0714.35029  Micheletti, A.M., A remark on the resonance set for a semilinear elliptic equation, Proc. roy. soc. Edinburgh, 124, 803-809, (1994) · Zbl 0808.35036  Nečas, J., Introduction to the theory of nonlinear elliptic equations, (1983), Teubner Leipzig · Zbl 0615.73047  Tolksdorf, P., On the Dirichlet problem for quasilinear equations in domains with conical boundary points, Comm. partial differential equations, 8, 773-817, (1983) · Zbl 0515.35024  Touzani, A., Quelques résultats sur le A_p-laplacien avec poids indéfini, (1992), Université Libre de Bruxelles
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