zbMATH — the first resource for mathematics

Resonance problems for the $$p$$-Laplacian. (English) Zbl 0940.35087
Using variational arguments the authors prove the existence of a weak solution for the boundary value problem $\begin{cases} -\Delta_p u-\lambda|u|^{p-2}u+f(x,u)=0\quad&\text{in }\Omega,\\ u=0\quad&\text{on }\partial\Omega,\end{cases}$ where $$\Delta_p u=$$div$$(|Du|^{p-2}Du)$$, $$p>1$$, $$\Omega$$ is a bounded domain of $$\mathbb R^N$$, $$\lambda\in\mathbb R$$ and $$f:\Omega\times\mathbb R \to \mathbb R$$ is a bounded Carathéodory function such that there exist $$\lim_{t\to\pm\infty}f(x,t)= f^\pm(x)$$ a.e. in $$\Omega$$, with either $\begin{gathered}\int_{v>0}f^+ v+\int_{v<0}f^- v>0, \qquad\text{or} \tag{(LL)}$$_\lambda^+$$\\ \int_{v>0}f^+ v+\int_{v<0}f^- v<0\phantom{\qquad\text{or}} \tag{(LL)}$$_\lambda^-$$\end{gathered}$ for all $$v\in$$Ker$$(-\Delta_p-\lambda)\setminus\{0\}$$, and also there is $$g\in L^{p/(p-1)}(\Omega)$$ such that $|f(x,t)|\leq g(x)\quad\text{in }\Omega\times\mathbb R.$
The conditions (LL)$$_\lambda^\pm$$ are the standard Landesman – Laser conditions for resonance problems, and of course are trivially satisfied whenever $$\lambda$$ is not an eigenvalue.
Reviewer: P.Pucci (Perugia)

MSC:
 35J65 Nonlinear boundary value problems for linear elliptic equations 35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs 35A15 Variational methods applied to PDEs 35A05 General existence and uniqueness theorems (PDE) (MSC2000)
Full Text:
References:
 [1] Anane, A.; Tsouli, N., On the second eigenvalue of the p-Laplacian, () · Zbl 0854.35081 [2] Arcoya, D.; Orsina, L., Landesman – lazer conditions and quasilinear elliptic equations, Nonlinear analysis T.M.A., 28, 1623-1632, (1997) · Zbl 0871.35037 [3] Binding, P.; Drábek, P.; Huang, Y.X., On the Fredholm alternative for the p-Laplacian, Proc. amer. math. soc., 125, 3555-3559, (1997) · Zbl 0882.35049 [4] Binding, P.; Drábek, P.; Huang, Y.X., On the range of the p-Laplacian, Appl. math. letters, 10, 77-82, (1997) · Zbl 0894.34013 [5] M. Cuesta, D. G. deFigueiredo, and, J.-P. Gossez, The beginning of the Fucik spectrum for the p-Laplacian, J. Differential Equations, to appear. · Zbl 0947.35068 [6] Deimling, K., Nonlinear functional analysis, (1985), Springer-Verlag Berlin/Heidelberg · Zbl 0559.47040 [7] Drábek, P., Solvability and bifurcations of nonlinear equations, Pitman research notes in mathematics 265, (1992), Longman Harlow · Zbl 0753.34002 [8] Drabek, P.; Kufner, A.; Nicolosi, F., Quasilinear elliptic equations with degenerations and singularities, (1997), Walter de Gruyter Berlin/New York · Zbl 0894.35002 [9] P. Drábek, and, S. B. Robinson, Resonance problems for the one-dimensional p-Laplacian, preprint. [10] P. Drábek, and, P. Takáč, A counterexample to the Fredholm alternative for the p-Laplacian, to appear in, Proc. Amer. Math. Soc. [11] Ghoussoub, N., Duality and perturbation methods in critical point theory, (1993), Cambridge University Press · Zbl 0790.58002 [12] Landesman, E.M.; Lazer, A.C., Nonlinear perturbations of linear elliptic boundary value problems at resonance, J. math. mech., 19, 609-623, (1970) · Zbl 0193.39203 [13] Lindqvist, P., On the equation ÷(|≠u|p−2≠u)+λ|u|p−2u=0, Proc. amer. math. soc., 109, 157-164, (1990) · Zbl 0714.35029 [14] Struwe, M., Variational methods; applications to nonlinear partial differential equations and Hamiltonian systems, (1990), Springer-Verlag New York · Zbl 0746.49010
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.