Electron. J. Diff. Eqns., Vol. 2005(2005), No. 08, pp. 1-10.

Strong resonance problems for the one-dimensional p-Laplacian

Jiri Bouchala

Abstract:
We study the existence of the weak solution of the nonlinear boundary-value problem
$$\displaylines{ -(|u'|^{p-2}u')'= \lambda |u|^{p-2}u + g(u)-h(x)\quad \hbox{in } (0,\pi) ,\cr u(0)=u(\pi )=0\,, }$$
where $p$ and $\lambda$ are real numbers, $p$ greater than 1, $h\in L^{p'}(0,\pi )$ ($p' =\frac{p}{p-1}$) and the nonlinearity $g:\mathbb{R} \to \mathbb{R}$ is a continuous function of the Landesman-Lazer type. Our sufficiency conditions generalize the results published previously about the solvability of this problem.

Submitted June 21, 2004. Published January 5, 2005.
Math Subject Classifications: 34B15, 34L30, 47J30.
Key Words: p-Laplacian; resonance at the eigenvalues; Landesman-Lazer type conditions.

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Jirí Bouchala
Department of Applied Mathematics
VSB-Technical University Ostrava, Czech republic
email: jiri.bouchala@vsb.cz

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