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A note on a nonresonance condition for a quasilinear elliptic problem. (English) Zbl 0816.35031
The authors study sufficient conditions for nonresonance of the problem $$-\text{div}(\vert \nabla u\vert\sp{p-2} \nabla u)= f(u)+ h(x) \quad (x\in \Omega), \qquad u=\varphi \quad (x\in \partial \Omega),$$ i.e. existence of solutions for any given $h\in L\sb \infty (\Omega)$ and $\varphi\in W\sb p\sp{1-1/p} (\partial \Omega)\cap L\sb \infty (\partial \Omega)$. Typically, these conditions are formulated in terms of the “interaction” of the asymptotic growth of the primitive of the nonlinearity $f$, on the one hand, and the first eigenvalue of the above differential operator, on the other.

35J65Nonlinear boundary value problems for linear elliptic equations
35A05General existence and uniqueness theorems (PDE) (MSC2000)
35P30Nonlinear eigenvalue problems for PD operators; nonlinear spectral theory
Full Text: DOI
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