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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.

MSC:
35J65Nonlinear boundary value problems for linear elliptic equations
35A05General existence and uniqueness theorems (PDE) (MSC2000)
35P30Nonlinear eigenvalue problems for PD operators; nonlinear spectral theory
Keywords:
nonresonance
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References:
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