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Nonuniform nonresonance of semilinear differential equations. (English) Zbl 0962.34062
Consider the Dirichlet problem $$(\phi_p(x'))'+f(t,x)=0,\quad x(0)=0=x(T), \tag 1$$ with $\phi_p(u)=|u|^{p-2}u$, $1<p<\infty$, and assume $$a(t)\leq\liminf_{|u|\to\infty}\frac{f(t,u)}{\phi_p(u)} \leq\limsup_{|u|\to\infty}\frac{f(t,u)}{\phi_p(u)}\leq b(t). \tag 2$$ The existence of solutions to (1) is obtained from conditions on the eigenvalues of the problem $$(\phi_p(x'))'+\lambda w(t)\phi_p(x)=0,\quad x(0)=0=x(T), \tag 3$$ with $w=a$ and $w=b$. It is first proved that for positive weights $w$, the problem (3) has a sequence of eigenvalues $0<\lambda_1(w)<\ldots<\lambda_k(w)<\ldots$ which depend monotonically on the weight. The existence of a solution to $$(\phi_p(x'))'+g(x)x'+f(t,x)=0,\quad x(0)=0=x(T),$$ is obtained assuming $$\limsup_{|u|\to\infty}\frac{f(t,u)}{\phi_p(u)}\leq b(t),$$ where $b\in L^1(0,T)$ is positive and $\lambda_1(b)>1$. Similarly, the existence of a solution to (1) follows assuming (2), where $a$ and $b\geq a$ are positive and for some $k\geq 2$, $\lambda_{k-1}(a)<1<\lambda_k(b)$. These results are based on degree arguments. In a last section, best Sobolev constants are obtained which imply estimates on the first eigenvalue of (3). These are used in examples.

##### MSC:
 34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds for OD operators 34B15 Nonlinear boundary value problems for ODE 34L30 Nonlinear ordinary differential operators
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##### References:
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