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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.

34L15Eigenvalues, estimation of eigenvalues, upper and lower bounds for OD operators
34B15Nonlinear boundary value problems for ODE
34L30Nonlinear ordinary differential operators
Full Text: DOI
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