We study the existence of positive solutions for perturbations of the classical eigenvalue problem for the Dirichlet p-Laplacian. We consider three cases. In the rst the perturbation is (p-1)-sublinear near $+\infty$, while in the second the perturbation is (p-1)-superlinear near $+infty$ and in the third we do not require asymptotic condition at $+\infty$. Using variational methods together with truncation and comparison techniques, we show that for $\lambda\in(0,\hat\lambda_1)$ ($\lambda> 0$ is the parameter and $\hat\lambda_1$ being the principal eigenvalue of $-\Delta_p,W_0^{1,p}(\Omega)$, we have positive solutions, while for $\lamba\geq \hat\lambda_1$, no positive solutions exist. In the \sublinear case" the positive solution is unique under a suitable monotonicity condition, while in the "superlinear case" we produce the existence of a smallest positive solution. Finally, we point out an existence result of a positive solution without requiring asymptotic condition at $+\infty$, provided that the perturbation is damped by a parameter.

Bonanno, G., Candito, P., Livrea, R., Papageorgiou, N. (2017). Existence, nonexistence and uniqueness of positive solutions for nonlinear eigenvalue problems. COMMUNICATIONS ON PURE AND APPLIED ANALYSIS, 16(4), 1169-1188 [10.3934/cpaa.2017057].

### Existence, nonexistence and uniqueness of positive solutions for nonlinear eigenvalue problems

#### Abstract

We study the existence of positive solutions for perturbations of the classical eigenvalue problem for the Dirichlet p-Laplacian. We consider three cases. In the rst the perturbation is (p-1)-sublinear near $+\infty$, while in the second the perturbation is (p-1)-superlinear near $+infty$ and in the third we do not require asymptotic condition at $+\infty$. Using variational methods together with truncation and comparison techniques, we show that for $\lambda\in(0,\hat\lambda_1)$ ($\lambda> 0$ is the parameter and $\hat\lambda_1$ being the principal eigenvalue of $-\Delta_p,W_0^{1,p}(\Omega)$, we have positive solutions, while for $\lamba\geq \hat\lambda_1$, no positive solutions exist. In the \sublinear case" the positive solution is unique under a suitable monotonicity condition, while in the "superlinear case" we produce the existence of a smallest positive solution. Finally, we point out an existence result of a positive solution without requiring asymptotic condition at $+\infty$, provided that the perturbation is damped by a parameter.
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http://aimsciences.org/journals/pdfs.jsp?paperID=13951&mode=full
Bonanno, G., Candito, P., Livrea, R., Papageorgiou, N. (2017). Existence, nonexistence and uniqueness of positive solutions for nonlinear eigenvalue problems. COMMUNICATIONS ON PURE AND APPLIED ANALYSIS, 16(4), 1169-1188 [10.3934/cpaa.2017057].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10447/247852