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Existence theorems for quasilinear elliptic eigenvalue problems in unbounded domains. (English) Zbl 1260.35085

Summary: The paper focuses on the existence of nontrivial solutions of a nonlinear eigenvalue problem depending on a real parameter \(\lambda\) under Robin boundary conditions in unbounded domains, with (possibly noncompact) smooth boundary. The problem involves a weighted \(p\)-Laplacian operator and subcritical nonlinearities, and even in the case \(p=2\) the main existence results are new. Denoting by \(\lambda_1\) the first eigenvalue of the underlying Robin eigenvalue problem, we prove the existence of (weak) solutions, with different methods, according to the case \(\lambda\geq \lambda_1\) or \(\lambda <\lambda_1\). In the first part of the paper we show the existence of a nontrivial solution for all \(\lambda \in \mathbb R\) for the problem under Ambrosetti-Rabinowitz-type conditions on the nonlinearities involved in the model. In detail, we apply the mountain-pass theorem of Ambrosetti and Rabinowitz if \(\lambda <\lambda_1\), while we use mini-max methods and linking structures over cones, as in [M. Degiovanni, J. Fixed Point Theory Appl. 7, No. 1, 85–102 (2010; Zbl 1205.58007); M. Degiovanni and S. Lancelotti, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 24, No. 6, 907–919 (2007; Zbl 1132.35040)], if \(\lambda \geq \lambda_1\). In the latter part of the paper we do not require any longer the Ambrosetti-Rabinowitz condition at \(\infty\), but the so-called Szulkin-Weth conditions, and we obtain the same result for all \(\lambda \in \mathbb R\). More precisely, using the Nehari-manifold method for \(C^1\) functionals developed by A. Szulkin and T. Weth [in: Handbook of nonconvex analysis and applications. Somerville, MA: International Press. 597–632 (2010; Zbl 1218.58010)], we prove existence of ground states, multiple solutions, and least-energy sign-changing solutions, whenever \(\lambda <\lambda_1\). On the other hand, in the case \(\lambda \geq \lambda_1\), we establish the existence of solutions again by linking methods.

MSC:

35P15 Estimates of eigenvalues in context of PDEs
35J66 Nonlinear boundary value problems for nonlinear elliptic equations
35J20 Variational methods for second-order elliptic equations
35J25 Boundary value problems for second-order elliptic equations
35J70 Degenerate elliptic equations
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Full Text: Euclid