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We are concerned with the existence of infinitely many radial symmetric solutions for a nonlinear stationary problem driven by a new class of nonhomogeneous differential operators. The proof relies on the symmetric version of the mountain pass theorem.

We consider a parametric semilinear Robin problem driven by the Laplacian plus an indefinite potential. The reaction term involves competing nonlinearities. More precisely, it is the sum of a parametric sublinear (concave) term and a superlinear (convex) term. The superlinearity is not expressed via the Ambrosetti-Rabinowitz condition. Instead, a more general hypothesis is used. We prove a bifurcation-type theorem describing the set of positive solutions as the parameter $\lambda > 0$ varies. We also show the existence of a minimal positive solution $\tilde{u}_\lambda$ and determine the monotonicity and continuity properties of the map $\lambda \mapsto \tilde{u}_\lambda$.

We study perturbations of the eigenvalue problem for the negative Laplacian plus an indefinite and unbounded potential and Robin boundary condition. First we consider the case of a sublinear perturbation and then of a superlinear perturbation. For the first case we show that for $ λ<\widehat{λ}_{1}$ ($ \widehat{λ}_{1}$ being the principal eigenvalue) there is one positive solution which is unique under additional conditions on the perturbation term. For $ λ≥q\widehat{λ}_{1}$ there are no positive solutions. In the superlinear case, for $ λ<\widehat{λ}_{1}$ we have at least two positive solutions and for $ λ≥q\widehat{λ}_{1}$ there are no positive solutions. For both cases we establish the existence of a minimal positive solution $ \bar{u}_{λ}$ and we investigate the properties of the map $ λ\mapsto\bar{u}_{λ}$.

We consider a nonlinear Robin problem driven by the *p*-Laplacian plus an indefinite potential. The reaction term is of arbitrary growth and only conditions near zero are imposed. Using critical point theory together with suitable truncation and perturbation techniques and comparison principles, we show that the problem admits a sequence of distinct smooth nodal solutions converging to zero in $C^1(\overline{Ω})$.

We consider perturbations of nonlinear eigenvalue problems driven by a nonhomogeneous differential operator plus an indefinite potential. We consider both sublinear and superlinear perturbations and we determine how the set of positive solutions changes as the real parameter $λ$ varies. We also show that there exists a minimal positive solution $\overline{u}_λ$ and determine the monotonicity and continuity properties of the map $λ\mapsto\overline{u}_λ$. Special attention is given to the particular case of the $p$-Laplacian.

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