We develop a variational framework to detect high energy solutions of the planar Schrödingerwith a positive function a ∈ L ∞ (R 2 ) and γ > 0. In particular, we deal with the periodic setting where the corresponding functional is invariant under Z 2 -translations and therefore fails to satisfy a global Palais-Smale condition. The key tool is a surprisingly strong compactness condition for Cerami sequences which is not available for the corresponding problem in higher space dimensions. In the case where the external potential a is a positive constant, we also derive, as a special case of a more general result, the existence of nonradial solutions (u, w) such that u has arbitrarily many nodal domains. Finally, in the case where a is constant, we also show that solutions of the above problem with u > 0 in R 2 and w(x) → −∞ as |x| → ∞ are radially symmetric up to translation. Our results are also valid for a variant of the above system containing a local nonlinear term in u in the first equation.
In this paper we obtain multiple positive solutions of the nonlinear elliptic equationare competing potential functions. We relate the number of solutions with the topology of the global minima set of a suitable ground energy function and improve a recent existence result of X.
Abstract. We consider the stationary nonlinear magnetic Choquard equationwhere A is a real valued vector potential, V is a real valued scalar potential, N ≥ 3, α ∈ (0, N ) and 2 − (α/N ) < p < (2N − α)/(N − 2). We assume that both A and V are compatible with the action of some group G of linear isometries of R N . We establish the existence of multiple complex valued solutions to this equation which satisfy the symmetry conditionwhere τ : G → S 1 is a given group homomorphism into the unit complex numbers.MSC2010: 35Q55, 35Q40, 35J20, 35B06.
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