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Many problems in science and engineering are described by nonlinear differential equations, which can be notoriously difficult to solve. Through the interplay of topological and variational ideas, methods of nonlinear analysis are able to tackle such fundamental problems. This graduate text explains some of the key techniques in a way that will be appreciated by mathematicians, physicists and engineers. Starting from elementary tools of bifurcation theory and analysis, the authors cover a number of more modern topics from critical point theory to elliptic partial differential equations. A series of Appendices give convenient accounts of a variety of advanced topics that will introduce the reader to areas of current research. The book is amply illustrated and many chapters are rounded off with a set of exercises.

Given a compact four dimensional manifold, we prove existence of conformal metrics with constant Q-curvature under generic assumptions. The problem amounts to solving a fourth-order nonlinear elliptic equation with variational structure. Since the corresponding Euler functional is in general unbounded from above and from below, we employ topological methods and min-max schemes, jointly with the compactness result of [35].

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We deal with a class on nonlinear Schrödinger equations (NLS) with potentials V (x) ∼ |x| −α , 0 < α < 2, and K(x) ∼ |x| −β , β > 0. Working in weighted Sobolev spaces, the existence of ground states v ε belonging to W 1,2 (R N ) is proved under the assumption that σ < p < (N + 2)/(N − 2) for some σ = σ N,α,β . Furthermore, it is shown that v ε are spikes concentrating at a minimum point of A = V θ K −2/(p−1) , where θ = (p + 1)/(p − 1) − 1/2.

We consider surfaces immersed in three-dimensional pseudohermitian manifolds. We define the notion of (p-)mean curvature and of the associated (p-)minimal surfaces, extending some concepts previously given for the (flat) Heisenberg group. We interpret the p-mean curvature not only as the tangential sublaplacian of a defining function, but also as the curvature of a characteristic curve, and as a quantity in terms of calibration geometry. As a differential equation, the p-minimal surface equation is degenerate (hyperbolic and elliptic). To analyze the singular set (i.e., the set where the (p-)area integrand vanishes), we formulate some extension theorems, which describe how the characteristic curves meet the singular set. This allows us to classify the entire solutions to this equation and to solve a Bernstein-type problem (for graphs over the x y-plane) in the Heisenberg group H 1 . In H 1 , identified with the Euclidean space R 3 , the p-minimal surfaces are classical ruled surfaces with the rulings generated by Legendrian lines. We also prove a uniqueness theorem for the Dirichlet problem under a condition on the size of the singular set in two dimensions, and generalize to higher dimensions without any size control condition. We also show that there are no closed, connected, C 2 smoothly immersed constant p-mean curvature or p-minimal surfaces of genus greater than one in the standard S 3 . This fact continues to hold when S 3 is replaced by a general pseudohermitian 3-manifold.

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In this paper we consider the following Toda system of equations on a compact surface:  