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Using the deformation lemmas from the previous chapter, various existence and multiplicity results concerning the critical points for nonsmooth functionals are provided in this chapter. We start with the Mountain Pass, Saddle Point and Linking theorems for locally Lipschitz functions which satisfy the generalized (φ − C)c condition. The “zero altitude case” and the symmetric version of the Mountain Pass Theorem are also discussed. Finally, minimax results for Szulkin functionals and some Schecter-type as well as Ricceri-type multiplicity results are given.

In this chapter we study various boundary value problems governed by the Laplace operator or a generalized Laplacian (p −Laplacian, variable exponent p(⋅) −Laplacian and Φ −Laplacian). The approach is variational, i.e., the solutions are the critical points of the corresponding energy functional defined on a suitable Sobolev space. Using the theoretical results from the previous chapter existence and multiplicity results are established for each considered problem.

Let (M,g) be an n-dimensional complete open Riemannian manifold with nonnegative Ricci curvature verifying ρΔgρ⩾n−5⩾0, where Δg is the Laplace–Beltrami operator on (M,g) and ρ is the distance function from a given point. If (M,g) supports a second-order Sobolev inequality with a constant C>0 close to the optimal constant K0 in the second-order Sobolev inequality in Rn, we show that a global volume noncollapsing property holds on (M,g). The latter property together with a Perelman-type construction established by Munn (J. Geom. Anal. (2010) 723–750) provide several rigidity results in terms of the higher order homotopy groups of (M,g). Furthermore, it turns out that (M,g) supports the second-order Sobolev inequality with the constant C=K0 if and only if (M,g) is isometric to the Euclidean space Rn.

In this paper we solve a problem raised by Guti\'errez and Montanari about comparison principles for $H-$convex functions on subdomains of Heisenberg groups. Our approach is based on the notion of the sub-Riemannian horizontal normal mapping and uses degree theory for set-valued maps. The statement of the comparison principle combined with a Harnack inequality is applied to prove the Aleksandrov-type maximum principle, describing the correct boundary behavior of continuous $H-$convex functions vanishing at the boundary of horizontally bounded subdomains of Heisenberg groups. This result answers a question by Garofalo and Tournier. The sharpness of our results are illustrated by examples., Comment: Published in: J. Funct. Anal. 269 (2015), no. 9, 2669-2708

In this Note, we present geodesic versions of the Borell–Brascamp–Lieb, Brunn–Minkowski and entropy inequalities on the Heisenberg group H n . Our arguments use the Riemannian approximation of H n combined with optimal mass-transportation techniques.

In this paper we study a sharp Sobolev interpolation inequality on Finsler manifolds. We show that Minkowski spaces represent the optimal framework for the Sobolev interpolation inequality on a large class of Finsler manifolds: (1) Minkowski spaces support the sharp Sobolev interpolation inequality; (2) any complete Berwald space with non-negative Ricci curvature which supports the sharp Sobolev interpolation inequality is isometric to a Minkowski space. The proofs are based on properties of the Finsler–Laplace operator and on the Finslerian Bishop–Gromov volume comparison theorem.

We prove that if a metric measure space satisfies the volume doubling condition and the Caffarelli–Kohn–Nirenberg inequality with the same exponent \(n \ge 3\), then it has exactly the \(n\)-dimensional volume growth. As an application, if an \(n\)-dimensional Finsler manifold of non-negative \(n\)-Ricci curvature satisfies the Caffarelli–Kohn–Nirenberg inequality with the sharp constant, then its flag curvature is identically zero. In the particular case of Berwald spaces, such a space is necessarily isometric to a Minkowski space.

In this paper, we study the coupled Schrodinger–Maxwell system ( S M λ ) { − Δ u + u + e ϕ u = λ α ( x ) f ( u ) in R 3 , − Δ ϕ = 4 π e u 2 in R 3 , where e > 0 , α ∈ L ∞ ( R 3 ) ∩ L 6 / ( 5 − q ) ( R 3 ) for some q ∈ ( 0 , 1 ) , and the continuous function f : R → R is superlinear at zero and sublinear at infinity, e.g., f ( s ) = min ( | s | r , | s | p ) with 0 r 1 p . First, for small values of λ > 0 , we prove a non-existence result for ( S M λ ) , while for λ > 0 large enough, a recent Ricceri-type result guarantees the existence of at least two non-trivial solutions for ( S M λ ) as well as the ‘stability’ of system ( S M λ ) with respect to an arbitrary subcritical perturbation of the Schrodinger equation.

We consider the discrete boundary value problem (P): where the nonlinear term has an oscillatory behaviour near the origin or at infinity. By a direct variational method, we show that (P) has a sequence of non-negative, distinct solutions which converges to 0 (respectively ) in the sup-norm whenever f oscillates at the origin (respectively at infinity).

We study an eigenvalue problem in the framework of difference equations. We show that there exist two positive constants λ0and λ1verifying λ0≤ λ1such that any λ ∈ (0, λ0) is not an eigenvalue of the problem, while any λ ∈ [λ1, ∞) is an eigenvalue of the problem. Some estimates for λ0and λ1are also given.

We study a semilinear elliptic problem on a compact Riemannian manifold with boundary, subject to an inhomogeneous Neumann boundary condition. Under various hypotheses on the nonlinear terms, depending on their behaviour in the origin and infinity, we prove multiplicity of solutions by using variational arguments.

We study the multiplicity of nonnegative solutions to the problem, (Pλ) − Δ u = λ a ( x ) u p + f ( u ) in Ω , u = 0 on ∂ Ω , where Ω is a smooth bounded domain in R N , f : [ 0 , ∞ ) → R oscillates near the origin or at infinity, and p > 0 , λ ∈ R . While oscillatory right-hand sides usually produce infinitely many distinct solutions, an additional term involving u p may alter the situation radically. Via a direct variational argument we fully describe this phenomenon, showing that the number of distinct non-trivial solutions to problem ( P λ ) is strongly influenced by u p and depends on λ whenever one of the following two cases holds: • p ⩽ 1 and f oscillates near the origin; • p ⩾ 1 and f oscillates at infinity (p may be critical or even supercritical). The coefficient a ∈ L ∞ ( Ω ) is allowed to change its sign, while its size is relevant only for the threshold value p = 1 when the behaviour of f ( s ) / s plays a crucial role in both cases. Various H 0 1 - and L ∞ -norm estimates of solutions are also given.

We guarantee the existence of infinitely many different pairs of solutions to the system { − Δ u = v p in Ω ; − Δ v = f ( u ) in Ω ; u = v = 0 on ∂ Ω , where 0 p 2 N − 2 , Ω is a bounded domain in R N and the continuous nonlinear term f has an unusual oscillatory behavior. The sequence of solutions tends to zero (resp., infinity) with respect to certain norms and the nonlinear term f may enjoy an arbitrary growth at infinity (resp., at zero) whenever f oscillates near zero (resp., at infinity). Our results provide the first applications of Ricceri's variational principle in the theory of coupled elliptic systems.

We guarantee the existence of infinitely many distinct non-negative solutions of the elliptic problem { − div ( | x | − 2 a ∇ u ) = | x | − 2 b f ( u ) in Ω , u = 0 on ∂ Ω , where Ω is a bounded open domain in R n ( n ≥ 3 ) containing the origin, a , b are positive numbers of a certain range, and f is a continuous function oscillating either at the origin or at infinity. The sequence of solutions in L ∞ -norm tends to 0 (resp., to + ∞ ) whenever f oscillates at the origin (resp., at infinity).

Multiple solutions are obtained for a variational inequality defined on the half line ( 0 , ∞ ) . Our approach is based on a key embedding result as well as on the non-smooth critical point theory for Szulkin-type functionals.

In this paper we study a non-homogeneous Neumann-type problem which involves a nonlinearity satisfying a non-standard growth condition. By using a recent variational principle of Ricceri, we establish the existence of at least two non-trivial solutions in an appropriate Orlicz–Sobolev space.

We extend a recent result of Ricceri concerning the existence of three critical points of certain non-smooth functionals. Two applications are given, both in the theory of differential inclusions; the first one concerns a non-homogeneous Neumann boundary value problem, the second one treats a quasilinear elliptic inclusion problem in the whole $${\mathbb R^N}$$ .

We consider semilinear elliptic problems in which the right-hand-side nonlinearity depends on a parameter λ > 0. Two multiplicity results are presented, guaranteeing the existence of at least three non-trivial solutions for this kind of problem, when the parameter λ belongs to an interval (0,λ*). Our approach is based on variational techniques, truncation methods and critical groups. The first result incorporates as a special case problems with concave–convex nonlinearities, while the second one involves concave nonlinearities perturbed by an asymptotically linear nonlinearity at infinity.

Let (M;g) be a compact Riemannian manifold without boundary, with dimM 3; and f : R! R a continuous function which is sublinear at innity. By various variational approaches, existence of multiple solutions of the eigenvalue problem g! + ( )! = ~ K(; )f(!); 2 M; !2 H 2 1 (M); is established for certain eigenvalues > 0, depending on further properties of f and on explicit forms of the function ~ K: Here, g stands for the Laplace{Beltrami operator on (M;g); and; ~ K are smooth positive functions. These multiplicity results are then applied to solve Emden{Fowler equations which involve sublinear terms at innity.

We consider nonlinear elliptic problems driven by the $p$-Laplacian with a nonsmooth potential depending on a parameter $\lambda\ >\ 0$. The main result guarantees the existence of two positive, two negative and a nodal (sign-changing) solution for the studied problem whenever $\lambda\ >\ 0$ belongs to a small interval (0, λ*) and $p$ ≥ 2. We do not impose any symmetry hypothesis on the nonlinear potential. The constant-sign solutions are obtained by using variational techniques based on nonsmooth critical point theory (minimization argument, Mountain Pass theorem, and a Brezis-Nirenberg type result for C1-minimizers), while the nodal solution is constructed by an upper-lower solutions argument combined with the Zorn lemma and a nonsmooth second deformation theorem.

Given f, g:[0, ∞) → ℝ two continuous nonlinearities with f(0) = g(0) = 0 and f having a suitable oscillatory behavior at zero or at infinity, we prove by a direct method that for every k ∊ ℕ, there exists e k > 0 such that the problem has at least k distinct nonnegative weak solutions in W 1, p (Ω) whenever |e|≤e k . We also give various W 1, p - and L ∞-estimates of the solutions. No growth assumption on g is needed, and α ∊ L ∞(Ω) may be sign-changing or even negative depending on the rate of the oscillation of f.

In this paper we establish the existence of three weak solutions of an equation which involves a general elliptic operator in divergence form (in particular, a p -Laplacian operator), while the nonlinearity has a ( p − 1 ) -sublinear growth at infinity. This result completes some recent papers, where mountain pass type solutions were obtained providing the nonlinear term via a ( p − 1 ) -superlinear growth at infinity (fulfilling an Ambrosetti–Rabinowitz type condition). In our case, an abstract critical point result is applied, proved by G. Bonanno [G. Bonanno, Some remarks on a three critical points theorem, Nonlinear Analysis 54 (2003) 651–665].

This paper deals with the problem − Δ p u + α(x)|u| p−2 u = β(x)f(|u|) in Ω, subjected to the zero Neumann boundary condition, where p > 1, Ω ⊂ ℝ N is bounded with smooth boundary, α, β ∊ L ∞(Ω), essinfΩβ > 0, and f:[0,+ ∞) → ℝ is a not necessarily continuous nonlinearity that oscillates either at the origin or at the infinity. By using nonsmooth variational methods, we establish in both cases the existence of infinitely many distinct non-negative solutions of the Neumann problem. In our framework, α:Ω → ℝ may be a sign-changing or even a nonpositive potential, which is not permitted usually in earlier works.

In this paper we study an elliptic problem in R N which involves the p-Laplacian, p>N 2, and the nonlinear term has an oscillatory behavior. By means of a direct variational approach, we establish the existence of infinitely many homoclinic solutions whose W 1,p (R N )-norms tend to zero (to infinity, respectively) whenever the nonlinearity oscillates at zero (at infinity, respectively). The solutions have invariance properties with respect to certain subgroups of the orthogonal group O(N) .

For certain positive numbers μ \mu and λ , \lambda , we establish the multiplicity of solutions to the problem \[ { − △ u = μ u | x | 2 + λ f ( u ) a m p ; a.e.\ in Ω , u = 0 a m p ; on ∂ Ω , \left \{ \begin {array}{lll} -\triangle u=\mu \frac {u}{|x|^2}+\lambda f(u)& \textrm {a.e.\ in} \ \Omega , u=0 & \textrm {on}\ \partial \Omega , \end {array}\right . \] where Ω \Omega is a bounded open domain in R N ( N ≥ 3 ) \mathbb {R}^N\ (N\geq 3) containing the origin with smooth boundary ∂ Ω , \partial \Omega , while f : R → R f:\mathbb {R}\to \mathbb {R} is continuous, superlinear at zero and sublinear at infinity.

In the present paper we prove a multiplicity result for an anisotropic sublinear elliptic problem with Dirichlet boundary condition, depending on a positive parameter λ. By variational arguments, we prove that for enough large values of λ, our anisotropic problem has at least two non-zero distinct solutions. In particular, we show that at least one of the solutions provides a Wulff-type symmetry.

We consider the following variational inclusions system of the form − △ u + u ∈ ∂ 1 F ( u , v ) in R N , − △ v + v ∈ ∂ 2 F ( u , v ) in R N , with u , v ∈ H 1 ( R N ) , where F : R 2 → R is a locally Lipschitz function and ∂ i F ( u , v ) ( i ∈ { 1 , 2 } ) are the partial generalized gradients in the sense of Clarke. Under various growth conditions on the nonlinearity F we study the existence of nonzero weak solutions of the above system (in the sense of hemivariational inequalities), which are critical points of an appropriate locally Lipschitz function defined on H 1 ( R N ) × H 1 ( R N ) . The main tool used in the paper is the principle of symmetric criticality for locally Lipschitz functions.

We show the equivalence of the non-positivity of the flag curvature with the non-positive curvature properties of Busemann and Pedersen for (not necessarily reversible) Berwald manifolds. So an analytical property is characterized by synthetic concepts of non-positively curved metric spaces.

In this paper we consider the differential inclusion problem (DI) - ▵ p u + | u | p - 2 u ∈ α ( x ) ∂ F ( u ( x ) ) , x ∈ R N , u ∈ W 1 , p ( R N ) , where 2 ⩽ N p + ∞ , α ∈ L 1 ( R N ) ∩ L ∞ ( R N ) is radially symmetric, and ∂ F stands for the generalized gradient of a locally Lipschitz function F : R → R . Under suitable oscillatory assumptions on the potential F at zero or at infinity, we show the existence of infinitely many, radially symmetric solutions of (DI). No symmetry requirement on F is needed. Our approach is based on a non-smooth Ricceri-type variational principle, developed by Marano and Motreanu (J. Differential Equations 182 (2002) 108–120).

In this paper a class ofeigenvalue problems f hemivariational inequalities is studied which is defined on domains ofthe type × R ( is a bounded open subset of R m , m 1) and it involves

In this paper we study the multiplicity of solutions of the quasilinear elliptic system\begin{equation} \left. \begin{aligned} -\Delta_pu\amp=\lambda F_u(x,u,v)\amp\amp\text{in }\varOmega, \\ -\Delta_qv\amp=\lambda F_v(x,u,v)\amp\amp\text{in }\varOmega, \\ u=v\amp=0\amp\amp\text{on }\partial\varOmega, \end{aligned} \right\} \end{equation} \tag{S$_\lambda$}where $\varOmega$ is a strip-like domain and $\lambda>0$ is a parameter. Under some growth conditions on $F$, we guarantee the existence of an open interval $\varLambda\subset(0,\infty)$ such that for every $\lambda\in\varLambda$, the system (S$_\lambda$) has at least two distinct, non-trivial solutions. The proof is based on an abstract critical-point result of Ricceri and on the principle of symmetric criticality.

In this paper we study the existence of nontrivial solutions for a class of gradient-type systems on strip-like domains where the nonlinear term is not necessarily continuously differentiable. The proof of the main result is based on a nonsmooth version of the Mountain Pass Theorem which involves the Cerami compactness condition and on the Principle of Symmetric Criticality for locally Lipschitz functions.

Various results based on some convexity assumptions (involving the exponential map along with affine maps, geodesics and convex hulls) have been recently established on Hadamard manifolds. In this paper we prove that these conditions are mutually equivalent and they hold if and only if the Hadamard manifold is isometric to the Euclidean space. In this way, we show that some results in the literature obtained on Hadamard manifolds are actually nothing but their well known Euclidean counterparts., Comment: 6 pages, 1 figure

This paper deals with a sublinear differential inclusion problem (Pλ) depending on a parameter λ > 0 which is defined on a strip-like domain subject to the zero Dirichlet boundary condition. By variational methods, we prove that for large values of λ, problem (Pλ) has at least two non-zero axially symmetric weak solutions.

In this paper we study the equation $-u''+V(x)u=W(x)f(u),\ x\in\mathbb{R},$ where the nonlinear term $f$ has certain oscillatory behaviour. Via two different variational arguments, we show the existence of infinitely many homoclinic solutions whose norms in an appropriate functional space which involves the potential $V$ tend to zero (resp. at infinity) whenever $f$ oscillates at zero (resp. at infinity). Unlike in classical results, neither symmetry property on $f$ nor periodicity on the potentials $V$ and $W$ are required.

We propose a direct approach for detecting arbitrarily many solutions for perturbed elliptic problems involving oscillatory terms. Although the method works in various frameworks, we illustrate it on the problem(Pε){−Δu+u=Q(x)[f(u)+εg(u)],x∈RN,N⩾2,u⩾0,u(x)→0as |x|→∞, where Q:RN→R is a radial, positive potential, f:[0,∞)→R is a continuous nonlinearity which oscillates near the origin or at infinity and g:[0,∞)→R is any arbitrarily continuous function with g(0)=0. Our aim is to prove that: (a) the unperturbed problem (P0), i.e. ε=0 in (Pε), has infinitely many distinct solutions; (b) the number of distinct solutions for (Pε) becomes greater and greater whenever |ε| is smaller and smaller. In fact, our method surprisingly shows that (a) and (b) are equivalent in the sense that they are deducible from each other. Various properties of the solutions are also described in L∞- and H1-norms. Our method is variational and a specific construction enforces the use of the principle of symmetric criticality for non-smooth Szulkin-type functionals.

In this paper we prove the principle of symmetric criticality for Motreanu–Panagiotopoulos type functionals, i.e., for convex, proper, lower semicontinuous functionals which are perturbed by a locally Lipschitz function. By means of this principle a variational–hemivariational inequality is studied on certain type of unbounded strips.

Some multiplicity results are presented for the eigenvalue problem(Pλ,μ){−div(|x|−2a∇u)=λ|x|−2bf(u)+μ|x|−2cg(u)in Ω,u=0on ∂Ω, where Ω⊂Rn (n⩾3) is an open bounded domain with smooth boundary, 0∈Ω, 0