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In this work we address the question of the existence of nonradial domains inside a nonconvex cone for which a mixed boundary overdetermined problem admits a solution. Our approach is variational, and consists in proving the existence of nonradial minimizers, under a volume constraint, of the associated torsional energy functional. In particular we give a condition on the domain D on the sphere spanning the cone which ensures that the spherical sector is not a minimizer. Similar results are obtained for the relative isoperimetric problem in nonconvex cones.

We study a new family of sign-changing solutions to the stationary nonlinear Schrödinger equation \[ -\Delta v +q v =|v|^{p-2} v, \qquad \text{in}\,{ {\mathbb{R}^{3}},} \] with $2 < p < \infty$ and $q \ge 0$. These solutions are spiraling in the sense that they are not axially symmetric but invariant under screw motion, i.e., they share the symmetry properties of a helicoid. In addition to existence results, we provide information on the shape of spiraling solutions, which depends on the parameter value representing the rotational slope of the underlying screw motion. Our results complement a related analysis of Del Pino, Musso and Pacard in their study (2012, Manuscripta Math., 138, 273–286) for the Allen–Cahn equation, whereas the nature of results and the underlying variational structure are completely different.

We establish weighted$$L^p$$Lp-Fourier extension estimates for$$O(N-k) \times O(k)$$O(N-k)×O(k)-invariant functions defined on the unit sphere$${\mathbb {S}}^{N-1}$$SN-1, allowing for exponentspbelow the Stein–Tomas critical exponent$$\frac{2(N+1)}{N-1}$$2(N+1)N-1. Moreover, in the more general setting of an arbitrary closed subgroup$$G \subset O(N)$$G⊂O(N)andG-invariant functions, we study the implications of weighted Fourier extension estimates with regard to boundedness and nonvanishing properties of the corresponding weighted Helmholtz resolvent operator. Finally, we use these properties to derive new existence results forG-invariant solutions to the nonlinear Helmholtz equation$$\begin{aligned} -\Delta u - u = Q(x)|u|^{p-2}u, \quad u \in W^{2,p}({\mathbb {R}}^{N}), \end{aligned}$$-Δu-u=Q(x)|u|p-2u,u∈W2,p(RN),whereQis a nonnegative bounded andG-invariant weight function.

The present paper is concerned with the half-space Dirichlet problem "Equation missing"where $$\mathbb {R}^{N}_{+}:= \{\,x \in \mathbb {R}^N: x_N > 0\, \}$$ R + N : = { x ∈ R N : x N > 0 } for some $$N \ge 1$$ N ≥ 1 and $$p > 1$$ p > 1 , $$c > 0$$ c > 0 are constants. We analyse the existence, non-existence and multiplicity of bounded positive solutions to ($$P_c$$ P c ). We prove that the existence and multiplicity of bounded positive solutions to ($$P_c$$ P c ) depend in a striking way on the value of $$c > 0$$ c > 0 and also on the dimension N. We find an explicit number $${c_p}\in (1,\sqrt{e})$$ c p ∈ ( 1 , e ) , depending only on p, which determines the threshold between existence and non-existence. In particular, in dimensions $$N \ge 2$$ N ≥ 2 , we prove that, for $$0< c < {c_p}$$ 0 < c < c p , problem ($$P_c$$ P c ) admits infinitely many bounded positive solutions, whereas, for $$c > {c_p}$$ c > c p , there are no bounded positive solutions to ($$P_c$$ P c ).

In this note, we study the asymptotic behavior of eigenvalues and eigenfunctions of the regional fractional Laplacian $(-{\Delta })^{s}_{\Omega }$ ( − Δ ) Ω s as $s\rightarrow 0^{+}.$ s → 0 + . Our analysis leads to a study of the regional logarithmic Laplacian, which arises as a formal derivative of regional fractional Laplacians at s = 0.

We derive a shape derivative formula for the family of principal Dirichlet eigenvalues $$\lambda _s(\Omega )$$ λ s ( Ω ) of the fractional Laplacian $$(-\Delta )^s$$ ( - Δ ) s associated with bounded open sets $$\Omega \subset \mathbb {R}^N$$ Ω ⊂ R N of class $$C^{1,1}$$ C 1 , 1 . This extends, with a help of a new approach, a result in Dalibard and Gérard-Varet (Calc. Var. 19(4):976–1013, 2013) which was restricted to the case $$s=\frac{1}{2}$$ s = 1 2 . As an application, we consider the maximization problem for $$\lambda _s(\Omega )$$ λ s ( Ω ) among annular-shaped domains of fixed volume of the type $$B\setminus \overline{B}'$$ B \ B ¯ ′ , where B is a fixed ball and $$B'$$ B ′ is ball whose position is varied within B. We prove that $$\lambda _s(B\setminus \overline{B}')$$ λ s ( B \ B ¯ ′ ) is maximal when the two balls are concentric. Our approach also allows to derive similar results for the fractional torsional rigidity. More generally, we will characterize one-sided shape derivatives for best constants of a family of subcritical fractional Sobolev embeddings.

The purpose of this paper is to study and classify singular solutions of the Poisson problem { L μ s u = f in Ω ∖ { 0 } , u = 0 in R N ∖ Ω \begin{equation*} \left \{ \begin {aligned} \mathcal {L}^s_\mu u = f \quad \ \text {in}\ \, \Omega \setminus \{0\},\\ u =0 \quad \ \text {in}\ \, \mathbb {R}^N \setminus \Omega \ \end{aligned} \right . \end{equation*} for the fractional Hardy operator L μ s u = ( − Δ ) s u + μ | x | 2 s u \mathcal {L}_\mu ^s u= (-\Delta )^s u +\frac {\mu }{|x|^{2s}}u in a bounded domain Ω ⊂ R N \Omega \subset \mathbb {R}^N ( N ≥ 2 N \ge 2 ) containing the origin. Here ( − Δ ) s (-\Delta )^s , s ∈ ( 0 , 1 ) s\in (0,1) , is the fractional Laplacian of order 2 s 2s , and μ ≥ μ 0 \mu \ge \mu _0 , where μ 0 = − 2 2 s Γ 2 ( N + 2 s 4 ) Γ 2 ( N − 2 s 4 ) > 0 \mu _0 = -2^{2s}\frac {\Gamma ^2(\frac {N+2s}4)}{\Gamma ^2(\frac {N-2s}{4})}>0 is the best constant in the fractional Hardy inequality. The analysis requires a thorough study of fundamental solutions and associated distributional identities. Special attention will be given to the critical case μ = μ 0 \mu = \mu _0 which requires more subtle estimates than the case μ > μ 0 \mu >\mu _0 .

We study the asymptotic (in time) behavior of positive and sign-changing solutions to nonlinear parabolic problems in the whole space or in the exterior of a ball with Dirichlet boundary conditions. We show that, under suitable regularity and stability assumptions, solutions are asymptotically (in time) foliated Schwarz symmetric, i.e., all elements in the associated omega-limit set are axially symmetric with respect to a common axis passing through the origin and are nonincreasing in the polar angle. We also obtain symmetry results for solutions of H\'enon-type problems, for equilibria (i.e. for solutions of the corresponding elliptic problem), and for time periodic solutions., Comment: 30 pages, 5 figures

Fluid flows around a symmetric obstacle generate vortices which may lead to symmetry breaking of the streamlines. We study this phenomenon for planar viscous flows governed by the stationary Navier-Stokes equations with constant inhomogeneous Dirichlet boundary data in a rectangular channel containing a circular obstacle. In such (symmetric) framework, symmetry breaking is strictly related to the appearance of multiple solutions. Symmetry breaking properties of some Sobolev minimizers are studied and explicit bounds on the boundary velocity (in terms of the length and height of the channel) ensuring uniqueness are obtained after estimating some Sobolev embedding constants and constructing a suitable solenoidal extension of the boundary data. We show that, regardless of the solenoidal extension employed, such bounds converge to zero at an optimal rate as the length of the channel tends to infinity.

We consider nonlinear second order elliptic problems of the type \[ -\Delta u=f(u) \text{ in } \Omega, \qquad u=0 \text{ on } \partial \Omega, \] where $\Omega$ is an open $C^{1,1}$-domain in $\mathbb{R}^N$, $N\geq 2$, under some general assumptions on the nonlinearity that include the case of a sublinear pure power $f(s)=|s|^{p-1}s$ with $01$ and $\lambda>\lambda_2(\Omega)$ (the second Dirichlet eigenvalue of the Laplacian). We prove the existence of a least energy nodal (i.e. sign changing) solution, and of a nodal solution of mountain-pass type. We then give explicit examples of domains where the associated levels do not coincide. For the case where $\Omega$ is a ball or annulus and $f$ is of class $C^1$, we prove instead that the levels coincide, and that least energy nodal solutions are nonradial but axially symmetric functions. Finally, we provide stronger results for the Allen-Cahn type nonlinearities in case $\Omega$ is either a ball or a square. In particular we give a complete description of the solution set for $\lambda\sim \lambda_2(\Omega)$, computing the Morse index of the solutions., Comment: 26 pages

In this work, we provide an estimate of the Morse index of radially symmetric sign changing bounded weak solutions $u$ to the semilinear fractional Dirichlet problem $$ (-\Delta)^su = f(u)\qquad \text{ in $\mathcal{B}$},\qquad \qquad u = 0\qquad \text{in $\quad\mathbb{R}^{N}\setminus \mathcal{B}$,} $$ where $s\in(0,1)$, $\mathcal{B}\subset \mathbb{R}^N$ is the unit ball centred at zero and the nonlinearity $f$ is of class $C^1$. We prove that for $s\in(1/2,1)$ any radially symmetric sign changing solution of the above problem has a Morse index greater than or equal to $N+1$. If $s\in (0,1/2],$ the same conclusion holds under additional assumption on $f$. In particular, our results apply to the Dirichlet eigenvalue problem for the operator $(-\Delta)^s$ in $\mathcal{B}$ for all $s\in (0,1)$, and it implies that eigenfunctions corresponding to the second Dirichlet eigenvalue in $\mathcal{B}$ are antisymmetric. This resolves a conjecture of Ba\~{n}uelos and Kulczycki., Comment: 18 pages

We obtain spectral inequalities and asymptotic formulae for the discrete spectrum of the operator $\frac12\, \log(-\Delta)$ in an open set $\Omega\in\Bbb R^d$, $d\ge2$, of finite measure with Dirichlet boundary conditions. We also derive some results regarding lower bounds for the eigenvalue $\lambda_1(\Omega)$ and compare them with previously known inequalities., Comment: 21 pages, 1 figure

In this paper, we are concerned with the Schrodinger–Poisson system {−Δu+u+u=|u|p−2uin Rd,Δ=u2in Rd. Due to its relevance in physics, the system has been extensively studied and is quite well understood in the case . In contrast, much less information is available in the planar case which is the focus of the present paper. It has been observed by Cingolani S and Weth T (2016 On the planar Schrodinger–Poisson system Ann. Inst. Henri Poincare 33 169–97) that the variational structure of (0.1) differs substantially in the case and leads to a richer structure of the set of solutions. However, the variational approach of Cingolani S and Weth T (2016 On the planar Schrodinger–Poisson system Ann. Inst. Henri Poincare 33 169–97) is restricted to the case which excludes some physically relevant exponents. In the present paper, we remove this unpleasant restriction and explore the more complicated underlying functional geometry in the case with a different variational approach.

We study the problem of maximizing the first nontrivial Steklov eigenvalue of the Laplace-Beltrami Operator among subdomains of fixed volume of a Riemannian manifold. More precisely, we study the expansion of the corresponding profile of this isoperimetric (or isochoric) problem as the volume tends to zero. The main difficulty encountered in our study is the lack of existence results for maximizing domains and the possible degeneracy of the first nontrivial Steklov eigenvalue, which makes it difficult to tackle the problem with domain variation techniques. As a corollary of our results, we deduce local comparison principles for the profile in terms of the scalar curvature on $\mathcal{M}$. In the case where the underlying manifold is a closed surface, we obtain a global expansion and thus a global comparison principle., 20 pages

We consider the Dirichlet problem for the Henon–Schrodinger system $$\begin{aligned} -\Delta u + \kappa _1 u = |x|^{\alpha }\partial _u F(u,v), \qquad -\Delta v + \kappa _2 v = |x|^{\alpha }\partial _v F(u,v) \end{aligned}$$ in the unit ball $$\Omega \subset \mathbb {R}^N, N\ge 2$$ , where $$\alpha \ge 0$$ is a parameter and $$F: \mathbb {R}^2 \rightarrow \mathbb {R}$$ is a p-homogeneous $$C^2$$ -function for some $$p>2$$ with $$F(u,v)>0$$ for $$(u,v) \not = (0,0)$$ . We show that, as $$\alpha \rightarrow \infty $$ , the Morse index of nontrivial radial solutions of this problem (positive or sign-changing) tends to infinity. This result is new even for the corresponding scalar Henon equation and extends a previous result by Moreira dos Santos and Pacella [19] for the case $$N=2$$ . In particular, the result implies symmetry breaking for ground state solutions, but also for other solutions obtained by an $$\alpha $$ -independent variational minimax principle.

For the stationary nonlinear Schr\"odinger equation $-\Delta u+ V(x)u- f(u) = \lambda u$ with periodic potential $V$ we study the existence and stability properties of multibump solutions with prescribed $L^2$-norm. To this end we introduce a new nondegeneracy condition and develop new superposition techniques which allow to match the $L^2$-constraint. In this way we obtain the existence of infinitely many geometrically distinct solutions to the stationary problem. We then calculate the Morse index of these solutions with respect to the restriction of the underlying energy functional to the associated $L^2$-sphere, and we show their orbital instability with respect to the Schr\"odinger flow. Our results apply in both, the mass-subcritical and the mass-supercritical regime.

We analyze the s-dependence of solutions u s to the family of fractional Poisson problems ( − Δ ) s u = f in Ω , u ≡ 0 on R N ∖ Ω in an open bounded set Ω ⊂ R N , s ∈ ( 0 , 1 ) . In the case where Ω is of class C 2 and f ∈ C α ( Ω ‾ ) for some α > 0 , we show that the map ( 0 , 1 ) → L ∞ ( Ω ) , s ↦ u s is of class C 1 , and we characterize the derivative ∂ s u s in terms of the logarithmic Laplacian of f. As a corollary, we derive pointwise monotonicity properties of the solution map s ↦ u s under suitable assumptions on f and Ω. Moreover, we derive explicit bounds for the corresponding Green operator on arbitrary bounded domains which are new even for the case s = 1 , i.e., for the local Dirichlet problem − Δ u = f in Ω, u ≡ 0 on ∂Ω.

We study a stationary scattering problem related to the nonlinear Helmholtz equation $-\Delta u - k^2 u = f(x,u) \ \ \text{in $\mathbb{R}^N$,}$ where $N \ge 3$ and $k>0$. For a given incident free wave $\varphi \in L^\infty(\mathbb{R}^N)$, we prove the existence of complex-valued solutions of the form $u=\varphi+u_{\text{sc}}$, where $u_{\text{sc}}$ satisfies the Sommerfeld outgoing radiation condition. Since neither a variational framework nor maximum principles are available for this problem, we use topological fixed point theory and global bifurcation theory to solve an associated integral equation involving the Helmholtz resolvent operator. The key step of this approach is the proof of suitable a priori bounds.

We consider the nonlinear stationary Schrodinger equation $$\begin{aligned} -\Delta u -\lambda u= Q(x)|u|^{p-2}u, \qquad \text {in }\mathbb {R}^N \end{aligned}$$ in the case where $$N \ge 3$$ , p is a superlinear, subcritical exponent, Q is a bounded, nonnegative and nontrivial weight function with compact support in $$\mathbb {R}^N$$ and $$\lambda \in \mathbb {R}$$ is a parameter. Under further restrictions either on the exponent p or on the shape of Q, we establish the existence of a continuous branch $$\mathcal {C}$$ of nontrivial solutions to this equation which intersects $$\{\lambda \} \times L^{s}(\mathbb {R}^N)$$ for every $$\lambda \in (-\infty , \lambda _Q)$$ and $$s> \frac{2N}{N-1}$$ . Here, $$\lambda _Q>0$$ is an explicit positive constant which only depends on N and $$\text {diam}(\text {supp }Q)$$ . In particular, the set of values $$\lambda $$ along the branch enters the essential spectrum of the operator $$-\Delta $$ .

We study the existence of nontrivial unbounded domains \({\Omega}\) in \({{\mathbb R}^{N}}\) such that the overdetermined problem $${-\Delta u = 1 \quad {\rm in} \, \Omega}, \quad u = 0, \quad \partial_{\nu} u = {\rm const} \quad {\rm on} \partial \Omega$$ admits a solution u. By this, we complement Serrin’s classification result from 1971, which yields that every bounded domain admitting a solution of the above problem is a ball in \({{\mathbb R}^{N}}\). The domains we construct are periodic in some variables and radial in the other variables, and they bifurcate from a straight (generalized) cylinder or slab. We also show that these domains are uniquely self Cheeger relative to a period cell for the problem.

We are concerned with hypersurfaces of ℝ N {\mathbb{R}^{N}} with constant nonlocal (or fractional) mean curvature. This is the equation associated to critical points of the fractional perimeter under a volume constraint. Our results are twofold. First we prove the nonlocal analogue of the Alexandrov result characterizing spheres as the only closed embedded hypersurfaces in ℝ N {\mathbb{R}^{N}} with constant mean curvature. Here we use the moving planes method. Our second result establishes the existence of periodic bands or “cylinders” in ℝ 2 {\mathbb{R}^{2}} with constant nonlocal mean curvature and bifurcating from a straight band. These are Delaunay-type bands in the nonlocal setting. Here we use a Lyapunov–Schmidt procedure for a quasilinear type fractional elliptic equation.

The present paper is devoted to geometric optimization problems related to the Neumann eigenvalue problem for the Laplace-Beltrami operator on bounded subdomains $\Omega$ of a Riemannian manifold $(\mathcal{M},g)$. More precisely, we analyze locally extremal domains for the first nontrivial eigenvalue $\mu_2(\Omega)$ with respect to volume preserving domain perturbations, and we show that corresponding notions of criticality arise in the form of overdetermined boundary problems. Our results rely on an extension of Zanger's shape derivative formula which covers the case when $\mu_2(\Omega)$ is not a simple eigenvalue. In the second part of the paper, we focus on product manifolds of the form $\mathcal{M} = \mathbb{R}^k \times \mathcal{N}$, and we classify the subdomains where an associated overdetermined boundary value problem has a solution., Comment: 26 pages

We set up a dual variational framework to detect real standing wave solutions of the nonlinear Helmholtz equation $$ -\Delta u-k^2 u =Q(x)|u|^{p-2}u,\qquad u \in W^{2,p}(\mathbb{R}^N) $$ with $N\geq 3$, $\frac{2(N+1)}{(N-1)}< p

We study Serrin’s overdetermined boundary value problem $$\begin{aligned} -\Delta _{S^N}\, u=1 \quad \text { in }\Omega ,\quad u=0, \; \partial _\eta u={\text {const}} \quad \text {on }\partial \Omega \end{aligned}$$ in subdomains $$\Omega $$ of the round unit sphere $$S^N \subset \mathbb {R}^{N+1}$$ , where $$\Delta _{S^N}$$ denotes the Laplace–Beltrami operator on $$S^N$$ . A subdomain $$\Omega $$ of $$S^N$$ is called a Serrin domain if it admits a solution of this overdetermined problem. In our main result, we construct Serrin domains in $$S^N$$ , $$N \ge 2$$ which bifurcate from symmetric straight tubular neighborhoods of the equator. Our result provides the first example of Serrin domains in $$S^{N}$$ which are not bounded by geodesic spheres.

We derive the unique continuation property of a class of semi-linear elliptic equations with non-Lipschitz nonlinearities. The simplest type of equations to which our results apply is given as $-\Delta u = |u|^{\sigma-1} u$ in a domain $\Omega \subset \mathbb{R}^N$, with $0 \le \sigma

We are concerned with unbounded sets of $\mathbb{R}^N$ whose boundary has constant nonlocal (or fractional) mean curvature, which we call CNMC sets. This is the equation associated to critical points of the fractional perimeter functional under a volume constraint. We construct CNMC sets which are the countable union of a certain bounded domain and all its translations through a periodic integer lattice of dimension $M\leq N$. Our CNMC sets form a $C^2$ branch emanating from the unit ball alone and where the parameter in the branch is essentially the distance to the closest lattice point. Thus, the new translated near-balls (or near-spheres) appear from infinity. We find their exact asymptotic shape as the parameter tends to infinity., 44 pages. Only the funding acknowledgements changed

In this paper, we study the logarithmic Laplacian operator $L_\Delta$, which is a singular integral operator with symbol $2\log |\zeta|$. We show that this operator has the integral representation $$L_\Delta u(x) = c_{N} \int_{\mathbb{R}^N } \frac{ u(x)1_{B_1(x)}(y)-u(y)}{|x-y|^{N} } dy + \rho_N u(x) $$ with $c_N = \pi^{- \frac{N}{2}} \Gamma(\frac{N}{2})$ and $\rho_N=2 \log 2 + \psi(\frac{N}{2}) -\gamma$, where $\Gamma$ is the Gamma function, $\psi = \frac{\Gamma'}{\Gamma}$ is the Digamma function and $\gamma= -\Gamma'(1)$ is the Euler Mascheroni constant. This operator arises as formal derivative $\partial_s \Big|_{s=0} (-\Delta)^s$ of fractional Laplacians at $s= 0$. We develop the functional analytic framework for Dirichlet problems involving the logarithmic Laplacian on bounded domains and use it to characterize the asymptotics of principal Dirichlet eigenvalues and eigenfunctions of $(-\Delta)^s$ as $s \to 0$. As a byproduct, we then derive a Faber-Krahn type inequality for the principal Dirichlet eigenvalue of $L_\Delta$. Using this inequality, we also establish conditions on domains giving rise to the maximum principle in weak and strong forms. This allows us to also derive regularity up to the boundary of solutions to corresponding Poisson problems., Comment: 34 pages, accepted by Comm. Part. Diff. Eq

In this paper we derive a strong maximum principle for weak supersolutions of nonlocal equations of the form $Iu=c(x) u$ in $\Omega$, where $\Omega\subset \mathbb{R}^N$ is a domain, $c\in L^{\infty}(\Omega)$ and $I$ is an operator of the form $Iu(x)=P.V.\int_{\mathbb{R}^N}(u(x)-u(y))j(x-y)\ dy$ with a nonnegative kernel function $j$. We formulate minimal positivity assumptions on $j$ corresponding to a class of operators which includes highly anisotropic variants of the fractional Laplacian. Somewhat surprisingly, this problem leads to the study of general lattices in $\mathbb{R}^N$. Our results extend to the regional variant of the operator $I$ and, under weak additional assumptions, also to the case of $x$-dependent kernel functions., Comment: To appear in Math. Z

We consider the nonlinear problem $$\begin{aligned} (P)\qquad \left\{ \begin{array}{ll} I u=f(x,u)&{} \quad \text { in } \ \Omega ,\\ u=0 &{}\quad \text { on } \ \mathbb {R}^{N}{\setminus }\Omega \\ \end{array}\right. \end{aligned}$$ in an open bounded set \(\Omega \subset \mathbb {R}^{N}\), where \(I\) is a nonlocal operator, which may be anisotropic and may have varying order. We assume mild symmetry and monotonicity assumptions on \(I, \Omega \) and the nonlinearity \(f\) with respect to a fixed direction, say \(x_1\), and we show that any nonnegative weak solution \(u\) of \((P)\) is symmetric in \(x_1\). Moreover, we have the following alternative: Either \(u\equiv 0\) in \(\Omega \), or \(u\) is strictly decreasing in \(|x_1|\). The proof relies on new maximum principles for antisymmetric supersolutions of an associated class of linear problems.

We study the local Szego–Weinberger profile in a geodesic ball \(B_g(y_0,r_0)\) centered at a point \(y_0\) in a Riemannian manifold \(({\mathcal {M}},g)\). This profile is obtained by maximizing the first nontrivial Neumann eigenvalue \(\mu _2\) of the Laplace–Beltrami Operator \(\Delta _g\) on \({\mathcal {M}}\) among subdomains of \(B_g(y_0,r_0)\) with fixed volume. We derive a sharp asymptotic bounds of this profile in terms of the scalar curvature of \({\mathcal {M}}\) at \(y_0\). As a corollary, we deduce a local comparison principle depending only on the scalar curvature. Our study is related to previous results on the profile corresponding to the minimization of the first Dirichlet eigenvalue of \(\Delta _g\), but additional difficulties arise due to the fact that \(\mu _2\) is degenerate in the unit ball in \(\mathbb {R}^N\) and geodesic balls do not yield the optimal lower bound in the asymptotics we obtain.

In this paper, we study real solutions of the nonlinear Helmholtz equation $$ - \Delta u - k^2 u = f(x,u),\qquad x\in \R^N $$ satisfying the asymptotic conditions $$ u(x)=O(|x|^{\frac{1-N}{2}}) \quad \text{and} \quad \frac{\partial^2 u}{\partial r^2}(x)+k^2 u(x)) =o(|x|^{\frac{1-N}{2}}) \qquad \text{as $r=|x| \to \infty$.} $$ We develop the variational framework to prove the existence of nontrivial solutions for compactly supported nonlinearities without any symmetry assumptions. In addition, we consider the radial case in which, for a larger class of nonlinearities, infinitely many solutions are shown to exist. Our results give rise to the existence of standing wave solutions of corresponding nonlinear Klein-Gordon equations with arbitrarily large frequency., Comment: Corrected version. To appear in Archive for Rational Mechanics and Analysis

We show that the fractional Sobolev inequality for the embedding $\H \hookrightarrow L^{\frac{2N}{N-s}}(\R^N)$, $s \in (0,N)$ can be sharpened by adding a remainder term proportional to the distance to the set of optimizers. As a corollary, we derive the existence of a remainder term in the weak $L^{\frac{N}{N-s}}$-norm for functions supported in a domain of finite measure. Our results generalize earlier work for the non-fractional case where $s$ is an even integer., 13 pages

We develop a variational framework to detect high energy solutions of the planar Schrodinger–Poisson system { − Δ u + a ( x ) u + γ w u = 0 , Δ w = u 2 in R 2 with 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 consider a class of gradient systems of type $$ -c_i \Delta u_i + V_i(x)u_i=P_{u_i}(u),\quad u_1,..., u_k>0 \text{in}\Omega, \qquad u_1=...=u_k=0 \text{on} \partial \Omega, $$ in a bounded domain $\Omega\subseteq \R^N$. Under suitable assumptions on $V_i$ and $P$, we prove the existence of ground-state solutions for this problem. Moreover, for $k=2$, assuming that the domain $\Omega$ and the potentials $V_i$ are radially symmetric, we prove that the ground state solutions are foliated Schwarz symmetric with respect to antipodal points. We provide several examples for our abstract framework., Comment: 21 pages, 0 figures

We consider solutions of some nonlinear parabolic boundary value problems in radial bounded domains whose initial profile satisfy a reflection inequality with respect to a hyperplane containing the origin. We show that, under rather general assumptions, these solutions are asymptotically (in time) foliated Schwarz symmetric, i.e., all elements in the associated omega limit set are axially symmetric with respect to a common axis passing through the origin and nonincreasing in the polar angle from this axis. In this form, the result is new even for equilibria (i.e. solutions of the corresponding elliptic problem) and time periodic solutions., 16 pages

We consider the principal Dirichlet eigenfunction u of the Laplacian in a bounded region in \({\mathbb{R}^2}\) which is convex in one direction, say in x1. It has been asked by Kawohl (Remarks on some old and current eigenvalue problems, Cambridge University Press, pp 165–183, 1994) whether in this case u is quasiconcave in x1, i.e., all superlevel sets of u are convex in x1. In this note we provide a negative answer to this question by giving an explicit counterexample.

For higher order Hilbertian Sobolev spaces, we improve the embedding inequality for the critical L p -space by adding a remainder term with a suitable weak norm.

We discuss some recent results on symmetry of solutions of nonlinear partial differential equations. We focus on elliptic and degenerate elliptic boundary value problems of second order with variational structure and the simple looking case where the underlying domain is radially symmetric. In this setting, we study solutions which are given as minimizers of constrained minimization problems or have low Morse index, and we examine which amount of symmetry of the data is inherited by these solutions. We highlight how the answer to this general question depends on specific assumptions on the data. The underlying techniques collected in this survey are elementary as they solely rely on hyperplane reflections and well known analytical and topological tools, but they yield surprisingly general results in situations where classical methods do not apply.

We study the set of solutions of the nonlinear elliptic system (P) { − Δ u + λ 1 u = μ 1 u 3 + β v 2 u in Ω , − Δ v + λ 2 v = μ 2 v 3 + β u 2 v in Ω , u , v > 0 in Ω , u = v = 0 on ∂ Ω , in a smooth bounded domain Ω ⊂ R N , N ⩽ 3 , with coupling parameter β ∈ R . This system arises in the study of Bose–Einstein double condensates. We show that the value β = − μ 1 μ 2 is critical for the existence of a priori bounds for solutions of (P) . More precisely, we show that for β > − μ 1 μ 2 , solutions of (P) are a priori bounded. In contrast, when λ 1 = λ 2 , μ 1 = μ 2 , (P) admits an unbounded sequence of solutions if β ⩽ − μ 1 μ 2 .

In this paper we prove symmetry results for classical solutions of semilinear elliptic equations in the whole R N or in the exterior of a ball, N ⩾ 2 , in the case when the nonlinearity is either convex or has a convex first derivative. More precisely we prove that solutions having Morse index j ⩽ N are foliated Schwarz symmetric, i.e. they are axially symmetric with respect to an axis passing through the origin and nonincreasing in the polar angle from this axis. From this we deduce some nonexistence results for positive or sign changing solutions in the case when the nonlinearity does not depend explicitly on the space variable.

We consider the 2 m -th order elliptic boundary value problem L u = f ( x , u ) on a bounded smooth domain Ω ⊂ R N with Dirichlet boundary conditions on ∂ Ω . The operator L is a uniformly elliptic linear operator of order 2 m whose principle part is of the form ( − ∑ i , j = 1 N a i j ( x ) ∂ 2 ∂ x i ∂ x j ) m . We assume that f is superlinear at the origin and satisfies lim s → ∞ f ( x , s ) s q = h ( x ) , lim s → − ∞ f ( x , s ) | s | q = k ( x ) , where h , k ∈ C ( Ω ¯ ) are positive functions and q > 1 is subcritical. By combining degree theory with new and recently established a priori estimates, we prove the existence of a nontrivial solution.