Multibump bound states for quasilinear Schrödinger systems with critical frequency

In this paper, we are concerned with the multibump solutions for the following quasilinear Schrödinger system in ${\mathbb{R}^N}$ : $$\left\{\begin{array}{ll}-\Delta{u} + \lambda{a(x)u} - \frac{1}{2}(\Delta|u|^2)u = \frac{2\alpha}{\alpha + \beta}|u|^{\alpha-2}|\upsilon|^\beta u, \\-\Delta{\upsilon} + \lambda{b(x)\upsilon} - \frac{1}{2}(\Delta|\upsilon|^2)\upsilon = \frac{2\beta}{\alpha + \beta}|u|^\alpha|\upsilon|^{\beta-2} \upsilon, \\u(x) \rightarrow 0, \upsilon(x) \rightarrow 0 \quad as|x| \rightarrow \infty,\end{array}\right.$$ where λ > 0 is a parameter, α, β > 2 satisfying α + β < 2 · 2*, here ${2^{*} = \frac{2N}{N-2}}$ is the critical Sobolev exponent for ${N \geq 3}$ and a(x), b(x) are nonnegative potentials. Using variational methods, we prove that if the zero sets of a(x) and b(x) have several common isolated connected components ${\Omega_{1}, . . . ,\Omega_{k}}$ such that the interior of ${\Omega_{i} (i = 1, 2, . . . , k)}$ is not empty and ${\partial\Omega_{i} (i = 1, 2, . . . , k)}$ is smooth, then for λ sufficiently large, the system admits, for any nonempty subset ${J \subset \{1, 2, . . . , k\}}$ , a solution which is trapped in a neighborhood of ${\cup_{j\epsilon{J}} \Omega_{j}}$ .     

Multiple positive solutions of singular and critical elliptic problem in {\mathbb{R}^2} with discontinuous nonlinearities

Let Ω be a bounded domain in ${\mathbb{R}^2}$ with smooth boundary. We consider the following singular and critical elliptic problem with discontinuous nonlinearity: $$(P_\lambda)\left \{\begin{array}{ll} - \Delta u = \lambda \left(\frac{m(x, u) e^{\alpha{u}^2}}{|x|^{\beta}} + u^{q}g(u - a)\right),\quad{u} > 0 \quad {\rm in} \quad \Omega\\u \quad \quad = 0\quad {\rm on} \quad \partial \Omega \end{array}\right.$$ where ${0\leq q < 1 ,0< \alpha\leq4\pi}$ and ${\beta \in [0, 2)}$ such that ${\frac{\beta}{2} + \frac{\alpha}{4\pi} \leq 1}$ and ${{g(t - a) = \left\{\begin{array}{ll}1, t \leq a\\ 0, t > a.\end{array}\right.}}$ Under the suitable assumptions on m(x, t) we show the existence and multiplicity of solutions for maximal interval for λ.     

{W^{1,1}_0} -solutions for elliptic problems having gradient quadratic lower order terms

In this paper we deal with solutions of problems of the type $$\left\{\begin{array}{ll}-{\rm div} \Big(\frac{a(x)Du}{(1+|u|)^2} \Big)+u = \frac{b(x)|Du|^2}{(1+|u|)^3} +f \quad &{\rm in} \, \Omega,\\ u=0 &{\rm on} \partial \, \Omega, \end{array} \right.$$ where ${0 < \alpha \leq a(x) \leq \beta, |b(x)| \leq \gamma, \gamma > 0, f \in L^2 (\Omega)}$ and Ω is a bounded subset of ${\mathbb{R}^N}$ with N ≥ 3. We prove the existence of at least one solution for such a problem in the space ${W_{0}^{1, 1}(\Omega) \cap L^{2}(\Omega)}$ if the size of the lower order term satisfies a smallness condition when compared with the principal part of the operator. This kind of problems naturally appears when one looks for positive minima of a functional whose model is: $$J (v) = \frac{\alpha}{2} \int_{\Omega}\frac{|D v|^2}{(1 + |v|)^{2}} + \frac{12}{\int_{\Omega}|v|^2} - \int_{\Omega}f\,v , \quad f \in L^2(\Omega),$$ where in this case a(x) ≡ b(x) = α > 0.     

Isolated boundary singularities of semilinear elliptic equations

Given a smooth domain ${\Omega\subset\mathbb{R}^N}$ such that ${0 \in \partial\Omega}$ and given a nonnegative smooth function ?? on ???, we study the behavior near 0 of positive solutions of ???u?=?u ^q in ?? such that u =? ?? on ???\{0}. We prove that if ${\frac{N+1}{N-1} < q < \frac{N+2}{N-2}}$ , then ${u(x)\leq C |x|^{-\frac{2}{q-1}}}$ and we compute the limit of ${|x|^{\frac{2}{q-1}} u(x)}$ as x ?? 0. We also investigate the case ${q= \frac{N+1}{N-1}}$ . The proofs rely on the existence and uniqueness of solutions of related equations on spherical domains.     

Quenched to annealed transition in the parabolic Anderson problem

We study limit behavior for sums of the form $\frac{1}{|\Lambda_{L|}}\sum_{x\in \Lambda_{L}}u(t,x),$ where the field $\Lambda_L=\left\{x\in {\bf{Z^d}}:|x|\le L\right\}$ is composed of solutions of the parabolic Anderson equation $$u(t,x) = 1 + \kappa \mathop{\int}_{0}^{t} \Delta u(s,x){\rm d}s + \mathop{\int}_{0}^{t}u(s,x)\partial B_{x}(s). $$ The index set is a box in Z ^d , namely $\Lambda_{L} = \left\{x\in {\bf Z}^{\bf d} : |x| \leq L\right\}$ and L = L(t) is a nondecreasing function $L : [0,\infty)\rightarrow {\bf R}^{+}. $ We identify two critical parameters $\eta(1) < \eta(2)$ such that for $\gamma > \eta(1)$ and L(t) = e^{γ t} , the sums $\frac{1}{|\Lambda_L|}\sum_{x\in \Lambda_L}u(t,x)$ satisfy a law of large numbers, or put another way, they exhibit annealed behavior. For $\gamma > \eta(2)$ and L(t) = e^{γ t} , one has $\sum_{x\in \Lambda_L}u(t,x)$ when properly normalized and centered satisfies a central limit theorem. For subexponential scales, that is when $\lim_{t \rightarrow \infty} \frac{1}{t}\ln L(t) = 0,$ quenched asymptotics occur. That means $\lim_{t\rightarrow \infty}\frac{1}{t}\ln\left (\frac{1}{|\Lambda_L|}\sum_{x\in \Lambda_L}u(t,x)\right) = \gamma(\kappa),$ where $\gamma(\kappa)$ is the almost sure Lyapunov exponent, i.e. $\lim_{t\rightarrow \infty}\frac{1}{t}\ln u(t,x)= \gamma(\kappa).$ We also examine the behavior of $\frac{1}{|\Lambda_L|}\sum_{x\in \Lambda_L}u(t,x)$ for L = e ^{γ t} with γ in the transition range $(0,\eta(1))$      

Existence and Multiplicity Results for Some Elliptic Systems in Unbounded Cylinders

We study the following nonlinear elliptic system of Lane–Emden type $$\left\{\begin{array}{ll} -\Delta u = {\rm sgn}(v) |v| ^{p-1} \qquad \qquad \qquad \; {\rm in} \; \Omega , \\ -\Delta v = - \lambda {\rm sgn} (u)|u| \frac{1}{p-1} + f(x, u)\; \; {\rm in}\; \Omega , \\ u = v = 0 \qquad \qquad \qquad \quad \quad \;\;\;\;\; {\rm on}\; \partial \Omega , \end{array}\right.$$ where ${\lambda \in \mathbb{R}}$ . If ${\lambda \geq 0}$ and ${\Omega}$ is an unbounded cylinder, i.e., ${\Omega = \tilde \Omega \times \mathbb{R}^{N-m} \subset \mathbb{R}^{N}}$ , ${N - m \geq 2, m \geq 1}$ , existence and multiplicity results are proved by means of the Principle of Symmetric Criticality and some compact imbeddings in partially spherically symmetric spaces. We are able to state existence and multiplicity results also if ${\lambda \in \mathbb{R}}$ and ${\Omega}$ is a bounded domain in ${\mathbb{R}^{N}, N \geq 3}$ . In particular, a good finite dimensional decomposition of the Banach space in which we work is given.     

Infinitely many solutions for an elliptic problem involving critical Sobolev growth and Hardy potential

In this paper, we will prove the existence of infinitely many solutions for the following elliptic problem with critical Sobolev growth and a Hardy potential: $$-\Delta u-\frac{\mu}{|x|^2}u = |u|^{2^{\ast}-2}u+a u\quad {\rm in}\;\Omega,\quad u=0 \quad {\rm on}\; \partial\Omega,\qquad (*)$$ under the assumptions that N ≥ 7, ${\mu\in \left[0,\frac{(N-2)^2}4-4\right)}$ and a > 0, where ${2^{\ast}=\frac{2N}{N-2}}$ , and Ω is an open bounded domain in ${\mathbb{R}^N}$ which contains the origin. To achieve this goal, we consider the following perturbed problem of (*), which is of subcritical growth, $$-\Delta u-\frac{\mu}{|x|^2}u = |u|^{2^{\ast}-2-\varepsilon_n}u+au \quad {\rm in}\,\Omega, \quad u=0 \quad {\rm on}\;\partial\Omega,\qquad(\ast\ast)_n$$ where ${\varepsilon_{n} > 0}$ is small and ${\varepsilon_n \to 0}$ as n → + ∞. By the critical point theory for the even functionals, for each fixed ${\varepsilon_{n} > 0}$ small, (**) _n has a sequence of solutions ${u_{k,\varepsilon_{n}} \in H^{1}_{0}(\Omega)}$ . We obtain the existence of infinitely many solutions for (*) by showing that as n → ∞, ${u_{k,\varepsilon_{n}}}$ converges strongly in ${H^{1}_{0}(\Omega)}$ to u _k , which must be a solution of (*). Such a convergence is obtained by applying a local Pohozaev identity to exclude the possibility of the concentration of ${\{u_{k,\varepsilon_n}\}}$ .     

Positive Solution for a Class of Degenerate Quasilinear Elliptic Equations in R N

We establish a result on the existence of a positive solution for the following class of degenerate quasilinear elliptic problems: $$(P)\quad \quad \left\{\begin{array}{ll}{-\Delta_{ap}u + V(x)|x|^{-ap^*} |u|^{p-2} u=K(x)f(x, u), {\rm in} \, R^N,}\\ {u > 0, {\rm in} \, R^N , \, u \in \mathcal{D}^{1,p}_a}{(R^N)},\end{array}\right. $$ denotes the Hardy-Sobolev’s \({{-\Delta_{ap}u = - div(|x|^{-ap}|\nabla u|^{p-2} \nabla u), 1 < p < N, -\infty < a < \frac{N-p}{p}, a \leq e \leq a+1, d=1+a-e}}\) , and \({{p^* := p^*(a,e)=\frac{Np}{N-dp}}}\) denotes the Hardy-Sobolev’s critical exponent, V and K are bounded nonnegative continuous potentials, K vanishes at infinity, and f has a subcritical growth at infinity. The technique used here is the variational approach.     

Multiple Solutions for a Semilinear Elliptic Equation with Critical Growth and Magnetic Field

In this paper, we are concerned with the multiplicity of nontrivial solutions for the following class of complex problems $$(-i\nabla - A(x))^2{u} = \mu|u|^{q-2}u + |u|^{2^*-2}u\, {\rm in}\, \Omega,\quad u=0\, {\rm on}\, \partial\, \Omega$$ where \({\Omega \subset \mathbb{R}^N(N \geq 4)}\) is a bounded domain with smooth boundary, \({A: \overline{\Omega} \rightarrow \mathbb{R}^N}\) is a continuous magnetic potential and \({2 \leq q < 2^* = \frac{2N}{N-2}}\) . Using the Lusternik-Schnirelman theory, we relate the number of solutions with the topology of Ω.     

Regularity of semi-stable solutions to fourth order nonlinear eigenvalue problems on general domains

We examine the fourth order problem $\Delta ^2 u = \lambda f(u) $ in $ \Omega $ with $ \Delta u = u =0 $ on $ {\partial \Omega }$ , where $ \lambda > 0$ is a parameter, $ \Omega $ is a bounded domain in $\mathbb{R }^N$ and where $f$ is one of the following nonlinearities: $ f(u)=e^u$ , $ f(u)=(1+u)^p $ or $ f(u)= \frac{1}{(1-u)^p}$ where $ p>1$ . We show the extremal solution is smooth, provided $$\begin{aligned} N < 2 + 4 \sqrt{2} + 4 \sqrt{ 2 - \sqrt{2}} \approx 10.718 \text{ when} f(u)=e^u, \end{aligned}$$ and $$\begin{aligned} N < \frac{4p}{p-1} + \frac{4(p+1)}{p-1} \left( \sqrt{ \frac{2p}{p+1}} + \sqrt{ \frac{2p}{p+1} - \sqrt{ \frac{2p}{p+1}}} - \frac{1}{2} \right) \end{aligned}$$ when $ f(u)=(u+1)^p$ . New results are also obtained in the case where $ f(u)=(1-u)^{-p}$ . These are substantial improvements to various results on critical dimensions obtained recently by various authors. To do that, we derive a new stability inequality satisfied by minimal solutions of the above equation, which is more amenable to estimates as it allows a method of proof reminiscent of the second order case.     

Invariant submanifolds of Sasakian space forms

In the present study, we consider isometric immersions ${f : M \rightarrow \tilde{M}(c)}$ of (2n + 1)-dimensional invariant submanifold M ²ⁿ⁺¹ of (2m + 1) dimensional Sasakian space form ${\tilde{M}^{2m+1}}$ of constant ${ \varphi}$ -sectional curvature c. We have shown that if f satisfies the curvature condition ${\overset{\_}{R}(X, Y) \cdot \sigma =Q(g, \sigma)}$ then either M ²ⁿ⁺¹ is totally geodesic, or ${||\sigma||^{2}=\frac{1}{3}(2c+n(c+1)),}$ or ${||\sigma||^{2}(x) > \frac{1}{3}(2c+n(c+1)}$ at some point x of M ²ⁿ⁺¹. We also prove that ${\overset{\_ }{R}(X, Y)\cdot \sigma = \frac{1}{2n}Q(S, \sigma)}$ then either M ²ⁿ⁺¹ is totally geodesic, or ${||\sigma||^{2}=-\frac{2}{3}(\frac{1}{2n}\tau -\frac{1}{2}(n+2)(c+3)+3)}$ , or ${||\sigma||^{2}(x) > -\frac{2}{3}(\frac{1}{2n} \tau (x)-\frac{1}{2} (n+2)(c+3)+3)}$ at some point x of M ²ⁿ⁺¹.     

Ground states for nonlinear Kirchhoff equations with critical growth

This paper is concerned with the existence, multiplicity and concentration behavior of positive solutions for the critical Kirchhoff-type problem $$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} -\left(\varepsilon ^2a+\varepsilon b\int _{\mathbb{R }^{3}}|\nabla u|^2\right)\Delta u+V(x)u=u^{2^*-1}+\lambda f(u)&\text{ in}~{\mathbb{R }^{3}},\\ u\in H^1({\mathbb{R }^{3}}), ~u(x)>0&\text{ in}~{\mathbb{R }^{3}}, \end{array}\right. \end{aligned}$$ where $\varepsilon $ and $\lambda $ are positive parameters, and $a,b>0$ are constants, $2^*(=6)$ is the critical Sobolev exponent in dimension three, $V$ is a positive continuous potential satisfying some conditions, and $f$ is a subcritical nonlinear term. We use the variational methods to relate the number of solutions with the topology of the set where $V$ attains its minimum, for all sufficiently large $\lambda $ and small $\varepsilon $ .     

Scattering of wave maps from {{\mathbb {R}^{2+1}}} to general targets

We show that smooth, radially symmetric wave maps U from ${\mathbb {R}^{2+1}}$ to a compact target manifold (N, where ? _r U and ? _t U have compact support for any fixed time, scatter. The result will follow from the work of Christodoulou and Tahvildar-Zadeh, and Struwe, upon proving that for ${(\lambda^{\prime} \in (0,1),}$ energy does not concentrate in the set $$K_{\frac{5}{8}T,\frac{7}{8}T}^{\lambda^{\prime}} = \{(x,t) \in \mathbb{R}^{2+1} \vert \quad|x| \leq \lambda^{\prime} t, t \in [(5/8)T,(7/8)T] \}.$$      

Partial regularity for non autonomous functionals with non standard growth conditions

We prove a C ^1,μ partial regularity result for minimizers of a non autonomous integral funcitional of the form $$\mathcal{F}(u; \Omega):=\int_{\Omega}f(x, Du)\ dx$$ under the so-called non standard growth conditions. More precisely we assume that $$c |z|^{p}\leq f(x ,z) \leq L (1+|z|^{q}),$$ for 2 ≤ p < q and that D _z f(x, z) is α-Hölder continuous with respect to the x-variable. The regularity is obtained imposing that ${\frac{p}{q} < \frac{n+\alpha}{n}}$ but without any assumption on the growth of ${D^{2}_{z}f}$ .     

Lower bounds on Ricci curvature and quantitative behavior of singular sets

Let Y ⁿ denote the Gromov-Hausdorff limit $M^{n}_{i}\stackrel{d_{\mathrm{GH}}}{\longrightarrow} Y^{n}$ of v-noncollapsed Riemannian manifolds with ${\mathrm{Ric}}_{M^{n}_{i}}\geq-(n-1)$ . The singular set $\mathcal {S}\subset Y$ has a stratification $\mathcal {S}^{0}\subset \mathcal {S}^{1}\subset\cdots\subset \mathcal {S}$ , where $y\in \mathcal {S}^{k}$ if no tangent cone at y splits off a factor ? ^k+1 isometrically. Here, we define for all η>0, 0<r≤1, the k-th effective singular stratum $\mathcal {S}^{k}_{\eta,r}$ satisfying $\bigcup_{\eta}\bigcap_{r} \,\mathcal {S}^{k}_{\eta,r}= \mathcal {S}^{k}$ . Sharpening the known Hausdorff dimension bound $\dim\, \mathcal {S}^{k}\leq k$ , we prove that for all y, the volume of the r-tubular neighborhood of $\mathcal {S}^{k}_{\eta,r}$ satisfies ${\mathrm {Vol}}(T_{r}(\mathcal {S}^{k}_{\eta,r})\cap B_{\frac{1}{2}}(y))\leq c(n,{\mathrm {v}},\eta)r^{n-k-\eta}$ . The proof involves a quantitative differentiation argument. This result has applications to Einstein manifolds. Let $\mathcal {B}_{r}$ denote the set of points at which the C ²-harmonic radius is ≤r. If also the $M^{n}_{i}$ are Kähler-Einstein with L ₂ curvature bound, $\| Rm\|_{L_{2}}\leq C$ , then ${\mathrm {Vol}}( \mathcal {B}_{r}\cap B_{\frac{1}{2}}(y))\leq c(n,{\mathrm {v}},C)r^{4}$ for all y. In the Kähler-Einstein case, without assuming any integral curvature bound on the $M^{n}_{i}$ , we obtain a slightly weaker volume bound on $\mathcal {B}_{r}$ which yields an a priori L _p curvature bound for all p<2. The methodology developed in this paper is new and is applicable in many other contexts. These include harmonic maps, minimal hypersurfaces, mean curvature flow and critical sets of solutions to elliptic equations.     

Poincaré–Sobolev equations in the hyperbolic space

We study the a priori estimates, existence/nonexistence of radial sign changing solution, and the Palais–Smale characterisation of the problem ${-\Delta_{{\mathbb B}^{N}}u - \lambda u = |u|^{p-1}u, u\in H^1({\mathbb B}^{N})}$ in the hyperbolic space ${{\mathbb B}^{N}}$ where ${1 < p\leq\frac{N+2}{N-2}}$ . We will also prove the existence of sign changing solution to the Hardy–Sobolev–Mazya equation and the critical Grushin problem.     

Pinned distance sets, k-simplices, Wolff’s exponent in finite fields and sum-product estimates

An analog of the Falconer distance problem in vector spaces over finite fields asks for the threshold α?>?0 such that ${|\Delta(E)| \gtrsim q}$ whenever ${|E| \gtrsim q^{\alpha}}$ , where ${E \subset {\mathbb {F}}_q^d}$ , the d-dimensional vector space over a finite field with q elements (not necessarily prime). Here ${\Delta(E)=\{{(x_1-y_1)}^2+\dots+{(x_d-y_d)}^2: x,y \in E\}}$ . Iosevich and Rudnev (Trans Am Math Soc 359(12):6127–6142, 2007) established the threshold ${\frac{d+1}{2}}$ , and in Hart et?al. (Trans Am Math Soc 363:3255–3275, 2011) proved that this exponent is sharp in odd dimensions. In two dimensions we improve the exponent to ${\tfrac{4}{3}}$ , consistent with the corresponding exponent in Euclidean space obtained by Wolff (Int Math Res Not 10:547–567, 1999). The pinned distance set ${\Delta_y(E)=\{{(x_1-y_1)}^2+\dots+{(x_d-y_d)}^2: x\in E\}}$ for a pin ${y\in E}$ has been studied in the Euclidean setting. Peres and Schlag (Duke Math J 102:193–251, 2000) showed that if the Hausdorff dimension of a set E is greater than ${\tfrac{d+1}{2}}$ , then the Lebesgue measure of Δ _y (E) is positive for almost every pin y. In this paper, we obtain the analogous result in the finite field setting. In addition, the same result is shown to be true for the pinned dot product set ${\Pi_y(E)=\{x\cdot y: x\in E\}}$ . Under the additional assumption that the set E has Cartesian product structure we improve the pinned threshold for both distances and dot products to ${\frac{d^2}{2d-1}}$ . The pinned dot product result for Cartesian products implies the following sum-product result. Let ${A\subset \mathbb F_q}$ and ${z\in \mathbb F^*_q}$ . If ${|A|\geq q^{\frac{d}{2d-1}}}$ then there exists a subset ${E'\subset A\times \dots \times A=A^{d-1}}$ with ${|E'|\gtrsim |A|^{d-1}}$ such that for any ${(a_1,\dots, a_{d-1}) \in E'}$ , $$ |a_1A+a_2A+\dots +a_{d-1}A+zA| > \frac{q}{2}$$ where ${a_j A=\{a_ja:a \in A\},j=1,\dots,d-1}$ . A generalization of the Falconer distance problem is to determine the minimal α?>?0 such that E contains a congruent copy of a positive proportion of k-simplices whenever ${|E| \gtrsim q^{\alpha}}$ . Here the authors improve on known results (for k?>?3) using Fourier analytic methods, showing that α may be taken to be ${\frac{d+k}{2}}$ .     

Bounding S(t) and S_1(t) on the Riemann hypothesis

Let $\pi S(t)$ denote the argument of the Riemann zeta-function, $\zeta (s)$ , at the point $s=\frac{1}{2}+it$ . Assuming the Riemann hypothesis, we present two proofs of the bound $$\begin{aligned} |S(t)| \le \left(\frac{1}{4} + o(1) \right)\frac{\log t}{\log \log t} \end{aligned}$$ for large $t$ . This improves a result of Goldston and Gonek by a factor of 2. The first method consists of bounding the auxiliary function $S_1(t) = \int _0^{t} S(u) \> \text{ d}u$ using extremal functions constructed by Carneiro, Littmann and Vaaler. We then relate the size of $S(t)$ to the size of the functions $S_1(t\pm h)-S_1(t)$ when $h\asymp 1/\log \log t$ . The alternative approach bounds $S(t)$ directly, relying on the solution of the Beurling–Selberg extremal problem for the odd function $f(x) = \arctan \left(\frac{1}{x}\right) - \frac{x}{1 + x^2}$ . This draws upon recent work by Carneiro and Littmann.     

Fractional eigenvalues

We study the non-local eigenvalue problem $$\begin{aligned} 2\, \int \limits _{\mathbb{R }^n}\frac{|u(y)-u(x)|^{p-2}\bigl (u(y)-u(x)\bigr )}{|y-x|^{\alpha p}}\,dy +\lambda |u(x)|^{p-2}u(x)=0 \end{aligned}$$ for large values of $p$ and derive the limit equation as $p\rightarrow \infty $ . Its viscosity solutions have many interesting properties and the eigenvalues exhibit a strange behaviour.     

{\mathcal{C}^{\alpha}}-regularity for a class of non-diagonal elliptic systems with p-growth

We consider weak solutions to nonlinear elliptic systems in a W ^1,p -setting which arise as Euler equations to certain variational problems. The solutions are assumed to be stationary in the sense that the differential of the variational integral vanishes with respect to variations of the dependent and independent variables. We impose new structure conditions on the coefficients which yield everywhere ${\mathcal{C}^{\alpha}}$ -regularity and global ${\mathcal{C}^{\alpha}}$ -estimates for the solutions. These structure conditions cover variational integrals like ${\int F(\nabla u)\; dx}$ with potential ${F(\nabla u):=\tilde F (Q_1(\nabla u),\ldots, Q_N(\nabla u))}$ and positively definite quadratic forms in ${\nabla u}$ defined as ${Q_i(\nabla u)=\sum_{\alpha \beta} a_i^{\alpha \beta} \nabla u^\alpha \cdot \nabla u^\beta}$ . A simple example consists in ${\tilde F(\xi_1,\xi_2):= |\xi_1|^{\frac{p}{2}} + |\xi_2|^{\frac{p}{2}}}$ or ${\tilde F(\xi_1,\xi_2):= |\xi_1|^{\frac{p}{4}}|\xi_2|^{\frac{p}{4}}}$ . Since the Q _i need not to be linearly dependent our result covers a class of nondiagonal, possibly nonmonotone elliptic systems. The proof uses a new weighted norm technique with singular weights in an L ^p -setting.