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Register a new userLocal boundedness for solutions of a class of nonlinear elliptic systems
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In this paper we are concerned with the regularity of solutions to a nonlinear elliptic system of $m$ equations in divergence form, satisfying $p$ growth from below and $q$ growth from above, with $p leq q$; this case is known as $p, q$-growth conditions. Well known counterexamples, even in the simpler case $p=q$, show that solutions to systems may be singular; so, it is necessary to add suitable structure conditions on the system that force solutions to be regular. Here we obtain local boundedness of solutions under a componentwise coercivity condition. Our result is obtained by proving that each component $u^alpha$ of the solution $u=(u^1,...,u^m)$ satisfies an improved Caccioppolis inequality and we get the boundedness of $u^{alpha}$ by applying De Giorgis iteration method, provided the two exponents $p$ and $q$ are not too far apart. Let us remark that, in dimension $n=3$ and when $p=q$, our result works for $frac{3}{2} < p < 3$, thus it complements the one of Bjorn whose technique allowed her to deal with $p leq 2$ only. In the final section, we provide applications of our result.
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We prove global essential boundedness for the weak solutions of divergence form quasilinear systems. The principal part of the differential operator is componentwise coercive and supports controlled growths with respect to the solution and its gradient, while the lower order term exhibits componentwise controlled gradient growth. The x-behaviour of the nonlinearities is governed in terms of Morrey spaces.
We study boundary blow-up solutions of semilinear elliptic equations $Lu=u_+^p$ with $p>1$, or $Lu=e^{au}$ with $a>0$, where $L$ is a second order elliptic operator with measurable coefficients. Several uniqueness theorems and an existence theorem are obtained.
We revisit the following nonlinear critical elliptic equation begin{equation*} -Delta u+Q(y)u=u^{frac{N+2}{N-2}},;;; u>0;;;hbox{ in } mathbb{R}^N, end{equation*} where $Ngeq 5.$ Although there are some existence results of bubbling solutions for problem above, there are no results about the periodicity of bubbling solutions. Here we investigate some related problems. Assuming that $Q(y)$ is periodic in $y_1$ with period 1 and has a local minimum at 0 satisfying $Q(0)=0,$ we prove the existence and local uniqueness of infinitely many bubbling solutions of the problem above. This local uniqueness result implies that some bubbling solutions preserve the symmetry of the potential function $Q(y),$ i.e. the bubbling solution whose blow-up set is ${(jL,0,...,0):j=0,1,2,...,m}$ must be periodic in $y_{1}$ provided that $L$ is large enough, where $m$ is the number of the bubbles which is large enough but independent of $L.$ Moreover, we also show a non-existence of this bubbling solutions for the problem above if the local minimum of $Q(y)$ does not equal to zero.
In this paper, the finite time extinction of solutions to the fast diffusion system $u_t=mathrm{div}(| abla u|^{p-2} abla u)+v^m$, $v_t=mathrm{div}(| abla v|^{q-2} abla v)+u^n$ is investigated, where $1<p,q<2$, $m,n>0$ and $Omegasubset mathbb{R}^N (Ngeq1)$ is a bounded smooth domain. After establishing the local existence of weak solutions, the authors show that if $mn>(p-1)(q-1)$, then any solution vanishes in finite time provided that the initial data are ``comparable; if $mn=(p-1)(q-1)$ and $Omega$ is suitably small, then the existence of extinction solutions for small initial data is proved by using the De Giorgi iteration process and comparison method. On the other hand, for $1<p=q<2$ and $mn<(p-1)^2$, the existence of at least one non-extinction solution for any positive smooth initial data is proved.
We consider finite Morse index solutions to semilinear elliptic questions, and we investigate their smoothness. It is well-known that: - For $n=2$, there exist Morse index $1$ solutions whose $L^infty$ norm goes to infinity. - For $n geq 3$, uniform boundedness holds in the subcritical case for power-type nonlinearities, while for critical nonlinearities the boundedness of the Morse index does not prevent blow-up in $L^infty$. In this paper, we investigate the case of general supercritical nonlinearities inside convex domains, and we prove an interior a priori $L^infty$ bound for finite Morse index solution in the sharp dimensional range $3leq nleq 9$. As a corollary, we obtain uniform bounds for finite Morse index solutions to the Gelfand problem constructed via the continuity method.
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