Supercritical Exponents Research Articles

Systems of nonlinear wave equations with damping and supercritical boundary and interior sources

utt −∆u + g1(ut) = f1(u, v) vtt −∆v + g2(vt) = f2(u, v), in a bounded domain Ω ⊂ R with Robin and Dirichlet boundary conditions on u and v respectively. The nonlinearities f1(u, v) and f2(u, v) are with supercritical exponents representing strong sources, while g1(ut) and g2(vt) act as damping. In addition, the boundary condition also contains a nonlinear source and a damping term. By employing nonlinear semigroups and the theory of monotone operators, we obtain several results on the existence of local and global weak solutions, and uniqueness of weak solutions. Moreover, we prove that such unique solutions depend continuously on the initial data.

Positive solutions of integral systems involving Bessel potentials

This paper is concerned with integral systems involving the Bessel potentials. Such integral systems are helpful to understand the corresponding PDE systems, such as some static Shrodinger systems with the critical and the supercritical exponents. We use the lifting lemma on regularity to obtain an integrability interval of solutions. Since the Bessel kernel does not have singularity at infinity, we extend the integrability interval to the whole $[1,\infty]$. Next, we use the method of moving planes to prove the radial symmetry for the positive solution of the system. Based on these results, by an iteration we obtain the estimate of the exponential decay of those solutions near infinity. Finally, we discuss the uniqueness of the positive solution of PDE system under some assumption.

Nonexistence and multiplicity of solutions to elliptic problems with supercritical exponents

We consider the supercritical problem -\Delta u = |u|^{p-2}u in \Omega, u=0 on \partial\Omega, where $\Omega$ is a bounded smooth domain in $\mathbb{R}^{N},$ $N\geq3,$ and $p\geq2^{*}:= 2N/(N-2).$ Bahri and Coron showed that if $\Omega$ has nontrivial homology this problem has a positive solution for $p=2^{*}.$ However, this is not enough to guarantee existence in the supercritical case. For $p\geq 2(N-1)/(N-3)$ Passaseo exhibited domains carrying one nontrivial homology class in which no nontrivial solution exists. Here we give examples of domains whose homology becomes richer as $p$ increases. More precisely, we show that for $p> 2(N-k)/(N-k-2)$ with $1\leq k\leq N-3$ there are bounded smooth domains in $\mathbb{R}^{N}$ whose cup-length is $k+1$ in which this problem does not have a nontrivial solution. For $N=4,8,16$ we show that there are many domains, arising from the Hopf fibrations, in which the problem has a prescribed number of solutions for some particular supercritical exponents.

Infinitely many positive solutions for an elliptic problem with critical or supercritical growth

We prove that for some supercritical exponents p > N + 2 N − 2 and for some smooth domains D in R N there are infinitely many (distinct) positive solutions to the following Lane–Emden–Fowler equation: { − Δ u = u p , u > 0 , in D , u = 0 , on ∂ D This seems to be the first result for such type of equations.

Quasilinear Riccati Type Equations with Super-Critical Exponents

We establish explicit criteria of solvability for the quasilinear Riccati type equation − Δ p u = |∇u| q + ω in a bounded 𝒞1 domain Ω ⊂ ℝ n , n ≥ 2. Here Δ p , p > 1, is the p-Laplacian, q is in the supper critical range q > p, and the datum ω is a measure. Our existence criteria are given in the form of potential theoretic or geometric estimates that are sharp when ω is nonnegative and compactly supported in Ω. Our existence results are new even in the case dω =f dx where f belongs to the weak Lebesgue space . Moreover, our methods allow the treatment of more general equations where the principal operators may have discontinuous coefficients. As a consequence of the solvability results, a characterization of removable singularities for the corresponding homogeneous equation is also obtained.

A sharp constant in a Sobolev-Nirenberg inequality and its application to the Schrödinger equation

We prove that the solution of the Cauchy problem for a non-linear Schrödinger evolution equation with critical and supercritical exponents can blow up at a finite time for some initial data, and this time is estimated from above and below. To this end, an interpolation Nirenberg-type inequality and a Sobolev-type inequality are proved and the values of sharp constants in these inequalities are calculated.

Local and global wellposedness of weak solutions for the wave equation with nonlinear boundary and interior sources of supercritical exponents and damping

We consider the wave equation with interior and boundary nonlinear sources and damping and we are interested in local and global wellposedness of finite energy solutions. The main difficulty is represented by the fact that the Lopatinski condition fails to hold (unless the dim ( Ω ) = 1 ), and thus the analysis of the boundary nonlinearities becomes a subtle issue.

The compactness theorem for inhomogeneous semilinear elliptic equations with supercritical exponents

In this paper, we consider the following inhomogeneous semilinear equation: Δ u + k ( x ) u α + f ( x ) = 0 in Ω ⊂ R n ( n ≥ 3 ) where α ≥ n + 2 n − 2 , k ( x ) , f ( x ) ∈ C 1 ( Ω ) ; a 1 < k ( x ) < b 1 , 0 < a 1 < b 1 and | ∇ log k ( x ) | ≤ C for x ∈ Ω ¯ ; f ( x ) > 0 , ‖ f ‖ L ∞ ( Ω ) ≤ a 2 and ‖ ∇ f ‖ L ∞ ( Ω ) ≤ b 2 for some constants a 2 , b 2 > 0 . Using the monotonicity inequality and ε -regularity result, we obtain the measure estimate of the blow up set for positive smooth solutions { u i } of the above equation with { ‖ u i ‖ H 1 ( Ω ) + ‖ u i ‖ L α + 1 ( Ω ) } bounded.

On the Lazer-McKenna Conjecture Involving Critical and Super-critical Exponents

On the Lazer-McKenna Conjecture Involving Critical and Super-critical Exponents

Existence of nontrivial solutions for some elliptic equations with supercritical nonlinearity in exterior domains

The existence of positive solutions is discussed for some nonlinear elliptic equations involving the nonlinear terms with the growth order of super-critical exponents in exterior domains of balls such as $ -\Delta u = u^\beta $ in $\Omega$, ($(N+2)/(N-2) < \beta $), $u = 0 $ on $\partial B$, with $\Omega = \mathbb{R}^N \setminus\overline\Omega_0$ where $\Omega_0$ is the open ball. To recover the compactness of the embedding $L^{\beta+1}(\Omega) \subset H^1_0(\Omega)$, we work in the class of radially symmetric functions and introduce a new transformation, which reduces our problems to some nonlinear elliptic equations in annuli but with coefficients which have some singularity on the boundary. The difficulty caused by the singularity on the boundary will be managed by the arguments developed in our previous work.

Nonlinear elliptic equations with large supercritical exponents

The paper deals with the existence of positive solutions of the problem -Δ u=up in Ω, u=0 on ∂Ω, where Ω is a bounded domain of \(\mathcal{R}^n\), n≥ 3, and p>2. We describe new concentration phenomena, which arise as p→ +∞ and can be exploited in order to construct, for p large enough, positive solutions that concentrate, as p→ +∞, near submanifolds of codimension 2. In this paper we consider, in particular, domains with axial symmetry and obtain positive solutions concentrating near (n-2)-dimensional spheres, which approach the boundary of Ω as p→ +∞. The existence and multiplicity results we state allow us to find positive solutions, for large p, also in domains which can be contractible and even arbitrarily close to starshaped domains (while no solution can exist if Ω is starshaped and \(p\ge {2n\over n-2}\), as a consequence of the Pohožaev's identity).

Partial regularity for weak solutions of nonlinear elliptic equations with supercritical exponents

We shall prove a partial regularity theorem for a stationary positive weak solution of the equation Δu+h1uα+h2uβ=0inΩ⊂Rnwith α⩾n+2n-2,α+1⩾2β>2.

Partial regularity for weak solutions of nonlinear elliptic equations with supercritical exponents

We shall prove a partial regularity theorem for a stationary positive weak solution of the equation Δ u + h 1 u α + h 2 u β = 0 in Ω ⊂ R n with α ⩾ n + 2 n - 2 , α + 1 ⩾ 2 β > 2 .

Partial regularity for weak solutions of semilinear elliptic equations with supercritical exponents

Let Ω be an open subset in R n (n > 3). In this paper, we study the partial regularity for stationary positive weak solutions of the equation (1.1) Δu + h 1 (x)u + h 2 (x)u α = 0 in Ω. We prove that if α > n+2/n-2, and u ∈ H 1 (Ω) n L α+1 (Ω) is a stationary positive weak solution of (1.1), then the Hausdorff dimension of the singular set of u is less than n - 2α+1/α-1, which generalizes the main results in Pacard 1993 and Pacard 1994.

The blow up locus of semilinear elliptic equations with supercritical exponents

The blow up locus of semilinear elliptic equations with supercritical exponents

Existence and nonexistence in a class of equations with supercritical growth

We study nonexistence regularity and asymptotic behaviour of solutions of problems of the type We point out that even supercritical exponents are allowed out technique combines the method of lower and upper-solutions with variational arguments and suitable elliptic estimates

Nonlinear elliptic equations involving critical Sobolev exponent on compact riemannian manifolds in presence of symmetries

In this paper, we study a nonlinear elliptic equation with critical exponent, invariant under the action of a subgroup G of the isometry group of a compact riemannian manifold. We obtain some existence results of positive solutions of this equation, and under some assumptions on G, we show that we can solve this equation for supercritical exponents.

Positive solutions and bifurcation of fully nonlinear elliptic equations involving super-critical Sobolev exponents

This paper discuss the existence of bifurcation point of positive solutions for the fully nonlinear elliptic equations involving super-critical Sobolev exponent which include semilinear, Monge-Ampere and Hessian equations as its examples, by setting abstract bifurcation theorem via the topological degree theory.

Limit behavior of saturated approximations of nonlinear Schrödinger equation

We consider the solutionuɛ(t) of the saturated nonlinear Schrodinger equation $$i\partial u/\partial t = - \Delta u - \left| u \right|^{4/N} u + \varepsilon \left| u \right|^{q - 1} uandu(0,.) = \varphi (.)$$ (1) where\(N \geqslant 2,\varepsilon > 0,1 + 4/N 0 such that\(\mathop {\lim }\limits_{t \to T} \left\| {u(t)} \right\|_{H^1 } = + \infty \), whereu(t) is solution of $$i\partial u/\partial t = - \Delta u - \left| u \right|^{4/N} uandu(0,.) = \varphi (.)$$ (1) For ɛ>0 fixed,uɛ(t) is defined for all time. We are interested in the limit behavior as ɛ→0 ofuɛ(t) fort≥T. In the case where there is no loss of mass inuɛ at infinity in a sense to be made precise, we describe the behavior ofuɛ as ɛ goes to zero and we derive an existence result for a solution of (1) after the blow-up timeT in a certain sense. Nonlinear Schrodinger equation with supercritical exponents are also considered.