Existence of nonnegative solutions for an anisotropic elliptic equation with singular weights and critical growth

Some general foundational issues of quantum mechanics are considered and are related to aspects of quantum computation. The importance of quantum entanglement and quantum information is discussed a...

Lagrangian coordinates in free boundary problems for parabolic equations

For a class of sub-elliptic equations on Heisenberg group \(\mathbb{H}^N\) with Hardy type singularity and critical nonlinear growth, we prove the existence of least energy solutions by developing new techniques based on the Nehari constraint. This result extends previous works, e.g., by Han et al. [Hardy-Sobolev type inequalities on the H-type group, Manuscripta Math. 118 (2005), 235–252].

On a Schrödinger equation involving fractional [formula omitted]-Laplacian with critical growth and Trudinger–Moser nonlinearity

In this paper, we first give a necessary and sufficient condition for the boundedness and the compactness of a class of nonlinear functionals in $$H^{2}\left( {\mathbb {R}}^{4}\right) $$ which are of their independent interests. (See Theorems 2.1 and 2.2.) Using this result and the principle of symmetric criticality, we can present a relationship between the existence of the nontrivial solutions to the semilinear bi-harmonic equation of the form $$\begin{aligned} (-\Delta )^{2}u+\gamma u=f(u) \text {in} {\mathbb {R}}^{4} \end{aligned}$$ and the range of $$\gamma \in {\mathbb {R}}^{+}$$ , where $$f\left( s\right) $$ is the general nonlinear term having the critical exponential growth at infinity. (See Theorem 2.7.) Though the existence of the nontrivial solutions for the bi-harmonic equation with the critical exponential growth has been studied in the literature, it seems that nothing is known so far about the existence of the ground-state solutions for this class of equations involving the trapping potential introduced by Rabinowitz (Z Angew Math Phys 43:27–42, 1992). Since the trapping potential is not necessarily symmetric, classical radial method cannot be applied to solve this problem. In order to overcome this difficulty, we first establish the existence of the ground-state solutions for the equation 0.1 $$\begin{aligned} (-\Delta )^{2}u+V(x)u=\lambda s\exp (2|s|^{2})) \text {in} {\mathbb {R}}^{4}, \end{aligned}$$ when V(x) is a positive constant using the Fourier rearrangement and the Pohozaev identity. Then we will explore the relationship between the Nehari manifold and the corresponding limiting Nehari manifold to derive the existence of the ground state solutions for the Eq. (2.5) when V(x) is the Rabinowitz type trapping potential, namely it satisfies $$\begin{aligned} 0<\inf _{x \in {\mathbb {R}}^{4}} V(x)<\sup _{x \in {\mathbb {R}}^{4}} V(x)=\lim _{|x| \rightarrow +\infty } V(x). \end{aligned}$$ (See Theorem 2.8.) The same result and proof applies to the harmonic equation with the critical exponential growth involving the Rabinowitz type trapping potential in $${\mathbb {R}}^2$$ . (See Theorem 2.9.)

The regularity properties of nonlocal anisotropic elliptic equations with parameters are investigated in abstract weighted Lp spaces. The equations include the variable coefficients and abstract operator function A = A (x) in a Banach space E in leading part. We find the sufficient growth assumptions on A and appropriate symbol polynomial functions that guarantee the uniformly separability of the linear problem. It is proved that the corresponding anisotropic elliptic operator is sectorial and is also the negative generator of an analytic semigroup. Byusing these results, the existence and uniqueness of maximal regular solution of the nonlinear nonlocal anisotropic elliptic equation is obtained in weighted Lp spaces. In application, the maximal regularity properties of the Cauchy problem for degenerate abstract anisotropic parabolic equation in mixed Lp norms, the boundary value problem for anisotropic elliptic convolution equation, the Wentzel-Robin type boundary value problem for degenerate integro-differential equation and infinite systems of degenerate elliptic integro-differential equations are obtained.

We consider a class of anisotropic elliptic differential equations of second order with divergent form and variable exponents. The corresponding elliptic operators are pseudo-monotone and coercive. We obtain solvability conditions for the Dirichlet problem in unbounded domains Ω ⊂ ℝ n , n ≥ 2. The proof of existence of solutions is free of restrictions on growth of data for |x| → ∞.

We study the existence and regularity of the solution to the multivalued equation -ΔΦu ∈ ∂j(u) + λh in Ω, where Ω ⊂ RN is a bounded smooth domain, Φ is an N-function, ΔΦ is the corresponding Φ-Laplacian, λ &gt; 0 is a parameter, h is a measurable function, and j is a continuous function with critical growth where ∂j(u) denotes its subdifferential. We apply the Ekeland Variational Principle to an associated locally Lipschitz energy functional. A major point in our study is that in order to deal with the obtained Ekeland sequence we developed a generalized version for the framework of Orlicz–Sobolev spaces of a well-known Brézis–Lieb lemma which was employed together with a variant of the Lions concentration-compactness theory to get a solution of the equation.

The author considers the semilinear elliptic equation $$ -\Delta^{m}u=g(x,u), $$ subject to Dirichlet boundary conditions $u=Du=\cdots=D^{m-1}u=0$, on a bounded domain $\Omega\subset\mathbb{R}^{2m}$. The notion of nonlinearity of critical growth for this problem is introduced. It turns out that the critical growth rate is of exponential type and the problem is closely related to the Trudinger embedding and Moser type inequalities. The main result is the existence of non trivial weak solutions to the problem.

We are concerned with divergence type quasilinear parabolic equation with measurable coefficients and lower order terms model of which is a doubly nonlinear anisotropic parabolic equations with absorption term. This class of equations has numerous applications which appear in modeling of electrorheological fluids, image precessing, theory of elasticity, theory of non-Newtonian fluids with viscosity depending on the temperature. But the qualitative theory doesn't construct for these anisotropic equations. So, naturally, that during the last decade there has been growing substantial development in the qualitative theory of second order anisotropic elliptic and parabolic equations. The main purpose is to obtain the pointwise upper estimates in terms of distance to the boundary for nonnegative solutions of such equations. This type of estimates originate from the work of J. B. Keller, R. Osserman, who obtained a simple upper bound for any solution, in any number of variables for Laplace equation. These estimates play a crucial role in the theory of existence or nonexistence of so called large solutions of such equations, in the problems of removable singularities for solutions to elliptic and parabolic equations. Up to our knowledge all the known estimates for large solutions to elliptic and parabolic equations are related with equations for which some comparison properties hold. We refer to I.I. Skrypnik, A.E. Shishkov, M. Marcus , L. Veron, V.D. Radulescu for an account of these results and references therein. Such equations have been the object of very few works because in general such properties do not hold. The main ones concern equations only in the precise choice of absorption term \(f(u)=u^q\). Among the people who published significative results in this direction are I.I. Skrypnik, J. Vetois, F.C. Cirstea, J. Garcia-Melian, J.D. Rossi, J.C. Sabina de Lis. The main result of the paper is a priori estimates of Keller-Osserman type for nonnegative solutions of a doubly nonlinear anisotropic parabolic equations with absorption term that have been proven despite of the lack of comparison principle. To obtain these estimates we exploit the method of energy estimations and De Giorgy iteration techniques.

In the present article we deal with the Dirichlet problem for a class of degenerate anisotropic elliptic second-order equations with L1-right-hand sides in a bounded domain of ℝn(n ⩾ 2) . This class is described by the presence of a set of exponents q1,…, qn and a set of weighted functions ν1,…, νn in growth and coercitivity conditions on coefficients of the equations. The exponents qi characterize the rates of growth of the coefficients with respect to the corresponding derivatives of unknown function, and the functions νi characterize degeneration or singularity of the coefficients with respect to independent variables. Our aim is to investigate the existence of entropy solutions of the problem under consideration.

In a previous work [6], we got an exact local behavior to the positive solutions of an elliptic equation. With the help of this exact local behavior, we obtain in this paper the existence of solutions of an equation with Hardy–Sobolev critical growth and singular term by using variational methods. The result obtained here, even in a particular case, relates with a partial (positive) answer to an open problem proposed in: A. Ferrero and F. Gazzola, Existence of solutions for singular critical growth semilinear elliptic equations, J. Differential Equations 177, 494–522 (2001). (© 2007 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Sharp existence and nonexistence results for positive solutions of quasilinear elliptic equations with critical growth in geodesic balls on spheres are established. The arguments are based on Pohozaev type identities and asymptotic estimates for Emden–Fowler type equations. By means of spherical symmetrization and the concentration‐compactness principle existence and nonexistence results for general domains on spheres are obtained. (© 2005 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

We use the Hardy-Sobolev inequality to study existence and non-existence results for a positive solution of the quasilinear elliptic problem -\Delta{p}u − \mu \Delta{q}u = \limda[mp(x)|u|p−2u + \mu mq(x)|u|q−2u] in \Omega driven by nonhomogeneous operator (p, q)-Laplacian with singular weights under the Dirichlet boundary condition. We also prove that in the case where μ &gt; 0 and with 1 &lt; q &lt; p &lt; \infinity the results are completely different from those for the usual eigenvalue for the problem p-Laplacian with singular weight under the Dirichlet boundary condition, which is retrieved when μ = 0. Precisely, we show that when μ &gt; 0 there exists an interval of eigenvalues for our eigenvalue problem.

Existence of a nontrivial weak solution to quasilinear elliptic equations with singular weights and multiple critical exponents