In this paper, we analyze an eigenvalue problem for quasi-linear elliptic operators involving hom... more In this paper, we analyze an eigenvalue problem for quasi-linear elliptic operators involving homogeneous Dirichlet boundary conditions in a open smooth bounded domain. We show that the eigenfunctions corresponding to the eigenvalues belong to $$L^{\infty }$$ L ∞ , which implies $$C^{1,\alpha }$$ C 1 , α smoothness, and the first eigenvalue is simple. Moreover, we investigate the bifurcation results from trivial solutions using the Krasnoselski bifurcation theorem and from infinity using the Leray–Schauder degree. We also show the existence of multiple critical points using variational methods and the Krasnoselski genus.
In this paper we study the existence of nontrivial solutions for the following system of coupled ... more In this paper we study the existence of nontrivial solutions for the following system of coupled semilinear Poisson equations:
Nonlinear Differential Equations and Applications NoDEA, 2017
A boundary value problem on the unit disk in R 2 is considered, involving an elliptic operator wi... more A boundary value problem on the unit disk in R 2 is considered, involving an elliptic operator with a singular weight of logarithmic type and nonlinearities which are subcritical or critical with respect to the associated gradient norm. The existence of non-trivial solutions is proved, relying on variational methods. In the critical case, the associated energy functional is non-compact. A suitable asymptotic condition allows to avoid the non-compactness levels of the functional.
Bulletin of the South Ural State University. Series "Mathematical Modelling, Programming and Computer Software", 2015
Trudinger Moser inequalities provide continuous embeddings in the borderline cases of the standar... more Trudinger Moser inequalities provide continuous embeddings in the borderline cases of the standard Sobolev embeddings, in which the embeddings into Lebesgue L p spaces break down. One is led to consider their natural generalization, which are embeddings into Orlicz spaces with corresponding maximal growth functions which are of exponential type. Some parameters come up in the description of these growth functions. The parameter ranges for which embeddings exist increase by the use of weights in the Sobolev norm, and one is led to consider weighted TM inequalities. Some interesting cases are presented for special weights in dimension two, with applications to mean eld equations of Liouville type.
The Riemann International School of Mathematics (RISM) was inaugurated in 2009, on the occasion o... more The Riemann International School of Mathematics (RISM) was inaugurated in 2009, on the occasion of the 150th anniversary of the formulation of Riemann's Hypothesis. RISM plans to organize periodically schools on various themes of mathematics to honor the memory of Bernhard Riemann. The schools take place in Verbania (Italy), where Riemann spent the last days of his life, and where he died in 1866 at the age of 39. He was buried in the small cemetery of Selasca (Verbania), where a simple stone plate was placed in his memory.
Calculus of Variations and Partial Differential Equations, 2016
Sobolev type embeddings for radial functions into variable exponent Lebesgue spaces are studied. ... more Sobolev type embeddings for radial functions into variable exponent Lebesgue spaces are studied. In particular, the following inequality is proved:
We investigate connections between Hardy's inequality in the whole space R^n and embedding in... more We investigate connections between Hardy's inequality in the whole space R^n and embedding inequalities for Sobolev-Lorentz spaces. In particular, we complete previous results due to [A. Alvino, Sulla diseguaglianza di Sobolev in spazi di Lorentz, (1977)] and [G. Talenti, An inequality between u^* and |grad u^*|, (1992)] by establishing optimal embedding inequalities for the Sobolev-Lorentz quasinorm ∇ · _p,q also in the range p < q<∞, which remained essentially open since the work of Alvino. Attainability of the best embedding constants is also studied, as well as the limiting case when q=∞. Here, we surprisingly discover that the Hardy inequality is equivalent to the corresponding Sobolev-Marcinkiewicz embedding inequality. Moreover, the latter turns out to be attained by the so-called "ghost" extremal functions of [Brezis-Vázquez, Blow-up solutions of some nonlinear elliptic problems, (1977)], in striking contrast with the Hardy inequality, which is never atta...
We consider the Cauchy problem for a two space dimensional parabolic equation with square exponen... more We consider the Cauchy problem for a two space dimensional parabolic equation with square exponential nonlinearity. More precisely, $$ \left\{ \begin{aligned} & \partial_{t} u = \Delta u - u + \lambda f(u)\;\;{\text{in}}\; ( 0 ,T )\times {\mathbb{R}}^{2} , \hfill \\ & u(0,x)\; = u_{0} (x)\;\;\;\;\;\;\;\;\;\;{\text{in}}\;{\mathbb{R}}^{2} , \hfill \\ \end{aligned} \right. $$ where λ > 0, and f (u) := \( 2\alpha_{0}ue^{\alpha_{0}u^{2}} \), for some α0 > 0. We take into account initial data in the energy space H1(\( {\mathbb{R}}^{2} \)), i.e. u0 \( \in \) H1(\( {\mathbb{R}}^{2} \)), and in view of the Trudinger-Moser inequality, the nonlinearity f (which has square exponential growth at infinity) is in the energy critical regime.
Generalizations of the Trudinger-Moser inequality to Sobolev-Lorentz spaces with weights are cons... more Generalizations of the Trudinger-Moser inequality to Sobolev-Lorentz spaces with weights are considered. The weights in these spaces allow for the addition of certain lower order terms in the exponential integral. We prove an explicit relation between the weights and the lower order terms; furthermore, we show that the resulting inequalities are sharp, and that there are related phenomena of concentration-compactness.
Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 2009
We study the existence, nonexistence and multiplicity of non-negative solutions for the family of... more We study the existence, nonexistence and multiplicity of non-negative solutions for the family of problems where Ω is a bounded domain in ℝ2 and λ > 0 is a parameter. The coefficient a(x) is permitted to change sign. The techniques used in the proofs are a combination of upper and lower solutions, the Trudinger–Moser inequality and variational methods. Note that when f(x, u) = 0 the equation is of Liouville type.
We consider nonlinear elliptic equations of the form −Δu = g(u) in Ω, u = 0 on ∂Ω, and Hamiltonia... more We consider nonlinear elliptic equations of the form −Δu = g(u) in Ω, u = 0 on ∂Ω, and Hamiltonian-type systems of the form −Δu = g(v) in Ω, −Δv = f(u) in Ω, u = 0 and v = 0 on ∂Ω, where Ω is a bounded domain in R and f, g ∈ C(R) are superlinear nonlinearities. In two dimensions the maximal growth (= critical growth) of f and g (such that the problem can be treated variationally) is of exponential type, given by Pohozaev–Trudinger-type inequalities. We discuss existence and nonexistence results related to the critical growth for the equation and the system. A natural framework for such equations and systems is given by Sobolev spaces, which provide in most cases an adequate answer concerning the maximal growth involved. However, we will see that for the system in dimension 2, the Sobolev embeddings are not sufficiently fine to capture the true maximal growths. We will show that working in Lorentz spaces gives better results. DOI: 10.1134/S0081543806040195
Let be a bounded, smooth domain in R2. We consider the functional I (u) = ∫ eu 2 dx in the superc... more Let be a bounded, smooth domain in R2. We consider the functional I (u) = ∫ eu 2 dx in the supercritical Trudinger–Moser regime, i.e. for ∫ |∇u|2dx > 4π . More precisely, we are looking for critical points of I (u) in the class of functions u ∈ H1 0 ( ) such that ∫ |∇u|2 dx = 4π k (1+α), for small α > 0. In particular, we prove the existence of 1-peak critical points of I (u) with ∫ |∇u|2dx = 4π(1 + α) for any bounded domain , 2-peak critical points with ∫ |∇u|2dx = 8π(1 + α) for non-simply connected domains , and k-peak critical points with ∫ |∇u|2dx = 4kπ(1 + α) if is an annulus. Mathematics Subject Classification (2000) Primary 46E35 · 35J15
We study optimal embeddings for the space of functions whose Laplacian u belongs to L 1 (Ω), wher... more We study optimal embeddings for the space of functions whose Laplacian u belongs to L 1 (Ω), where Ω ⊂ R N is a bounded domain. This function space turns out to be strictly larger than the Sobolev space W 2,1 (Ω) in which the whole set of second-order derivatives is considered. In particular, in the limiting Sobolev case, when N = 2, we establish a sharp embedding inequality into the Zygmund space L exp (Ω). On one hand, this result enables us to improve the Brezis-Merle (Brezis and Merle (1991) [13]) regularity estimate for the Dirichlet problem u = f (x) ∈ L 1 (Ω), u = 0 on ∂Ω; on the other hand, it represents a borderline case of D.R. Adams' (1988) [1] generalization of Trudinger-Moser type inequalities to the case of higher-order derivatives. Extensions to dimension N 3 are also given. Besides, we show how the best constants in the embedding inequalities change under different boundary conditions.
We consider elliptic equations in bounded domains R N with nonlinearities which have critical gro... more We consider elliptic equations in bounded domains R N with nonlinearities which have critical growth at +1 and linear growth at1 , with > 1, the rst eigenvalue of the Laplacian. We prove that such equations have at least two solutions for certain forcing terms provided N 6. In dimensions N = 3; 4; 5 an additional lower order growth term has to be added to the nonlinearity, similarly as in the famous result of Brezis-Nirenberg for equations with critical growth.
The Trudinger-Moser inequality states that for functions $u \in H_0^{1,n}(\Omega)$ ($\Omega \subs... more The Trudinger-Moser inequality states that for functions $u \in H_0^{1,n}(\Omega)$ ($\Omega \subset \mathbb R^n$ a bounded domain) with $\int_\Omega |\nabla u|^ndx \le 1$ one has $\int_\Omega (e^{\alpha_n|u|^{\frac n{n-1}}}-1)dx \le c |\Omega|$, with $c$ independent of $u$. Recently, the second author has shown that for $n = 2$ the bound $c |\Omega| $ may be replaced by a uniform constant $d$ independent of $\Omega$ if the Dirichlet norm is replaced by the Sobolev norm, i.e. requiring $\int_\Omega (|\nabla u|^n + |u|^n)dx \le 1$. We extend here this result to arbitrary dimensions $n > 2$. Also, we prove that for $\Omega = \mathbb R^n$ the supremum of $\int_{\mathbb R^n} (e^{\alpha_n|u|^{\frac n{n-1}}}-1)dx$ over all such functions is attained. The proof is based on a blow-up procedure.
We first survey some recent results on optimal embeddings for the space of functions with ∆u ∈ L(... more We first survey some recent results on optimal embeddings for the space of functions with ∆u ∈ L(Ω), where Ω ⊂ R is a bounded domain. The target space in the embeddings turns out to be a Zygmund space and the best constants are explicitly known. Remarkably, the best constant in the case of zero boundary data is twice the best constant in the case of compactly supported functions. Then, following the same strategy, we establish a new version of the celebrated Trudinger–Moser inequality, as embedding into the Zygmund space Z 1/2 0 (Ω), and we prove that, in contrast to the Moser case, here the best embedding constant is not attained.
In this paper, we analyze an eigenvalue problem for quasi-linear elliptic operators involving hom... more In this paper, we analyze an eigenvalue problem for quasi-linear elliptic operators involving homogeneous Dirichlet boundary conditions in a open smooth bounded domain. We show that the eigenfunctions corresponding to the eigenvalues belong to $$L^{\infty }$$ L ∞ , which implies $$C^{1,\alpha }$$ C 1 , α smoothness, and the first eigenvalue is simple. Moreover, we investigate the bifurcation results from trivial solutions using the Krasnoselski bifurcation theorem and from infinity using the Leray–Schauder degree. We also show the existence of multiple critical points using variational methods and the Krasnoselski genus.
In this paper we study the existence of nontrivial solutions for the following system of coupled ... more In this paper we study the existence of nontrivial solutions for the following system of coupled semilinear Poisson equations:
Nonlinear Differential Equations and Applications NoDEA, 2017
A boundary value problem on the unit disk in R 2 is considered, involving an elliptic operator wi... more A boundary value problem on the unit disk in R 2 is considered, involving an elliptic operator with a singular weight of logarithmic type and nonlinearities which are subcritical or critical with respect to the associated gradient norm. The existence of non-trivial solutions is proved, relying on variational methods. In the critical case, the associated energy functional is non-compact. A suitable asymptotic condition allows to avoid the non-compactness levels of the functional.
Bulletin of the South Ural State University. Series "Mathematical Modelling, Programming and Computer Software", 2015
Trudinger Moser inequalities provide continuous embeddings in the borderline cases of the standar... more Trudinger Moser inequalities provide continuous embeddings in the borderline cases of the standard Sobolev embeddings, in which the embeddings into Lebesgue L p spaces break down. One is led to consider their natural generalization, which are embeddings into Orlicz spaces with corresponding maximal growth functions which are of exponential type. Some parameters come up in the description of these growth functions. The parameter ranges for which embeddings exist increase by the use of weights in the Sobolev norm, and one is led to consider weighted TM inequalities. Some interesting cases are presented for special weights in dimension two, with applications to mean eld equations of Liouville type.
The Riemann International School of Mathematics (RISM) was inaugurated in 2009, on the occasion o... more The Riemann International School of Mathematics (RISM) was inaugurated in 2009, on the occasion of the 150th anniversary of the formulation of Riemann's Hypothesis. RISM plans to organize periodically schools on various themes of mathematics to honor the memory of Bernhard Riemann. The schools take place in Verbania (Italy), where Riemann spent the last days of his life, and where he died in 1866 at the age of 39. He was buried in the small cemetery of Selasca (Verbania), where a simple stone plate was placed in his memory.
Calculus of Variations and Partial Differential Equations, 2016
Sobolev type embeddings for radial functions into variable exponent Lebesgue spaces are studied. ... more Sobolev type embeddings for radial functions into variable exponent Lebesgue spaces are studied. In particular, the following inequality is proved:
We investigate connections between Hardy's inequality in the whole space R^n and embedding in... more We investigate connections between Hardy's inequality in the whole space R^n and embedding inequalities for Sobolev-Lorentz spaces. In particular, we complete previous results due to [A. Alvino, Sulla diseguaglianza di Sobolev in spazi di Lorentz, (1977)] and [G. Talenti, An inequality between u^* and |grad u^*|, (1992)] by establishing optimal embedding inequalities for the Sobolev-Lorentz quasinorm ∇ · _p,q also in the range p < q<∞, which remained essentially open since the work of Alvino. Attainability of the best embedding constants is also studied, as well as the limiting case when q=∞. Here, we surprisingly discover that the Hardy inequality is equivalent to the corresponding Sobolev-Marcinkiewicz embedding inequality. Moreover, the latter turns out to be attained by the so-called "ghost" extremal functions of [Brezis-Vázquez, Blow-up solutions of some nonlinear elliptic problems, (1977)], in striking contrast with the Hardy inequality, which is never atta...
We consider the Cauchy problem for a two space dimensional parabolic equation with square exponen... more We consider the Cauchy problem for a two space dimensional parabolic equation with square exponential nonlinearity. More precisely, $$ \left\{ \begin{aligned} & \partial_{t} u = \Delta u - u + \lambda f(u)\;\;{\text{in}}\; ( 0 ,T )\times {\mathbb{R}}^{2} , \hfill \\ & u(0,x)\; = u_{0} (x)\;\;\;\;\;\;\;\;\;\;{\text{in}}\;{\mathbb{R}}^{2} , \hfill \\ \end{aligned} \right. $$ where λ > 0, and f (u) := \( 2\alpha_{0}ue^{\alpha_{0}u^{2}} \), for some α0 > 0. We take into account initial data in the energy space H1(\( {\mathbb{R}}^{2} \)), i.e. u0 \( \in \) H1(\( {\mathbb{R}}^{2} \)), and in view of the Trudinger-Moser inequality, the nonlinearity f (which has square exponential growth at infinity) is in the energy critical regime.
Generalizations of the Trudinger-Moser inequality to Sobolev-Lorentz spaces with weights are cons... more Generalizations of the Trudinger-Moser inequality to Sobolev-Lorentz spaces with weights are considered. The weights in these spaces allow for the addition of certain lower order terms in the exponential integral. We prove an explicit relation between the weights and the lower order terms; furthermore, we show that the resulting inequalities are sharp, and that there are related phenomena of concentration-compactness.
Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 2009
We study the existence, nonexistence and multiplicity of non-negative solutions for the family of... more We study the existence, nonexistence and multiplicity of non-negative solutions for the family of problems where Ω is a bounded domain in ℝ2 and λ > 0 is a parameter. The coefficient a(x) is permitted to change sign. The techniques used in the proofs are a combination of upper and lower solutions, the Trudinger–Moser inequality and variational methods. Note that when f(x, u) = 0 the equation is of Liouville type.
We consider nonlinear elliptic equations of the form −Δu = g(u) in Ω, u = 0 on ∂Ω, and Hamiltonia... more We consider nonlinear elliptic equations of the form −Δu = g(u) in Ω, u = 0 on ∂Ω, and Hamiltonian-type systems of the form −Δu = g(v) in Ω, −Δv = f(u) in Ω, u = 0 and v = 0 on ∂Ω, where Ω is a bounded domain in R and f, g ∈ C(R) are superlinear nonlinearities. In two dimensions the maximal growth (= critical growth) of f and g (such that the problem can be treated variationally) is of exponential type, given by Pohozaev–Trudinger-type inequalities. We discuss existence and nonexistence results related to the critical growth for the equation and the system. A natural framework for such equations and systems is given by Sobolev spaces, which provide in most cases an adequate answer concerning the maximal growth involved. However, we will see that for the system in dimension 2, the Sobolev embeddings are not sufficiently fine to capture the true maximal growths. We will show that working in Lorentz spaces gives better results. DOI: 10.1134/S0081543806040195
Let be a bounded, smooth domain in R2. We consider the functional I (u) = ∫ eu 2 dx in the superc... more Let be a bounded, smooth domain in R2. We consider the functional I (u) = ∫ eu 2 dx in the supercritical Trudinger–Moser regime, i.e. for ∫ |∇u|2dx > 4π . More precisely, we are looking for critical points of I (u) in the class of functions u ∈ H1 0 ( ) such that ∫ |∇u|2 dx = 4π k (1+α), for small α > 0. In particular, we prove the existence of 1-peak critical points of I (u) with ∫ |∇u|2dx = 4π(1 + α) for any bounded domain , 2-peak critical points with ∫ |∇u|2dx = 8π(1 + α) for non-simply connected domains , and k-peak critical points with ∫ |∇u|2dx = 4kπ(1 + α) if is an annulus. Mathematics Subject Classification (2000) Primary 46E35 · 35J15
We study optimal embeddings for the space of functions whose Laplacian u belongs to L 1 (Ω), wher... more We study optimal embeddings for the space of functions whose Laplacian u belongs to L 1 (Ω), where Ω ⊂ R N is a bounded domain. This function space turns out to be strictly larger than the Sobolev space W 2,1 (Ω) in which the whole set of second-order derivatives is considered. In particular, in the limiting Sobolev case, when N = 2, we establish a sharp embedding inequality into the Zygmund space L exp (Ω). On one hand, this result enables us to improve the Brezis-Merle (Brezis and Merle (1991) [13]) regularity estimate for the Dirichlet problem u = f (x) ∈ L 1 (Ω), u = 0 on ∂Ω; on the other hand, it represents a borderline case of D.R. Adams' (1988) [1] generalization of Trudinger-Moser type inequalities to the case of higher-order derivatives. Extensions to dimension N 3 are also given. Besides, we show how the best constants in the embedding inequalities change under different boundary conditions.
We consider elliptic equations in bounded domains R N with nonlinearities which have critical gro... more We consider elliptic equations in bounded domains R N with nonlinearities which have critical growth at +1 and linear growth at1 , with > 1, the rst eigenvalue of the Laplacian. We prove that such equations have at least two solutions for certain forcing terms provided N 6. In dimensions N = 3; 4; 5 an additional lower order growth term has to be added to the nonlinearity, similarly as in the famous result of Brezis-Nirenberg for equations with critical growth.
The Trudinger-Moser inequality states that for functions $u \in H_0^{1,n}(\Omega)$ ($\Omega \subs... more The Trudinger-Moser inequality states that for functions $u \in H_0^{1,n}(\Omega)$ ($\Omega \subset \mathbb R^n$ a bounded domain) with $\int_\Omega |\nabla u|^ndx \le 1$ one has $\int_\Omega (e^{\alpha_n|u|^{\frac n{n-1}}}-1)dx \le c |\Omega|$, with $c$ independent of $u$. Recently, the second author has shown that for $n = 2$ the bound $c |\Omega| $ may be replaced by a uniform constant $d$ independent of $\Omega$ if the Dirichlet norm is replaced by the Sobolev norm, i.e. requiring $\int_\Omega (|\nabla u|^n + |u|^n)dx \le 1$. We extend here this result to arbitrary dimensions $n > 2$. Also, we prove that for $\Omega = \mathbb R^n$ the supremum of $\int_{\mathbb R^n} (e^{\alpha_n|u|^{\frac n{n-1}}}-1)dx$ over all such functions is attained. The proof is based on a blow-up procedure.
We first survey some recent results on optimal embeddings for the space of functions with ∆u ∈ L(... more We first survey some recent results on optimal embeddings for the space of functions with ∆u ∈ L(Ω), where Ω ⊂ R is a bounded domain. The target space in the embeddings turns out to be a Zygmund space and the best constants are explicitly known. Remarkably, the best constant in the case of zero boundary data is twice the best constant in the case of compactly supported functions. Then, following the same strategy, we establish a new version of the celebrated Trudinger–Moser inequality, as embedding into the Zygmund space Z 1/2 0 (Ω), and we prove that, in contrast to the Moser case, here the best embedding constant is not attained.
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