We study the nonlocal Schr\"odinger-Poisson-Slater type equation $$ - \Delta u + (I_\alpha \... more We study the nonlocal Schr\"odinger-Poisson-Slater type equation $$ - \Delta u + (I_\alpha \ast \vert u\vert^p)\vert u\vert^{p - 2} u= \vert u\vert^{q-2}u\quad\text{in \(\mathbb{R}^N\),} $$ where $N\in\mathbb{N}$, $p>1$, $q>1$ and $I_\alpha$ is the Riesz potential of order $\alpha\in(0,N).$ We introduce and study the Coulomb-Sobolev function space which is natural for the energy functional of the problem and we establish a family of associated optimal interpolation inequalities. We prove existence of optimizers for the inequalities, which implies the existence of solutions to the equation for a certain range of the parameters. We also study regularity and some qualitative properties of solutions. Finally, we derive radial Strauss type estimates and use them to prove the existence of radial solutions to the equation in a range of parameters which is in general wider than the range of existence parameters obtained via interpolation inequalities.
Communications in Partial Differential Equations, 2008
We consider a class of semilinear elliptic equations− Δ u= f (x, u) in all of ℝ N with nonlineari... more We consider a class of semilinear elliptic equations− Δ u= f (x, u) in all of ℝ N with nonlinearities of the form where λ, μ are positive parameters, a (x), h (x) are positive functions, and g (u) is a super-linearly increasing function in a more general fashion than ...
Calculus of Variations and Partial Differential Equations, 2010
We study the existence of positive solutions for a class of nonlinear Schrödinger equations of th... more We study the existence of positive solutions for a class of nonlinear Schrödinger equations of the type $$-{\varepsilon}^2\Delta u + Vu = u^p\quad{{\rm in}\,{\mathbb R}^N},$$ where N ≥ 3, p > 1 is subcritical and V is a nonnegative continuous potential. Amongst other results, we prove that if V has a positive local minimum, and \({\frac{N}{N-2} , then for small ε the problem admits positive solutions which concentrate as ε → 0 around the local minimum point of V. The novelty is that no restriction is imposed on the rate of decay of V. In particular, we cover the case where V is compactly supported.
We study the nonlocal Schr\"odinger-Poisson-Slater type equation $$ - \Delta u + (I_\alpha \... more We study the nonlocal Schr\"odinger-Poisson-Slater type equation $$ - \Delta u + (I_\alpha \ast \vert u\vert^p)\vert u\vert^{p - 2} u= \vert u\vert^{q-2}u\quad\text{in \(\mathbb{R}^N\),} $$ where $N\in\mathbb{N}$, $p>1$, $q>1$ and $I_\alpha$ is the Riesz potential of order $\alpha\in(0,N).$ We introduce and study the Coulomb-Sobolev function space which is natural for the energy functional of the problem and we establish a family of associated optimal interpolation inequalities. We prove existence of optimizers for the inequalities, which implies the existence of solutions to the equation for a certain range of the parameters. We also study regularity and some qualitative properties of solutions. Finally, we derive radial Strauss type estimates and use them to prove the existence of radial solutions to the equation in a range of parameters which is in general wider than the range of existence parameters obtained via interpolation inequalities.
Communications in Partial Differential Equations, 2008
We consider a class of semilinear elliptic equations− Δ u= f (x, u) in all of ℝ N with nonlineari... more We consider a class of semilinear elliptic equations− Δ u= f (x, u) in all of ℝ N with nonlinearities of the form where λ, μ are positive parameters, a (x), h (x) are positive functions, and g (u) is a super-linearly increasing function in a more general fashion than ...
Calculus of Variations and Partial Differential Equations, 2010
We study the existence of positive solutions for a class of nonlinear Schrödinger equations of th... more We study the existence of positive solutions for a class of nonlinear Schrödinger equations of the type $$-{\varepsilon}^2\Delta u + Vu = u^p\quad{{\rm in}\,{\mathbb R}^N},$$ where N ≥ 3, p > 1 is subcritical and V is a nonnegative continuous potential. Amongst other results, we prove that if V has a positive local minimum, and \({\frac{N}{N-2} , then for small ε the problem admits positive solutions which concentrate as ε → 0 around the local minimum point of V. The novelty is that no restriction is imposed on the rate of decay of V. In particular, we cover the case where V is compactly supported.
Uploads