# On a quasilinear Poisson equation in the plane

On a quasilinear Poisson equation in the plane We study the Dirichlet problem for the quasilinear partial differential equation $$\triangle u(z) = h(z)\cdot f(u(z))$$ ▵ u ( z ) = h ( z ) · f ( u ( z ) ) in the unit disk $${\mathbb {D}}\subset {\mathbb {C}}$$ D ⊂ C with arbitrary continuous boundary data $$\varphi :\partial {\mathbb {D}}\rightarrow {\mathbb {R}}$$ φ : ∂ D → R . The multiplier $$h:{\mathbb {D}}\rightarrow {\mathbb {R}}$$ h : D → R is assumed to be in the class $$L^p({\mathbb {D}}),$$ L p ( D ) , $$p>1,$$ p > 1 , and the continuous function $$f:\mathbb {R}\rightarrow {\mathbb {R}}$$ f : R → R is such that $$f(t)/t\rightarrow 0$$ f ( t ) / t → 0 as $$t\rightarrow \infty .$$ t → ∞ . Applying the potential theory and the Leray–Schauder approach, we prove the existence of continuous solutions u of the problem in the Sobolev class $$W^{2,p}_{\mathrm{loc}}({\mathbb {D}})$$ W loc 2 , p ( D ) . Furthermore, we show that $$u\in C^{1,\alpha }_{\mathrm{loc}}({\mathbb {D}})$$ u ∈ C loc 1 , α ( D ) with $$\alpha = (p-2)/p$$ α = ( p - 2 ) / p if $$p>2$$ p > 2 and, in particular, with arbitrary $$\alpha \in (0,1)$$ α ∈ ( 0 , 1 ) if the multiplier h is essentially bounded. In the latter case, if in addition $$\varphi$$ φ is Hölder continuous of some order $$\beta \in (0,1)$$ β ∈ ( 0 , 1 ) , then u is Hölder continuous of the same order in $$\overline{{\mathbb {D}}}$$ D ¯ . We extend these results to arbitrary smooth ( $$C^1$$ C 1 ) domains. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Analysis and Mathematical Physics Springer Journals

# On a quasilinear Poisson equation in the plane

, Volume 10 (1) – Jan 1, 2020
14 pages

/lp/springer-journals/on-a-quasilinear-poisson-equation-in-the-plane-lPvhv0f6CL
Publisher
Springer Journals
Subject
Mathematics; Analysis; Mathematical Methods in Physics
ISSN
1664-2368
eISSN
1664-235X
DOI
10.1007/s13324-019-00345-3
Publisher site
See Article on Publisher Site

### Abstract

We study the Dirichlet problem for the quasilinear partial differential equation $$\triangle u(z) = h(z)\cdot f(u(z))$$ ▵ u ( z ) = h ( z ) · f ( u ( z ) ) in the unit disk $${\mathbb {D}}\subset {\mathbb {C}}$$ D ⊂ C with arbitrary continuous boundary data $$\varphi :\partial {\mathbb {D}}\rightarrow {\mathbb {R}}$$ φ : ∂ D → R . The multiplier $$h:{\mathbb {D}}\rightarrow {\mathbb {R}}$$ h : D → R is assumed to be in the class $$L^p({\mathbb {D}}),$$ L p ( D ) , $$p>1,$$ p > 1 , and the continuous function $$f:\mathbb {R}\rightarrow {\mathbb {R}}$$ f : R → R is such that $$f(t)/t\rightarrow 0$$ f ( t ) / t → 0 as $$t\rightarrow \infty .$$ t → ∞ . Applying the potential theory and the Leray–Schauder approach, we prove the existence of continuous solutions u of the problem in the Sobolev class $$W^{2,p}_{\mathrm{loc}}({\mathbb {D}})$$ W loc 2 , p ( D ) . Furthermore, we show that $$u\in C^{1,\alpha }_{\mathrm{loc}}({\mathbb {D}})$$ u ∈ C loc 1 , α ( D ) with $$\alpha = (p-2)/p$$ α = ( p - 2 ) / p if $$p>2$$ p > 2 and, in particular, with arbitrary $$\alpha \in (0,1)$$ α ∈ ( 0 , 1 ) if the multiplier h is essentially bounded. In the latter case, if in addition $$\varphi$$ φ is Hölder continuous of some order $$\beta \in (0,1)$$ β ∈ ( 0 , 1 ) , then u is Hölder continuous of the same order in $$\overline{{\mathbb {D}}}$$ D ¯ . We extend these results to arbitrary smooth ( $$C^1$$ C 1 ) domains.

### Journal

Analysis and Mathematical PhysicsSpringer Journals

Published: Jan 1, 2020